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What might seem like an obscure topic is actually one in which we are all affected: The Science of Lines.

Reflect for a minute: We wait at the retail checkout line – in all stores in which we buy stuff. In retail, the congestion is worse at the end of year holidays. A line at an amusement park is likely busy during the Summer months. So on and so forth.

So, how does a customer experience waiting and what is the effect of waiting on commerce and buying?

Clock Watching

In general, once a wait last longer than 3 minutes, the perceived wait time multiplies with each passing minute. Shoppers who actually waited five minutes reported having waited twice as long. This is the Psychology of Queueing.

Impulse Buying

If customers have to wait, why not make it a revenue opportunity. That’s exactly what most of us experience at the retail checkout line. Need a chap stick? What about that stuffed animal you never would have thought of buying, but are now thinking about it while you have to wait in line. Oh, wait, what about those batteries you don’t need for anything – yeah, just buy some just in case.

That’s what retailers are capitalizing on: if you have to wait, then make the wait feel less so and make it a revenue opportunity.

Line Jockeying

If a line is short, it’s probably because of a customer or other circumstances that is making the ability to serve take longer. For example, today I was at Sam’s Club where I was in position #4 from the checkout. The customer at the checkout counter was arguing about something, making the line longer. So what happened? Customers on my line left to other lines that were perceived to move faster – not necessarily shorter lines, but lines that were perceived to have higher velocity.

Balking

Balking is technically the state wherin a person who would otherwise have entered a line, decides not to enter it. This is typically a customer who is about to enter a line and surveys the available lines. They consider the wait time, look down at what they’re buying, does a cost or benefit analysis on waiting versus the benefit of buying your items. Sometimes, the customer will put down their items and just leave the store, concluding that waiting isn’t worth it.

Reneging

The technical definition of Reneging in the context of Waiting Lines is when a person enters a line, then leaves it before being served. This is similar to Balking, but where the customer has chosen a line, but then leaves it after they do the cost and benefit analysis in their mind on whether the wait is worth it.

Below is a nice infographic on the Science of Waiting Lines ¹

Queueing Systems and the Shopping Experience is a post from: Lean Six Sigma Consulting

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I’ve been on a recent search recently for an iPhone App that conveniently calculates the cycle time of an operation. I found one that looked interesting and helpful enough. So, I decided to take it out on a test drive while sitting at local McDonald’s Franchise – in this case, I’m measuring a critical process step in a restaurant operation.  

To my surprise, the iPhone App did more than I expected. I was able to calculate Server Time at Point of Sale (Cycle Time from order beginning to paid), and the iPhone App then calculated an Individuals Moving Range (IMR) Control Chart as well as a Run Chart. Pretty cool. And, to have the ability to do on-the-spot time studies like this is invaluable for me – while at the Gemba, I can do a time study and have the data calculated and visualized for me immediately. Very helpful.

Here’s the Point of Sale distribution at the McDonald’s. Each unit is a paying customer and I calculated their time from the beginning of giving their order to the time when they received their receipt.

Here is the same data, but the iPhone App calculated the IMR Control Chart, showing Upper Control Limit (UCL) and Lower Control Limit (LCL). I’m not sure how useful this is for the operation I was measuring, but it’s pretty cool that the iPhone App conveniently does this.

And, here is the run-of-the-mill run chart for the time at the point of sale.

And, below is the table of the data and the cycles I was able to measure.

Technology is amazing. What I normally would have done with a stopwatch and then plotting that data in Minitab or Microsoft Excel I was able to do on my iPhone. What do you think? Do you think an iPhone App like this is practical and helpful?

McDonald’s Franchise Queueing Study is a post from: Lean Six Sigma Consulting

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The other day I took the kids to our local grocery store – Kroger’s. After spending 30 minutes trying to manage my kids through the shopping experience, we were finally ready to pay for 1 Gallon of Milk, Bananas, and some Ice Cream. Yes – when you have a large family as I do, doing the simplest things take a long, long time to accomplish.

While waiting at the checkout line, I looked up and noticed something interesting – a large flat screen television that said the following:

• Lanes Open: 4
• Action Now: 4
• Next 30 Minutes: 2

From the sounds of it, it looks like a point of sale waiting line management system. It piqued my interest. A picture of what I saw is below:

So I came home and did some research and learned a few interesting facts about the point of sale queueing system at Kroger’s. Apparently, Kroger’s developed this system and is conducting pilot tests across several mid-Atlantic stores.

Here are a few details about the queueing system¹:

It consists of a flat panel television screen that displays three yellow circles and is placed on walls in front of the store’s lineup of checkout lanes. Inside these circles, changing numbers indicate to employees how many checkout lanes should be open immediately and how many should open up within 15 or 30 minutes.

Sensors are positioned over a store’s entrance and exit doors and above checkout lanes to indicate the number of customers who are shopping at specific times. These sensors send signals to a computerized monitoring system that registers the number of checkout lanes to open.

According to the same source, this new system is designed to create a better customer and a better shopping experience. They’ve branded this system “1-plus-1 queuing” and Kroger’s ultimate goal with QueVision is to have only one customer waiting in each line. By doing so, it allows the cashier at the point of sale to build a relationship with the customer.

I’m not sure how they can pull this off. There is a fixed cost to open a line. Managing that cost is a challenge.

But, there are clearly other reasons besides the customer experience at stake here. We’re talking about MONEY.

According to Progressive Grocer,

The front end of the supermarket represents more than $7 billion in sales and as much as 2 percent of a retailer’s total profits. The front area is the one department that every shopper visits. And it is the last chance for a retailer to make a good impression on consumers.

[The Progressive Grocer report goes on to state that] 75 percent of more than 1,000 shoppers interviewed said a positive experience at the front area of the supermarket makes their overall opinion of the store “much better,” while 80 percent also said the front-end experience makes them more likely to shop there again.

And, the checkout line and waiting time at the checkout counter also acts as a litmus test for whether a prospective shopper will shop at the store. According the report,

Concerns about checking out are so strong that many shoppers will, upon entering the store, glance at the front end and base the length of their trip (and therefore, the size of their transaction) on the conditions they observe . . .

And, regarding average wait time in a grocery store,

The average time spent in a grocery store’s checkout lane is 2 minutes 52 seconds, based on a survey of retailers in the report. But consumers often perceive the wait time to be much longer.

We know why the wait feels longer – the Psychology of Queueing is a play here.

Retail Checkout Counter: Waiting Line Management is a post from: Lean Six Sigma Consulting

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I had an opportunity to visit a local Jiffy Lube for an oil change recently. I needed to change the oil in our car before moving to Tennessee. In the process of getting an oil change, I learned how Jiffy Lube approaches work and their adherence to metrics.

This Jiffy Lube had 4 bays and at the time I was there, 3 were occupied.

While my car was waiting, I had a good conversation with the shift supervisor at Jiffy Lube. I asked him about their approach to work and how they hold themselves accountable. Here is what I learned:

• Instead of dividing up his team and have a few work on each car, he has the whole team work on 1 car at a time. Without knowing it, he was applying the principle of single-piece flow. And then I asked him why not divide up your team and work on all 3 cars at the same time. He said something profound and very pragmatic:

“It’s just faster to work on one car at a time. We get ‘er done that much quicker.”

• They religiously keep track of time the second the car enters the bay until the car exits the bay. This supervisor said they monitor how they do per week and have a daily standup every morning to discuss how the previous day went. In that meeting, they go through some simple problem solving on how to go faster that day. Again, without knowing it, this team is applying some level of PDCA.
• For this team, it’s not good enough to track time based data internally, they publish it in a large screen in the waiting room. As the picture below shows, my van was in Bay 4 and was in the Bay for 17:21 minutes and a total time of 19:56 minutes. It begs the question, though, why the Isuzo in Bay 2 was in the bay for just a few minutes but was there for 1 hour. Interesting.

Additionally, without knowing it, this supervisor – or this Jiffy Lube for that matter – was applying some other best practices for how to manage queues, such as the application of one piece flow I explained earlier. It’s great to see effective and practical applications of Lean and Queueing.

Have you seen any lately? Share your thoughts here.

Jiffy Lube Oil Change and Queueing is a post from: Lean Six Sigma Consulting

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About two weeks ago, we went on a family vacation – on a Carnival Cruise. En route to the Long Beach and San Pedro port, we stopped at Las Vegas for the night. If you ignore the gambling, the permissiveness, and the shear excess of the city, it is really an engineering marvel. That got me thinking: what does it take to operate a mega hotel in Las Vegas? Indeed, how do Mega Hotels manage Hotel Queueing and Waiting Lines?

In the Wall Street Journal ¹, there is a feature on The MGM Grand in Las Vegas, which is the largest – the biggest hotel – in the United States. To give you a sense of the magnitude of running this hotel, here are some facts:

• The MGM Grand is 18 years old
• The MGM Grand has 2,995 Rooms
• Over 8,000 employees
• Hotel Driveway is 14 lanes wide
• 70,000 on average enter the front doors daily
• The MGM Grand parking lot holds 9,487 cars
• 370 Housekeepers are employed
• The MGM Grand parking lot Valet parked over 1 million cars in 2010
• The MGM Grand has 3,000 Pool Deck Chairs
• Every 3 days, the MGM Grand produces 886 floral arrangements

Do those numbers feel overwhelming? The MGM Grand also thinks so, which is why they have established a number of business processes to make MGM Grand customer experience better by helping the hotel guest feel that the hotel is smaller than it is. In their words,

“The challenge is to cater to everybody without alienating anybody,” says Mr. Scott Sibella, President and Chief Operating Officer of MGM Grand.

So, they employ a number of strategies that make their operations feel smaller and they do this in almost every customer touchpoint. They do this through establishing metrics that foster a “smaller hotel” feel and also in their hotel layout and design. Here are some:

Hotel Service Metrics

• Room Service: The MGM Grand helps the customer feel special and fosters a “small hotel” psychology by delivering room service within 30 minutes from order to room delivery.
• Hotel Maintenance and Facilities Management: Service calls to fix equipment or service other items such as air conditioners or kitchen equipment must be answered within 15 minutes.
• Valet Car Delivery: A car must be retrieved every 8 minutes (this would be their Takt Time).
• Housecleaning: Each room must be cleaned within 30 minutes of Guest checkout.

Hotel Management: Queueing and Waiting Lines

Contrary to traditional queueing theory of implementing a strategy of a Single Queue but with Multiple Servers, the MGM Grand Hotel has implemented 36 smaller lines and 36 front desk staff to help guest check-in.

In their words,

For years, hotel guests stood in two long, snaking lines to check in. Now, on busy days, there are 36 small lines, one in front of each front-desk worker. Guests “are third, not 23rd,” says Shawna Cabrera, front desk manager. “There’s the perception that they’re going to get through quickly.”

So, the MGM Grand is employing an old tactic from Psychology of Waiting Lines.

Below are some pictures that shed light on the above facts about MGM Grand:

Las Vegas Vacations: How the Largest Hotel Feels Small is a post from: Lean Six Sigma Consulting

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We are pleased to bring you the second part of Cornelio Abellanas’ first article on modeling the impact of variability on systems and processes. In the first article, Cornelio showed the impact of variability, in general, on processes as it relates to lead time, cycle time, work in process (WIP), and the impact to the customer. In this article, he applies the same approach, but with a focus on Just in Time Logistics.

Enjoy this article and learn more about Cornelio at the end of this article.

In the previous article we saw some effects of variability: accumulation of WIP before and after the high variability step and overall Lead Time increase.

In a situation like this we might decide to apply Just-In-Time Inventory in order to reduce the excess inventory and in this way reduce Lead Time. We can try this with the Monte Carlo Simulator.

We first reproduce the previous situation:

• Variation of step 2 = 40 and run for a while to reach stability and then apply JIT in the whole line by limiting Max queue in all steps to 200 (double the throughput of 100). This means that we are adding a constraint: each step will only process if there is room to store its output (output WIP + processed in next step ≤ 200).

Analyzing the results we see we have achieved our objective of reducing excess WIP and Lead Time but have caused a bigger problem: Customer dissatisfaction due to a decrease in % On Time and average Thruput.

By doing this we eventually eliminate the excess WIP which had accumulated before and after the high variability step (2). The elimination of the excess WIP After removes the protection it was providing to customer deliveries and therefore % On time drops: the customer is dissatisfied.

Another consequence of this action is that thruput drops in every single step to around 90% of capacity. Notice that all this is due to the variability in step 2 combined with the application of JIT in the line.

This drop in thruput starts in step 2 due to the fact that although its theoretical capacity is between 60 and 140, due to the JIT restriction, it will never receive more than 100 units from the previous step and will not be allowed to supply more than 100 to the next step this makes the effective capacity to vary from 60 to 100. Therefore this step has become a bottleneck for the whole line which explains why all steps downstream from 2 will be equally restricted.

The amazing fact is that also the steps upstream are restricted by this bottleneck in step 2. The reason is that when step 2 processes only 60 units it leaves only 60 holes in the input buffer which allows the previous step to process only 60 units.

So this variability bottleneck, combined with JIT, degrades the productivity of the whole line: both upstream and downstream.

Imagine you go to the line in this situation: the cause of the problem will not be evident. If you try to convince step 2 that it is causing a problem to the whole line the reply might be: Why do you say that? I am producing on average like everyone else!

This is again the “Virus” of variability at work.

By applying JIT in this case we have therefore exposed the customer to all the variability of step 2 and dropped the productivity of the complete line: a complete disaster!

Maybe you can work out alternative solutions to reduce WIP in this case of high variability and propose them in your comments.

About Cornelio Abellanas

Has a PhD in Telecommunications Engineering degree from the University of Madrid and a Master of Science from the University of Kent at Canterbury (UK). He is a Lean Six Sigma Black Belt and EFQM (European Foundation for Quality Management) assessor.

He has been Lean Six Sigma manager and a member of the Management Committee in Celestica Spain where he implemented 80 improvement projects per year with a total savings for the Company of 1% of the yearly revenue. Improvement projects yielded these savings by reducing operator time, optimising plant layouts, reducing equipment setup time, reducing admin and production lead times, increasing equipment availability, reducing operator and equipment defect rate, reducing work-in-process and component inventory, reducing scrap and rework, improving on-time delivery, etc.

He designed and implemented a line data collection system and real-time operator feedback which enabled self control in autonomous electronic board production cells.
He performed internal audits and achieved success in external audits for the Company with standards ISO 9000: 2000, AS9100 (Aerospace), ISO/TS 16949 (Automotive) and ISO 13485 (electro-medicine). He has developed and maintained the Quality System: Quality Manual, Process Value Stream Maps, Procedures, Corrective actions, Suggestion program, etc. for Celestica Spain. He has delivered EFQM training and assessment in companies Volvo Truck and Guzman.

He currently delivers Lean and Six Sigma training and leads Kaizen workshops in companies around Europe: IBM, AT&T, MSL, Philips, Volvo Truck, Ford, Celestica, IBC, Pt Pro, BP Solar, Italgres, PCS, General Dynamics, Telefónica International Wholesale Services, EMT Valencia, Importaco, Faurecia, Asac Pharma, etc. in France, Spain, Portugal, Belgium, Netherlands, Germany, Austria, Italy, Ireland, Greece, Switzerland, Sweden, etc.

He lectures at MBAs in local Universities.

He has managed and delivered courses at the IBM International Education Center in La Hulpe, Belgium on the topics: Total Quality Management, ISO 9000, EFQM, Stress Screening, Six Sigma, Statistical Process Control and Capability, Design of Experiments, Design for Manufacturability, Business Process Improvement, Lean Production, Kaizen, Theory of Constraints, etc.

He managed an Independent Business Unit which was responsible for production of a tape drive unit designed in IBM Tucson, USA and manufactured in a local IBM Spain subcontractor. He was Manufacturing Engineering Manager for banking control units in IBM Spain. He designed a matrix ticket printer for British Railways in Ventek Ltd., London (Datapoint Computers representative for UK). He worked as a Systems Engineer in logic design for Burroughs Machines Ltd. (now Unisys) in Cumbernauld (Scotland).

Contact information:
Polyhedrika CB
Valdelinares 2, 11a
46015 VALENCIA (Spain)
Mobile: (34) 678464624

Just In Time Inventory: Modeling The Impact of Variability with Monte Carlo Simulation is a post from: Lean Six Sigma Consulting

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Where’s my legal stuff?

Yes, even in the industry of law, we ask the fundamental question of “where’s my stuff?”

Similar to the supply chain and service chain of iPhone Supply Chain, Order Pipeline of Events, the Pizza Supply Chain, Ecommerce Supply Chain, the Domino’s Pizza Supply Chain, the BMW Supply Chain,  the Pharmaceutical Supply Chain, the iPhone 4S and Fedex Tracker, and even the Supply Chain of the White Truffle Mushroom - we ask a similar question for legal documents – Where is my stuff? When will my work be done? What is expected of me?

And, we know that informing the client or the customer helps him or her to feel as if the waiting isn’t so bad. That’s a fundamental law of queueing psychology.

I recently incorporated a company using Legalzoom. Legal work is very complicated, as can be read on this article on Lean and the Law. But, what makes it less complicated is informing the customer where they are in the process – in the service process. That’s what Legalzoom does incredibly well.

What makes legal work complicated, in general, are the following:

1. A lot of stakeholders involved, which leads to waiting.
2. Many stakeholders means there are several hand-offs. Hand-offs leads to defects.
3. Hand-offs involves communication and there is very little of it in the legal industry. In other words, the client or the customer is almost always in the dark.

And, we know, that a supply chain or a service chain with little or almost no information flow is a poor one.

Having said that, now take a look at Legalzoom’s information flow – their attempt at keeping me, the client, up-to-date on where my service is in the order pipeline of events:

• Check to see if Limited Liability Corporation (LLC) name is available
• Awaiting signatures from me, which I had to FAX to Legalzoom
• Signature received
• Application sent to the state in which my LLC filing was made
• Awaiting for the state to approve my filing
• Notification that the state approved my filing
• Processing final documents for review
• Awaiting approval from me
• Approved

Legal Forms Online: Service Supply Chain of Legalzoom is a post from: Lean Six Sigma Consulting

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We are pleased to bring you this article by Cornelio Abellanas, a practitioner of lean manufacturing who is based in Spain. His strong technical background validates what many practitioners of lean manufacturing take for granted, but he knows through mathematical validation how and why certain lean methods work.

Enjoy this article and learn more about Cornelio at the end of this article.

The purpose of Lean is the elimination of waste while Six Sigma is centered in the reduction of variability. One of the forms of waste we try to eliminate with Lean is excess inventory and particularly Work-In-Process which is responsible for long lead times and poor customer responsiveness.

Variability was described by Myron Trybus as a virus which infects our processes: it causes chaos and is often undetected. Variability is in fact often the root cause of our problems and Six Sigma allows us to detect it and reduce it at the source.

The best way to understand the effects of variability is by using a Monte Carlo simulator. In my teaching I use a simple 3 step process simulator in Excel (shown above) to let participants experience for themselves the effects of variation, try different solutions to the problems presented and see the side effects by downloading the monte carlo simulation yourself.

We want to see the effects of variation in one step on the total process. To do that we first run the simulator for a while by pressing start and see what the ideal process would look like: 100 items committed and delivered to the customer on every period with an average WIP per step of 100 and average lead time of 1.

Now let us key in a variation of 40 in step 2 (random variation of thruput between 60 and 140 with an average of 100). When we run the first thing we experience is customer dissatisfaction due to missed deliveries (% On time drops).

Eventually some WIP starts accumulating before and after step 2 (the one causing the problem). If we continue we will see WIP moving between Before and After step 2. When the excess WIP is After it protects the customer from the variability so % On time recovers to 100% but when the excess WIP moves to Before we start missing deliveries again.

In the graph in the bottom we can see the evolution of average lead time (which is related to total WIP) as well as the instances of missed deliveries. The first thing we notice, as a direct consequence of variability is an increase of WIP which in turn causes longer lead times. The “virus” effect of variability can also be noticed when we look at the average thruput of step 2: it is still the same as that of the other steps, therefore it is not obvious where the problem is coming from. We can also see that on some occasions the excess WIP accumulates AFTER step 2 and this can mislead us to blame step 3 (it has the big pile in front of it).

This result confirms that unless we measure variability we will not improve it (Six Sigma) and the effects of variability is excess WIP and long lead time (Waste).

In an upcoming article, we will later experience the simulation will show when we apply just in time (JIT) to the model.

About Cornelio Abellanas

Has a PhD in Telecommunications Engineering degree from the University of Madrid and a Master of Science from the University of Kent at Canterbury (UK). He is a Lean Six Sigma Black Belt and EFQM (European Foundation for Quality Management) assessor.

He has been Lean Six Sigma manager and a member of the Management Committee in Celestica Spain where he implemented 80 improvement projects per year with a total savings for the Company of 1% of the yearly revenue. Improvement projects yielded these savings by reducing operator time, optimising plant layouts, reducing equipment setup time, reducing admin and production lead times, increasing equipment availability, reducing operator and equipment defect rate, reducing work-in-process and component inventory, reducing scrap and rework, improving on-time delivery, etc.

He designed and implemented a line data collection system and real-time operator feedback which enabled self control in autonomous electronic board production cells.
He performed internal audits and achieved success in external audits for the Company with standards ISO 9000: 2000, AS9100 (Aerospace), ISO/TS 16949 (Automotive) and ISO 13485 (electro-medicine). He has developed and maintained the Quality System: Quality Manual, Process Value Stream Maps, Procedures, Corrective actions, Suggestion program, etc. for Celestica Spain. He has delivered EFQM training and assessment in companies Volvo Truck and Guzman.

He currently delivers Lean and Six Sigma training and leads Kaizen workshops in companies around Europe: IBM, AT&T, MSL, Philips, Volvo Truck, Ford, Celestica, IBC, Pt Pro, BP Solar, Italgres, PCS, General Dynamics, Telefónica International Wholesale Services, EMT Valencia, Importaco, Faurecia, Asac Pharma, etc. in France, Spain, Portugal, Belgium, Netherlands, Germany, Austria, Italy, Ireland, Greece, Switzerland, Sweden, etc.

He lectures at MBAs in local Universities.

He has managed and delivered courses at the IBM International Education Center in La Hulpe, Belgium on the topics: Total Quality Management, ISO 9000, EFQM, Stress Screening, Six Sigma, Statistical Process Control and Capability, Design of Experiments, Design for Manufacturability, Business Process Improvement, Lean Production, Kaizen, Theory of Constraints, etc.

He managed an Independent Business Unit which was responsible for production of a tape drive unit designed in IBM Tucson, USA and manufactured in a local IBM Spain subcontractor. He was Manufacturing Engineering Manager for banking control units in IBM Spain. He designed a matrix ticket printer for British Railways in Ventek Ltd., London (Datapoint Computers representative for UK). He worked as a Systems Engineer in logic design for Burroughs Machines Ltd. (now Unisys) in Cumbernauld (Scotland).

Contact information:
Polyhedrika CB
Valdelinares 2, 11a
46015 VALENCIA (Spain)
Mobile: (34) 678464624

Modeling The Impact of Variability with Monte Carlo Simulation is a post from: Lean Six Sigma Consulting

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I’m fascinated by supply chains of all types. I’ve written before on the iPhone Supply Chain, Order Pipeline of Events, the Pizza Supply Chain, Ecommerce Supply Chain, the Domino’s Pizza Supply Chain, the BMW Supply Chain,  the Pharmaceutical Supply Chain, and even the Supply Chain of the White Truffle Mushroom.

Today, I want to highlight how a company meets a basic need of knowing “Where’s my Stuff?”.

I recently purchased an iPhone 4S. My carrier is AT&T and I was finally able to upgrade from my dinosaur of a phone – the original iPhone 3G, which I’ve had for 4 years. My iPhone 3G has a battery life of about 2 minutes, but I’ve endured and I don’t let it bother me. But, I’m thankful that I can finally upgrade.

So, in my anticipation for my new iPhone 4S, I’ve been checking the Fedex Tracker. As of this morning, my iPhone 4S is in Memphis Tennessee.

The Fedex Shipment Tracker isn’t just a novel tool, but I see it as a basic feature that companies need to have in order to be competitive. It’s basic because it answers the basic human need to know where his or her stuff is – and providing an answer to the customer alleviates any pain of waiting, for we know that known waits feel shorter than unknown waits – which is a basic tenet in the psychology of queueing.

So, whenever you can, let your customer know their anticipated wait time. Don’t worry if it’s long. They are adults and they can handle it. The important lesson is that, as a company, you are being transparent and honest. For most companies, that’s a big win in itself.

Where’s My Stuff: iPhone 4S and Fedex Tracker is a post from: Lean Six Sigma Consulting

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Applying Queueing Theory in a restaurant operation might be helpful to those who proactively wish to manage revenue. After all, the drivers of revenue in a restaurant is how many guests a restaurant can serve in a given shift, as well as average order value.

So how does one apply Queueing Theory in a restaurant operation? Suppose the following information below for a typical restaurant:

Here are some assumptions:

• The restaurant operates 180 minutes on the evening shift, which starts at 7:30 PM and closes at 10:30 PM.
• This restaurant is unusual because the guests arrive and leave exactly at the half hour mark (this is to make it simpler)
• Read the data above like: The number of guests in the restaurant between 8:00 and 8:30 is 85. The number of arrivals at 8:30 is exactly 25. The number of guests that depart at 8:30 is 30.

So, given our restaurant data, we arrive at the following questions:

1. What is the average throughput of customers, using the unit of (customers per minute) of operation?
2. What is the average cycle time of a customer in minutes?

Restaurant Operations Management

The questions above are important because:

• The lower the cycle time per restaurant customer, then the restaurant manager can accept more guests and increase the restaurant revenue.
• Knowing the throughput of the restaurant can give the restaurant manager more ability to manage the restaurant to a drum beat. This is typically called Takt Time in the language of lean manufacturing.

1. What is the average throughput of customers, using the unit of (customers per minute) of operation?

Answer: We get 240 / 180, which means 1.33 customers per minute.

2. What is the average cycle time of a customer in minutes?

Answer: To get cycle time, we first calculate total inventory. By inventory in this setting, we’re talking about restaurant guests. This means we sum the number of guests in the restaurant during the evening shift:

(60+85+135+155+125+45) / 6 = 605 / 6 = 100.83 customers.

So, cycle time can be calculated:

100.83 / 1.33 = 75.64 minutes is the average cycle time of a restaurant customer

So, if one’s goal is to increase restaurant revenue, then the restaurant manager can do the following, given the data above:

• lower the average cycle time of the restaurant guests, which allows for other guests to occupy that spot and thereby increase revenue
• do nothing to cycle time, but try to increase the average order value per guests through menu up-sell or cross-sell
• do nothing to cycle time, but investigate menu pricing and increase menu item prices as appropriate, but done with wisdom and care, otherwise customers will leave if prices aren’t in line with their expectations

Additionally, if one is interested in reducing cycle time per restaurant guest, this is also an area in which Queueing Theory and Lean can work together. Perhaps there are process steps that do not add value, which contribute to the current cycle time calculation. Perhaps eliminating wastes in the restaurant operations processes can thereby reduce cycle time per customer.

Queueing Theory in a Restaurant Operation is a post from: Lean Six Sigma Consulting

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Queueing Theory, if used appropriately, can describe the a system and its dynamics accurately in order to pave the way for eventually improving the system. Let me illustrate.

Let’s assume the following scenario.

Widgets are made to order in an assembly line that consists of three steps, each performed by a single worker. So, there are a total of 3 workers on the assembly line. Here are a few facts and data for this process:

• Wait-time between process steps is caused by buildup of work-in-process inventory.
• Wait-time in front of Step 1 represents the wait-time from an arrival of an order to the start of production.
• After step 3, the product is delivered immediately to the customer.
• On average, orders for widgets arrive every 15 minutes.
• We define Capacity is its maximum sustainable throughput:

The # of resources simultaneously performing the activity / The time duration of the activity

Below is the data:

So, given the scenario above,

• What is the average number of jobs in the system, including orders waiting to be processing PLUS work-in-process?

To answer the above question, we need the following:

• Cycle Time: 28+9+20+8+30+10 = 105 Minutes
• Average Throughput Rate = 1/15

So, the average number of jobs in the system including orders waiting and current work-in-process is:

• 1/15*(105) = 7 Widgets

Application

This simple example shows the power of Queueing Theory. I purposely chose the generic term “widgets” because you can substitute anything you want. For example, the process above can be done to answer the following questions:

• On average, how many patients are in the system, including those in the waiting room that haven’t been seen yet and those currently being seen by a doctor?
• On average, how many bags are in the airport carousel, including those that are enroute from the landed airplane as well as those currently on the carousel?
• On average, how many projects are there, including those that whose business case is still being developed and those projects currently in process?
• On average, how many custom bikes are being built including the bike orders that haven’t been started and those currently in process?

As you can see, my example above answers a variety of business questions, that can shed greater light on resource constraints, strategic planning, where constraints are in the system, and many  more.

Queueing Theory Application in Service and Manufacturing is a post from: Lean Six Sigma Consulting

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Due to an overly-busy life, I forgot to apply for a social security card for my youngest daughter, the baby. She’s 1.5 years and so I thought I probably should just grin and bear it and head over to our local social security office.

So, I got the youngest 4 kids ready and we went on an adventure to the social security office. I made sure that I had completed the social security card application at home and brought the necessary proof of identification.

When we got to the social security office, it was crowded. Then, right by the front door was a machine where one could get a ticket. Mine is below:

The ticket I got was K223. I asked one of the people waiting what number was just called and she said K211. Not bad. So, I decided to wait.

Psychology of Queueing

Based on my ticket, I knew that I was 11 people away from getting served. So, I made sure that I was aware and alert of the numbers being called. This tactic in queueing satisfies one of the principles in the Psychology of Queueing; namely,

• Uncertain Waits Seem Longer than Certain Waits

For me, while the time was not certain (I didn’t know how long I’d have to wait), I did know that I was 11 people away. But then something interesting happened. Instead of hearing “K212″, “K213″, I heard “9″, “10″. So, what is going on?

It turns out that “K” was for new social security cards and there was a different ticketing and numbering system for other services, such as social security disability and business licenses.

So, while the ticketing system satisfied one principle in the psychology of queueing, it violated another; namely,

• Unfair Waits Feel Longer than Equitable Waits

To some degree, the two different numbering and ticketing schemes added confusion for me. Eventually, I got used to hearing two different ticketing numbers and after 40 minutes, it was finally our time to be served.

Social Security Office: A Lesson on Queueing is a post from: Lean Six Sigma Consulting

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We know that Little’s Law is the average number of customers in a system (over some interval) is equal to their average arrival rate, multiplied by their average time in the system.

• TH = throughput (arrival rate). This is the velocity or speed of production and is calculated by determining how many items are produced and dividing it by the length of time it took to produce them. It can, of course, be computed from Little’s Law as TH = WIP/CT.
• CT = cycle time (average time in the system). This is the time it takes to complete the production cycle or the average time it takes to produce one unit. Generally, determining cycle time requires either direct measurement or can be computed from Little’s Law as CT = WIP/TH.
• WIP = work in process (average number of units or customers in a system). This is the number of items currently in production or being serviced in some way. This figure must be measured (counted) directly or can be computed from Little’s Law.

In other words,

And, you can also transform the equation below to find the average wait time for a system, whereas the above tells us the average length of the queue, or how many people are waiting in line.

So what are the applications of Little’s Law in business? Considering a typical production situation of accepting new orders into a production process, let’s assume we are running a process in which throughput (TH), the number of units we produce, equals 50 units per day. Our work in process (WIP), the number of units in various stages of production, remains relatively constant at 200. Given these conditions, our cycle time, the average time it takes to complete one unit, would be four days because of the calculation below:

• (CT = WIP/TH, CT = 200/50, CT = 4).

This means we can accommodate new orders of 50 units each day and the system will remain balanced. But suppose we receive an order for 85 units, 35 more than the standard order of 50. The WIP would increase from 200 units to 235, and because TH would remain constant at 50 units per day, CT would immediately increase from four days per unit to 5.4 days per unit.

Why do we care?

Well, the increase in orders, which is normally a good thing, immediately causes a decrease in production efficiency, which would be a bad thing. This is one of the strange but accurate implications of Little’s Law:

• Significant levels of new orders cause production efficiency to decrease.

How to Reduce Cycle Time

To improve cycle time, only two options are available:

1. Increase throughput.
• This can be done through process improvement. That is, identify waste in a system – especially waste that is present in the system constraint – then systematically eliminate it.
• Commit significant investment to increase the scale of the system to better handle the increase of WIP. This is fine when considering longer-term system capacity. It is impractical, however, when addressing relatively short-term variations in demand.
2. Reduce WIP.
• Only this option can be used to effectively address these short-term demand variations. Reducing WIP may require some counterintuitive actions, such as temporarily pulling projects or orders out of the workflow and setting them aside.
• With the resulting decline in WIP, cycle time drops and the remaining projects get done better and faster, so much so that projects originally pulled out of the workflow can then be reinserted and completed on or before the original target date. In other words, by stopping work on a project, it gets done faster.

Little’s Law is a wonderful tool that can help us better understand what are seemingly complicated and dynamic processes. But, underneath it all, Little’s Law describes the connections in dynamic systems well and allows us to proactively improve the system.

Update

I posted this article on the Operations Management group on Linkedin, which generated a flood of opinion, which I post here:

• Bob Echlin • Excellent review Peter. Thanks.
• Albert Rodriguez • Great stuff Peter! thank you
• Richard Shipperbottom • Thanks, Peter. The title is a little “over-marketing”: it isn’t an explanation of how to reduce cycle time, it is an explanation of how to apply Little’s Law. Forgive you for that, your post is beautiful and simple: should be adopted as a standard text
• Dan Trietsch • Quoting from the post: ”With the resulting decline in WIP, cycle time drops and the remaining projects get done better and faster, so much so that projects originally pulled out of the workflow can then be reinserted and completed on or before the original target date. In other words, by stopping work on a project, it gets done faster.” Are you talking about reducing the effect of congestion? That is, making it easier to walk and work on the floor? If so, you got a point albeit without having said so clearly and depending on whether queues are indeed physically present (so shifting them physically helps). Other than that, it is simply not true that by postponing a job you can finish it in time. One of the things people don’t understand about limiting WIP in the system is that it automatically shifts waiting to some conceptual queue outside the system. I am not opposed to this shifting: it is good in terms of postponing actual investments in those items and reducing said congestion. It may also be justified sometimes to preempt a very long job to promote several small jobs (even though it is against the religion of some). But I am opposed to ignoring the external queue. In conclusion, I agree wtih Richard up to “Forgive you for that” and on the positive side the graphic L = “lambda” W looking like “LAW” is something I have never seen/noticed before. Did Little intend it to be seen that way?
• Isha Dubey • I am posting this comment on the basis of my knowledge about Reduction in Cycle Time. Cycle time consists of the summation of the process time at various workstations in a manufacturing layout. In very simple terms in order to reduce cycle time for a manufacturing process we need to redesign the layout in such a fashion so as to eliminate 1.Unnecessary movement of material 2.Long waiting lines 3.Too many inspection procedures at different levels of production 4.Unnecessary storage and the WIP can be directly transferred to the next process. The other reason for focusing on the variabilities is because this constitutes approximately 25% of the sales generated as cost to employ them. The important point here is that, we need not improve the processes at the workstations as that constitutes very little part of the cycle time and we try to improve the efficiency of the workstations, which actually donot contribute much to the cycle time. The most important measures we need to focus for a world class manufacturing system are: 1. On Time Delivery cycle time versus lead time(committed to the customer) 2. What is the fraction of the total cycle time is the processing time ( sum of process time at each workstations) 
• Prasad Velaga • Isha, you seem to have a good understanding of lean manufacturing. However, I want to comment on one of your views. Improvement in the efficiency of bottleneck workstations can not only increase throughput but also reduce both WIP and lead times. The proponents of Little’s Law can help me understand how the simple law captures this point. In this case, lead time (W) and queue size (L) decrease and lambda increases. Some WIP is necessary to protect throughput against the impact of randomness (natural variation) in the system. For specific throughput and WIP, you have the corresponding lead time as given by Little’s Law. When we reduce the natural variation in the system, both WIP and lead time will decrease. Six sigma is necessary for this purpose.
• Bob Sproull, Lean Six Sigma Master BB/Jonah • Peter, excellent portrayal of how Little’s Law (it’s actually a tautology) can be used to improve throughput. My work is based in an MRO operation and one of the first things we did shortly after I arrived was to use Little’s Law to calculate the necessary WIP in the system to achieve a given level of throughput for a necessary cycle time. This was our first step prior to implementing Critical Chain Project Management. We achieved a C/T reduction of nearly 40% on our first try. We too focused our Lean and Six Sigma efforts on the system constraint and for each product line the results got better and better. It’s amazing to me that more organizations/people aren’t using both Little’s Law and an integrated Lean, Six Sigma and Theory of Constraints (a.k.a. TLS) to maximize profitability. Maybe with more discussions like the one you started here will move this along. Thanks Peter for stimulating this discussion.
• Prasad Velaga • Bob, could you please explain more clearly “Calculate the necessary WIP in the system to achieve a given level of throughput for a necessary cycle time”? For example, the necessary cycle time is 14 days and the given level of throughput is 10 per day. Do you want to keep the average WIP in the system at 140 for this purpose? In my view, WIP is to protect throughput against variation but not to “achieve” certain level of throughput.
• Bob Sproull, Lean Six Sigma Master BB/Jonah • Prasad, I would be happy to explain. Think of Little’s Law as an equation that must be balanced (i.e. the left side must equal the right side of the equation). The equation is, WIP = C/T x TP. If the TP is known (i.e. the customer demand) and, for example, we know what the C/T is, then we automatically know what the WIP should be. Likewise, if we have a given amount of WIP and we know what the TP must be, then we can calculate the required C/T (i.e. TP = WIP/(C/T). One of the key take-aways from this law is that excessive WIP lengthens the C/T, so WIP explosions should always be avoided. I want to recommend a great book for you entitled, Factory Physics by Hopp and Spearman. What you’re suggesting is that WIP should be used as a buffer, but I believe that buffers should be a combination of time and product. Understand, I am a disciple of the Theory of Constraints, so automatically, we think differently. Great questions Prasad!
• Prasad Velaga • Bob, thanks for the explanation. Dan, I wonder what else I am going to read in this thread.
• Abhinav Sinha • I am totally agreeing with Ms. Isha Dubey on her views of cycle time reduction. But also i want to add some more on reduction of cycle time. Companies could always go for trying to achieve line balancing method which would help the whole process to pull together in the same way. It is like the concept of Drum-Buffer-Rope. Again the unnecessary movements could be removed by redesigning the layout of the flow process and trying to achieve U shaped instead of straight line used everywhere. This would drastically reduce the travelling time of the material and also more visibility is maintained between different work stations simultaneously. The so called undesirable WIP also gets reduced upto 75% without going for anything drastic. Just simple steps helps to achieve so called standard set by the process of World Class Manufacturing companies.
• Bob Sproull, Lean Six Sigma Master BB/Jonah • Abhinav, I am a bit confused by your posting. How in the world do you equate line balancing to Drum-Buffer-Rope. DBR recognizes the existence of a constraint and pulls material into the line as product exits the constraint. If you’ve read the book Velocity, line balancing is not a good approach if you are trying to optimize throughput.
• Dan Trietsch • Hi all, Subject to Prasad’s correct reservation, I also agree with Isha and with Abhinav too. I also agree with Prasad that Little’s Law is not relevant, a point on which I elaborate below. All in all, that’s about all I agree with in this post. Factory Physics (FP) was written by two colleagues of mine who understand queueing theory. They are not supporters of the so-called “Theory” of Constraints (“TOC”) AT ALL, although it is on public record that one of them took a Jonah course from the late Eli Goldratt. Later he lamented the fact that he was the only one in the class who questioned the correctness of Goldratt’s teachings, whereas the other participants, mostly practitioners [who do not really understand queueing theory], swallowed everything hook, line and sinker (my words, not his). The same author also said that the “Theory” of Constraints is not a theory, and he was right. FP is a solid attempt to present a correct version of “TOC,” not one that is based on blindly following a mesmerizing guru. They suggest, correctly, is that if we limit the maximal WIP (by an approach they call CONWIP, which has similarities to DBR but IS NOT identical to it), then the system will behave better. But they do NOT suggest that you can use Little’s Law to calculate the necessary WIP limit. CONWIP–which I prefer to call CAPWIP (because you can’t keep WIP constant, you can only CAP it)–was later adopted by the TOC crowd under the title S-DBR (simplified DBR), but to this day “TOC” disciples refuse to admit that the original DBR is junk (although Goldratt himself admitted that it requires a major overhaul in a 1988 paper that he published in an academic journal). Now, the necessary WIP to assure a particular TH level is a function of variability in the system and it is ALWAYS higher than what Little’s Law suggests. Why? Because Little’s Law gives the AVERAGE WIP and the average is always smaller than the maximum! Therefore, if I understood the post about using Little’s Law to calculate the necessary WIP in the system, I must say that it is simply incorrect. On the bright side, again, the posts about proactive actions necessary to reduce variations are more relevant to reality. However, I prefer to think about the discipline that teaches us how to do that as JIT rather than Lean/SixSigma or any other modern buzzword. For instance, U lines are a classic JIT development (based on GT ideas originally developed in the Soviet Union during WW II). Give credit where it is due: the JIT discipline does it all, including pointing out which resources need to be improved to enable cycle time reductions. Those U lines also sport the basic CAPWIP idea, because there are no kanbans within them, so even that is not originally a “TOC” idea. Wherever we stray from JIT–e.g., in the DBR direction or even in the arbitrary SixSigma direction–we reap suboptimization. Lean/SixSigma only work where they essentially use JIT. Indeed, my understanding is that “lean” is just a new term for JIT, and likewise, for a while, “agile,” but I can’t say the same for Six Sigma. Six Sigma is an attempt to revise the quite stupid Zero Defects idea into something more realistic, but it is not sufficiently smarter than ZD and thus it is not a correct way to focus variation reduction efforts in a system. I’d like to finish on a personal note. There must be many “TOC” groups out there, as likewise re Six Sigma, but I never considered joining them. I figure that disciples have the right to join like-minded people and reinforce their beliefs by interacting with those other disciples, all without interference from opponents like me. Similarly, I would not dream of going into a church to argue against the religion or sect of the disciples. But this group is not committed to any such approach and thus if one or more of them is promoted here it is almost the duty of opponents to speak up. I don’t expect to convince any disciple, but at least the criticism should be on record.
• Bob Sproull, Lean Six Sigma Master BB/Jonah • So Dan….I assume that your mind is closed on TOC…even though there are many examples of very successful implementations? Of course your argument will be since you don’t believe it…how could it possibly work. I still hold firm in my belief that TLS is the methodology that delivers which of course includes Lean, Six Sigma and TOC. Have fun with this discussion Dan.
• Dan Trietsch • Bob, Please don’t assume you know what I think and how open my mind is. Before I start, there is another name for the correct aspects of “TOC” that I can write without quotation marks: Management by Constraints (MBC). I will use it in this thread from now on where I can. If you read my post carefully you will see that I am supportive of FP’s aim to build a correct approach on the basis of “TOC.” What I didn’t say is that it is very close to MBC. In fact, I personally used MBC extremely successfully in my consulting in the US Navy almost 20 years ago. However, I never accepted the “single or very few bottlenecks” hypothesis of Goldratt’s so, independently from the FP authors, I used CAPWIP. MBC allows and eventually achieves balance in systems (by repeatedly implementing step 4–elevate–PRUDENTLY). Hence it works. Another thing I stated starting 25 years ago was that MBC is isomorphic to PERT/CPM (which also has the focusing on constraints approach–you crash only critical activities). However, in 2002 I discovered two things which turned me into an opponent of “TOC” (but I still consider MBC solid). It was then that I read Goldratt’s Critical Chain for the first time (but at the time, I did not know that CC was due to Pittman 1994). First, not knowing about Pittman, I thought Goldratt cynically turned around and used the focusing principle of PERT/CPM to invent a “NEW” way to focus in project scheduling. Second, Goldratt “proved” that balance is detrimental, thus proving that he did not understand what he was talking about. Since then I have developed the MBC idea into the concept of Management by Criticalities and I developed CC into PERT 21. So my mind is not “closed” to MBC, but rather I am beyond MBC, and onto a much better way. But it’s true that my mind is closed to “TOC,” because it is FALSE. Here is a link to what I think and what I wrote on “TOC.” (It goes as far as Management by Criticalities but not yet to PERT 21.) You don’t have to read it, but unless you do there is no point to continuing this debate. Let me know, however, if you want to see a PERT 21 link. 
• Isha Dubey • Thanks Dan, Bob, Prasad, Abhinav for providing immense knowledge. It is certainly true that we need to maintain a certain level of WIP in the production processes so as to avoid the problems of constraints and natural variations.But we are not considering a major factor here that increasing or maintaining a certain level of inventory WIP would anyways increase the cost and that would directly affect the RETURN ON CAPITAL EMPLOYED.ROCE is calculated on the basis of the cost of inventory(Fixed assets and current assets) which if increased to maintain to a certain level for a changing customer need as per discussed in LITTLE”S LAW would surely decrease the returns and hence less profit margin to the company. I understand that fact that customer should be treated like a KING and should be made available with the finished products on time as committed (LEAD TIME) but to match this increasing WIP isnt an appropriate solution. Thus what I can understand the topic of discussion is to just try to improve the layout and monitor the measures as mentioned in the previous post by me. Please provide more suggestions on this…I am learning a lot from this forum!! Thanks
• Isha Dubey • Another point which I would like to make here..is that SIX SIGMA I suppose would be appropriate in a situation when we are having a production process where more of rejections are there..Reduction in cycle time needs optimization of the whole system. If we are considering SIX Sigma or lean manufacturing, that should be employed in the whole system all the processes right from the procurement to product delivery. Let me explain by this example: Suppose we received an order of supplying a product X in a lead time of 25 days to the customer. The cycle time (time needed to produce one unit) is say 1 day.The order quantity is 10. Since we understand that the processing time at each workstation totally make up the cycle time and that is one day, why did we committed to 25 days. This is because we have also included the extra time needed for purchasing, procurement,etc. Thus we try to optimize the extra time ACTIVITIES instead of the activities included in the cycle time. Please correct me if I am wrong…Awaiting for a response 
• Dan Trietsch • Isha, let me respond to your first post of the last two. For this purpose I will invoke two very basic queueing formulas. I am not suggesting they reflect reality but they show the behavior I want to explain. The first model is M/M/1: Poisson arrivals, exponential processing times, one server, NO limit on WIP. The average WIP in this case obeys Little’s Law, of course, but it depends ENTIRELY on the rate of demand (arrivals to the system) as compared to the rate of supply (service). You can think of those rates as the products of the sales arm, which generates demand, and the production arm, which actually delivers the physical items or services required. If demand exceeds supply, the classical theory says WIP will explode. Well, it won’t explode but it will grow without control at an average rate that is a tad above the difference in rates. If you want to control WIP in such a system you have no choice but to reduce service time or reduce service variance. In the M/M/1 context that can only be done by increasing the server’s rate of service (which reduces both the mean and the variance in this instance). If you don’t, the customers will balk. Which leads us to the second, more interesting model: M/M/1/K. That model assumes that if there are K customers in the system than any additional customer balks immediately (which means that this customer is lost). Now, I wish I could draw you a picture but you must do it yourself. Draw two small circles at some vertical distance. Call the top one of them “supply” and put a nice Greek “mu” in it (for the service rate). Call the other “demand” and put a nice Greek lambda in it (for the arrival rate). From the top, supply, circle draw a circular arrow towards the bottom, demand, circle. On the other side, draw a counter-symmetric arrow from demand to supply. The result so far should be two small circles connected by outside arrows in a circular shape. The arrows show flow directions: the one from demand to supply shows sales that queue behind the server (supply). The one from supply to demand reflects product output. Finally, inside the big circular shape, put the number K. This is an unconventional but simple diagram of the M/M/1/K queue. Now notice this: there are exactly K units in the system. Those between demand and supply represent a physically visible queue of orders awaiting service. Those between supply and demand represent potential new sales. If there are no units awaiting or receiving service, then all K units are awaiting the “demand server” — i.e., the lower small circle. Symmetrically, if all units are served or awaiting service, no new sales can be generated. In this model supply and demand are completely symmetrical, and there is no rule that one of them should exceed the other. Now, in this system, throughput depends on K. Reduce K and there will be a lot of idling and WASTED CAPACITY. Increase K and throughput will increase, but at the expense of increasing lead time. So what you must seek is the correct balance between lost capacity and increased lead time. By your post, you only see one side of the coin.
• Dan Trietsch • One final point: in this system K represents the cap on WIP. So if you increase your cap, you lose throughput and gain lead-time reduction. Of course, here too variance reductions as well as speeding the server up will help. Thus the model shows that Prasad’s comment way above that improving speeds and variances helps is correct. If anybody tries to teach you differently they do not understand what they are talking about. Sorry, but it’s that simple. 
• Isha Dubey • Comment from my friend Abhinav as he is not able to post comments here: Please read!! Thanks again to Dan, Bob, Isha for providing so much insight on the topic. Its really very enlightening for me to know so much from practioneers like u. Again to comment on my earlier post i would say that line balancing is very much required to pull the process taking into account all the constraints present. The customer is the epicenter around whom all are policies should be present, so if we are doing something which is not going to add in sort of value to him then he is not going to pay us. This ultimately would hit the profitability of the companies as rightly pointed by Isha. What is the use of WIP in the system if that is only increasing our costs. Little Law gives us the basis which says that WIP directly proportional to Throughput. So in order to increase throughput we need to increase our WIP. But it really contradicts with a simple fact that WIP is inversely proportional to ROCE (return on capital employed). We need to think these aspects in a more mathematical way which could give us some base n logic instead of going by theory and books written by all. Just ask a customer whether he really cares whether my process has WIP or not, is lean or not, he/she wants only the best at lowest price. So wouldnt it be better if we ourselves reduce our costs with all unnecessary WIP and only maintain a certain level which would help us to keep the line moving in case of any defect arising in previous stations and is requiring time to get it right. This could be done by analyzing data present with company. Please comment on this. I want to know all the deficiencies in this concept and correct my thinking and approach 
• Isha Dubey • Dan: Thanks for the detailed explanation about M/M/1 model in queuing theory I am very much aware of the this theory and has been taught to us by the experienced professionals in India.I understood well your example of drawing two circles one as supply and the other as demand. In M/M/1 model we say the optimum situation is one in which the service rate and the arrival rate should be same. Keeping up inventory levels for constraints is also true.But the fact is we should limit this inventory to the required quantity at each workstations so that they can be utilized at the time of any constraining problem or halt to keep up the throughput constant.Another thing if the demand keeps on fluctuating like 50 to 85 we need to work on the line balancing aspect rather than building up inventory to meet the fluctuations. As stated by you ” K represents the cap on WIP. So if you increase your cap, you lose throughput and gain lead-time reduction. Of course, here too variance reductions as well as speeding the server up will help.” Why do we need to increase the capital when the optimum WIP can be maintained at each stations to avoid the constraints…like what is the concept of DBR!!
• Isha Dubey • From Abhinav Dan: I am not able to understand why you are so keen not to accept the above points. This is so very simple of not complicating the matters but to accept them in the most simple way. You are saying about applying the concept of queuing theory but that has a flaw. Queuing theory models are based on the exponential distribution, thus the model has the traits of the exponential distribution. The main problem lies in that the exponential distribution has a coefficient of variation of one. This fact precludes the modeling of any process that has a coefficient of variation significantly different from one. Now because of the low likelihood that any random process will have coefficient of variation as ONE, thus they become rather impractical to apply. Also in support of my above comment posted by Isha and hers above comment, keeping a certain level of WIP in the system doesnt make any problem, rather it helps to maintain certain buffer in case of failure of any workstation. This has been made cleared above only. So instead to going for some radical changes and lots of thinking its better to accept small easy solutions which are easily understood by all in the production floor instead of going for complex models and charts.
• Dan Trietsch • Abhinav, Unfortunately, we are going nowhere fast. For instance, what exactly is the statement “This is so very simple of not complicating the matters but to accept them in the most simple way” supposed to mean? You call that “simple”? And what simple thing am I so keen not accept? In short, I am totally confused by your message. In addition, you clearly misunderstood my example and the caveats that went with it. Your comments about the limits of the exponential and the need to balance supply and demand exactly demonstrate that, but I doubt I can explain it here. A good source would be the same Factory Physics text recommended here previously; please consult them especially with respect to the approximate simple transformations from the exponential to general distributions: nothing changes qualitatively, ONLY quantitatively. Anyway, here is what counts: Lead time should be reduced for many reasons, mostly to do with customer satisfaction, avoiding harmful systems dynamics and avoiding cancellations and changes to orders. That requires reducing processing time and reducing variance. In addition we must THEN put a limit on the maximal WIP to avoid congestion and to be able to provide reliable due date performance (except to those customers who find the system booked and whose orders must wait outside in a conceptual queue: they may have to be quoted higher due dates). Do not confuse the need to reduce lead time with trying to save inventory holding costs, a noble but usually minor objective. For instance, to reduce lead time it may be necessary to INCREASE inventories of raw materials and spare parts. You can decrease them only if you have reliable JIT suppliers, preferably nearby. Goodnight, over and out.
• Prasad Velaga • Abhinav said, “Little Law gives us the basis which says that WIP directly proportional to Throughput. So in order to increase throughput we need to increase our WIP”. This interpretation is correct only from mathematical point of view. Some mathematical relations like Little’s Law or the famous Einstein’s equation E = m*c^2 are quite often deceptive. Many people many not see which variables are independent and which ones are dependent. How many people see c (the light speed) as the square root of the ratio E/m? Can we increase the energy E or decrease the mass m in order to increase the light speed? Abhinav’s interpretation of Little’s Law is similar. In production environment, I do not want to see throughput as the ratio of WIP to lead time although it is mathematically correct. Otherwise, I will be tempted to increase this ratio for improving throughput. The following is my approach. I first want to enhance the “effective” processing rates at bottleneck workstations for improving throughput. Waiting times, transport times between operations and many other non value added activities (as indicated by Isha) do not affect throughput. They simply increase WIP and prolong the lead times. Therefore, they must be reduced for controlling WIP (inventory on shop floor) and lead times. Note that cutting a pipe of uniform length by half will not increase flow rate through the pipe. As the next step, I will try to find right WIP level to protect the achieved throughput. Both DBR and kanban control consist of this step. I take the lead time as a consequence of the two efforts. 
• Isha Dubey • Prasad, me and abhinav never stated that “Waiting times, transport times between operations and many other non value added activities (as indicated by Isha) do not affect throughput. ” But what we always tried to stick to the notion here…reduction in cycle time. We are not discussing here what is the idea behind throughput or something, we were just trying to elaborate more on the strategy behind reducing cycle time. One cannot decrease or increase throughput so easily until unless there is some change incorporated in the whole layout or implementing line balancing for an increased demand(please make a note of this point). The other points which I have quoted above strictly relate to the reduction in cycle time and has nothing to do with throughput. Exactly as you are saying that in mathematical formula one should not try to make relations in different manner, similarly throughput is not the concern here. The main thing is cycle time, which is actually large in companies. The reason behind is not just the processing times but some time extra needed for procurement, inspection, storing, material handling, material check, storage counting etc. These involve time and Cost as well. Thus overall the time of delivery or the lead time to the customer is quoted higher than the cycle time. One more thing to be noted here is that CYCLE time is very much different from LEAD time. In real situations Cycle time is less than the Lead time. I think this discussion is leading us nowhere, may be we are not able to make our point clear here.We are really thankful to Bob, Dan and other members to cooperate with us. Hope to participate again in next discussion. Thanks!
• Daniele Rosati • Hi all, I spent great part of my recent past working of these concepts, at the beginning working to a scheduling system based on the Factory Phisics theories and I’m working now with the Executive to evaluate the best CT according with the required throughput depending on the Customer Demand. Today the CT as you know could be estimated based on the a lot of factory variables (equipment, availability, Mean time to repair, arrival rate etc. ) , assumed as historical data (tipically a quarter or semester data). Ignoring for a second the variability of these measures depending on the MIX proposed by the Customer. I would propose after choosing the best financial measure to calculatee the optimum CT according with the Best value of the financial chosen measure. I don’t know if this model has been implemented by some software house (like SAP, Primavera, etc),let me know if there is something like that. This is cross functional activity and takes a lot of time , it is an iterated process and it should be done for each feasibility study or long-mid term scenarios activity, so very often it is to be prefer some approximate method related to the several kind of business (for instance we use the minimum CT corrispondent to max Capital Asset Utilization). These approximate methods adopt very strong Boundary Conditions and poor in flexibility but they are easy to formulate and use case by case, hope this can be useful comment – Daniele
• Bob Sproull, Lean Six Sigma Master BB/Jonah • Isha, I am always anxious to participate in these discussion and thanks for making your points known. I may not agree with all of them, but I am certainly opening to listening to other points of view.
• Prasad Velaga • Isha, you said, “In M/M/1 model, we say the optimum situation is one in which the service rate and the arrival rate should be same”. Home assignment: Find out the theoretical implication of the service rate being equal to the arrival rate in M/M/1 queue. ”If the demand keeps on fluctuating like 50 to 85, we need to work on the line balancing aspect rather than building up inventory to meet the fluctuations.” Yes, to deal with fluctuating rates at workstations, one can maintain certain WIP or some protective capacity or both. Adopt the most economical approach for this purpose in any given situation. I am not sure Peter started this thread with a focus on lead time reduction. You and Abhinav seem to have gone in that direction. 
• Isha Dubey • Yes, I think the topic says so…Reduction in Cycle time and not Lead time..therefore we started this discussion on reduction thing…Well thanks so much for this discussion forum, I really gained a lot.. hope it turns out to be a healthy one!!
• Dan Trietsch • I said I won’t comment any more because I felt my explanations were not coming through anyway. However, let me say this: in his two last messages Prasad could have been speaking directly for me. I agree with every word.
• Richard Shipperbottom • Some sparks flying in this discussion, eh?
• Isha/Abhinav – you have it spot on.
• Dan, I thought i was the only one with a mission to debunk the rediculous aura of six sigma, toc et al (I’m a frustrated JIT boy of old). However, the great thing about the original post by Peter, and why I liked it, is that it makes people THINK (evidenced by the above posts). His examples are typically ‘theoretical’ – i.e. rubbish, but in applying it to real situations you’d soon find that out…Interesting that at the bottom of the original post some chap from a call centre was commenting about how to use it…I just hope he wasn’t from a call centre I’ll be using: Little’s “Law” isn’t an equation you can use out of the box (I admit never having used it), and if anyone tried to use it like that they’d need to hand the factory keys back. Perhaps a more pertinent ‘theoretical’ debate might start with “I don’t want ANY wip” , or “I want to increase throughput without spending money on new kit” – I don’t think I’ve ever had a client demand a reduction in ‘cycle time’ (lead time, yes) – or “i want to improve the service level to my customers”. It perhaps also helps to try thinking about trading, not manufacturing (where it is possible to get negative throughput if you applied Little’s Law out of the box). What I love most about all this is how it illustrates the huge difference between academic/theoretical knowhow (Poisson arrivals, Dan? You need to get out more:)!), and business reality. I am impressed by Isha/Abhinav’s maturity given that you’re both students. 
• Isha Dubey • Richard, thanks for appreciation, I am not in a view to discourage any one for his/her discussion or post, actually I am being so junior to all, it is an opportunity for me to learn. I have acknowledged so many facts here which even I was unaware of specially the ones quoted by Dan and Prasad. We have a subject called World Class Manufacturing, where we have been taught this reduction in cycle time as a topic, in which we focused majorly on the reduction of cycle time. The most important principle which we were taught was to make the customers happy!! and serve them like a king!, for this reason we need to measure our ON TIME DELIVERY parameter. This is cycle time vs Lead time. The lead time(Time taken in executing all the activities right from procurement to delivery) is the time of delivery committed to the customer and the cycle time is the processing time or the time needed to obtain a value deliverable to the customer. Generally these two time are very different from each other. In an ideal situation both of them should be equal for keeping the customers happy as well as saving costs. One might be thinking how does cost comes into picture. As the main objective of any business is to make money now and in future…thus the return on capital employed should be high. The costs associated with the inventory (fixed+current, WIP) is inversely proportional to the returns we get. Little’s law says we need to modify our inventory levels to meet the changing customer demand what I understood from the example of 50 to 85 units and thus change of 200 to 235. This is like if the demand increases should we also increase our inventory?? Can’t we optimize the production process like what Abhinav commented as using LINE BALANCING along with maintaining little buffer at each stations (Drum Buffer Rope) concept. This is the case when the demand increased, but the point which I am trying to make is that we just need to reduce the unnecessary operations(Material handling, storage, inspection at each level, transportation). This can be only done when the design of the whole layout is changed. This also involves human factor, there work-ability and their motivational level. There are long queues, waiting times and long transportation routes to be followed unnecessarily when the two workstations can be kept close by. Generally a plant doesn’t have the design section, in some plants there might be, this is the reason why it takes time for implementing any design changes instantly and doing FMEA for any default in the production process. I am not experienced at all, but what I am commenting here, strictly relates to the life examples. I am thankful to all, for their generous cooperation. 
• Dan Trietsch • Richard, I quoted M/M/1/K as a basic model that shows something general. I said right away that the exponential is not realistic. I resent being attacked on a point I made myself. Factory Physics is a good source that explains how the same “something” generalizes to any queueing model. NOW, IF YOU ARE SUGGESTING THAT QUEUEING THEORY IN GENERAL IS NOT PRACTICAL, WE LIVE IN DIFFERENT WORLDS. If you are not suggesting that, please don’t fault me for using the simplest relevant example when trying to explain something fundamental. It is strange, though, that in the same post you say (correctly) that Little’s Law cannot be used the way people here were suggesting and then you endorse posts that include that junk. Granted, they do so only indirectly, but they do say WIP should be reduced without any recognition that there may be a limit and a downside to that. For instance, the posts you endorse in such a wholesale manner contain the following little gem (by Isha, addressing me): “As stated by you ‘K represents the cap on WIP. So if you increase your cap, you lose throughput and gain lead-time reduction.’ ” Well, the whole purpose of my example was to demonstrate the EXACT OPPOSITE: increase K and throughput goes UP (albeit at a marginally decreasing rate). Now, the WIP Little’s Law speaks to is the part of K that is visible in front of the server, but it is an increasing function of K. Therefore throughput is an increasing function of K, and so is WIP (and Little’s Law is working correctly). The error in Abhinav and Isha’s posts that I commented on was that they ignored the beneficial aspect of increasing WIP and instead preached zealously for reduction ONLY. Given the misinterpretation that increasing the WIP cap HELPS throughput, it is understandable that they think decreasing the WIP cap can ONLY HELP. But they are wrong there. As I said in my first post on the matter, they are right with respect to potential action to reduce variance, but that too is not always very easy in practice. Talking about practice, in very flexible plants one can change the layout quickly. Advanced JIT plants in Japan has been doing that for decades. But in most plants it takes a long time and major efforts to change the layout. In such situations you do not “increase WIP” to support throughput but rather you avoid decreasing WIP too much so throughput does not suffer. The way to go is to decrease WIP prudently, “expose rocks,” remove the “rocks,” and only then decrease more. If you encounter a “rock” that you can’t remove, do not decrease the WIP cap any more until you can remove it. You may even have to ratchet the WIP cap back up a bit to accommodate a “big rock.” That is the JIT approach. One last comment: Factory Physics is by far the best textbook on the subject out there. Nonetheless, they too missed the point about sales being part of the bigger system. Perhaps as a result, they do not say anything about adjusting the demand level as part of managing a system. Nobody else does either. The circular queueing model I gave is NOT from FP, or from any other book. (so I am very surprised to hear, Isha, that you know that model–to my knowledge it has never been published. However, such ideas do tend to emerge independently.) Be that as it may, it is a model that I personally use to teach my students how WIP limits work and the trade-off between the WIP limit and throughput. Then I take the next step and teach them how to adjust the whole system. I planned to get there in this thread too, but finding myself under attack I decided to abort that part.

Cycle Time Reduction: Use Little’s Law is a post from: Lean Six Sigma Consulting

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I find Customer Satisfaction Surveys interesting. It gives us a sense of what is important to the firm and the type of information they are attempting on gathering gives us a clue as to what is important to the company.

I was recently sent a survey from The Olive Garden, which is an Italian Restaurant franchise and has locations all over the United States. The customer satisfaction questionnaire had 20 questions and 5 of which had to do with Wait Time. If 25% of the questions on a customer satisfaction survey has to do with wait time, then I’m guessing that’s an important data point for the company.

The first question in the survey was:

• If you waited for dining room seating, approximately how long did you wait after arriving at the restaurant?

The next question is on wait time accuracy provided by lobby staff at the restaurant:

• How accurate was the wait time the lobby staff quoted you?

Then,

• Approximately how much longer than the quoted time did you wait to be seated?

Then, a series of questions on order accuracy:

• Did we prepare what you wanted, exactly as you wanted?

Followed by a series of questions on my activities before coming to the restaurant:

• Where were you immediately before coming to the Olive Garden?
• Where were you going immediately after leaving the Olive Garden?

This goes to show that a key driver in customer satisfaction is wait time and, in general, reducing wait time for customers is the right thing for the business.

You may now play the “Captain Obvious” theme song.

Restaurant Wait Time and Customer Satisfaction is a post from: Lean Six Sigma Consulting

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Joe Palca, a National Public Radio (NPR) correspondent recently published a book entitled “Annoying: The Science of What Bugs Us”. In the book, Joe Palca and his co-author share a number of common things that are annoying and also some research on why those things are annoying to most of us. In his research, he shares something familiar to those who study Queueing Theory – the annoying delay.

In an interview with NPR, Joe Palca shares some facts about what humans find annoying about delays – delays of all types: flight delays and travel delays. This appears to be a universal problem but one we have come to expect. Based on a simple Google query of the word “delay”, below are some of the results:

• Flight Delays
• Newark Airport Delays
• Atlanta Airport Delays
• FAA Flight Delays
• JFK Airport Delays
• Laguardia Airport Delays
• Chicago Airport Delays

And on and on. You get the point. Delays of all types are universally annoying and it bugs us all.

But Joe Palca shares an interesting and subtle point: it’s not just the delay that bugs us, but it is unexplained delays. Think about it. For the most part, we’ve come to expect a delay in some things, such as flight travel and waiting in line at Disneyland. But, the type of Delay that really bugs us are the ones for which there is no explanation.

A lesson from the Psychology of Queueing is helpful here:

1. Unoccupied time feels longer than occupied time.
2. Process-waits feel longer than in-process waits.
3. Anxiety makes waits seem longer.
4. Uncertain waits seem longer than known, finite waits.
5. Unfair waits are longer than equitable waits.
6. The more valuable the service, the longer the customer is willing to wait.
7. Solo waits feel longer than group waits.

Yes, specifically, a delay with no explanation leads to uncertainty, creating further anxiety and concern, which makes the wait feel even longer than it really is.

What’s the lesson for businesses?

Explain why your customers need to wait longer than expected. Just tell them and give them a sense of when they’ll be served. They’re adults. They can take it. But, worse of all – don’t lie to the customer about wait times and certainly don’t leave them uncertain by not explaining to them the reason for them to wait.

Below is a video promotion for Joe Palca’s book “Annoying: The Science of What Bugs Us”, followed by Amazon.com’s description of the book:

Two crackerjack science journalists from NPR look at why some things (and some people!) drive us crazy
It happens everywhere?offices, schools, even your own backyard. Plus, seemingly anything can trigger it?cell phones, sirens, bad music, constant distractions, your boss, or even your spouse. We all know certain things get under our skin. Can science explain why? Palca and Lichtman take you on a scientific quest through psychology, evolutionary biology, anthropology, and other disciplines to uncover the truth about being annoyed. What is the recipe for annoyance? For starters, it should be temporary, unpleasant, and unpredictable, like a boring meeting or mosquito bites

• Gives fascinating, surprising explanations for why people react the way they do to everything from chili peppers to fingernails on a blackboard
• Explains why irrational behavior (like tearing your hair out in traffic) is connected to worthwhile behavior (like staying on task)
• Includes tips for identifying your own irritating habits!

How often can you say you’re happily reading a really Annoying book? The insights are fascinating, the exploration is fun, and the knowledge you gain, if you act like you know everything, can be really annoying.

Queueing Psychology and Unexplained Delays is a post from: Lean Six Sigma Consulting

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Most of you are aware of the terrible earthquake and tsunami that devastated Japan last week. What I want to highlight today, however, isn’t the disaster, but (1) the manner in which the Japanese people are responding to this terrible crisis and (2) share several ways we can help Japan during this time of adversity.

No Signs of Crowd Craze and Mob Violence

We all know the typical response immediately after a tragedy – chaos, looting, violence, mob mentality, and other look-out-for-yourself behaviors. But, for now anyway, that seems to be absent in Japan. Instead, we’re seeing the spirit of the Japanese people stand strong despite this adversity.

Why?

“Looting simply does not take place in Japan. I’m not even sure if there’s a word for it that is as clear in its implications as when we hear ‘looting,’” said Gregory Pflugfelder, director of the Donald Keene Center of Japanese Culture at Columbia University.

Japanese have “a sense of being first and foremost responsible to the community,” he said (source: http://goo.gl/oacpa)

Keep in mind that the normal course of action is to enforce order and enforce crowd control, but since there’s no mob-like behavior that takes key personnel away from truly pressing needs, everyone can focus on the highest priority items – that is, nobody is distracted by unbecoming and mob-like behavior.

Think of it this way: if there’s looting and mob violence, medical and security personnel will be on the scene to assist victims and to enforce order. But, since there isn’t any of that in the Japan earthquake so far, medical and security personnel can continue to focus on finding real victims from the tsunami and the earthquake.

This picture demonstrates my point, courtesy of CNN:

Instead of violence and looting, we see people that continue to behave the way they behaved before the earthquake. This rapid return to normalcy, psychologists say, will help Japan more quickly recover.

Mob Mentality

A cultural anthropologist, interviewed by CNN, commented that the real question is not why Japan is responding so dignified and orderly, but

the real question is why looting and disorder exist in American society [and the West].

Consider this picture of a food line in Haiti after it was hit by an earthquake in 2010:

There is a stark difference between the queue behavior in Haiti versus the queue behavior in Japan.

They continue:

“Such social order and discipline are so enforced in ordinary times that I think it’s very easy for Japanese to kind of continue in the manner that they’re accustomed to, even under an emergency.”

The communitarian spirit at the foundation of Japanese culture seems to function even more efficiently under the stress of disaster, he said.

The natural American inclination is to operate independently.

“So you do everything you can to protect your own interests with the understanding that, in a rather free-market way, everybody else is going to do the same. And that order will come out of this sort of invisible hand.

“And Japanese don’t function that way. Order is seen as coming from the group and from the community as a sort of evening out of various individual needs.”

Commenting on whether this behavior is a type of self-protection to the Japanese society and whether it will help Japan more quickly recover from this tragedy,

Will this social attitude help Japan recover from this disaster? “In a word, YES.”

How You Can Help

Here are several ways we can help in the aftermath of the Japan earthquake and tsunami, courtesy of CNN and Associated Press:

• The U.S. State Department is urging any U.S. citizens to contact friends and family as soon as possible. They can also e-mail the State Department at JapanEmergencyUSC@state.gov. Those seeking information on security in or travel to Japan can call 1-888-407-4747 or 1-202-501-4444.
• Google also is assisting in helping victims touch base with friends and loved ones. Its People Finder, which was tracking about 114,000 records as of Sunday morning, allows users to look for victims or post information about people. It works in five languages.
• As myriad nations offer monetary aid, condolences and rescue teams, many people around the world are seeking ways to ease the burden on the Japanese government and people.
• The humanitarian group World Vision is rushing personnel into the affected areas and providing food, water, medical supplies and shelter for victims.
• It also plans to establish one or more “child-friendly spaces” for kids “affected by disasters to resume normal childhood activities and experience structure and security that are often lost following emergency situations.
• Follow World Vision’s blog for updates, and visit its website or text “4japan” to 20222 to send a $10 donation to the group. It will show up on your next mobile phone bill.
• The American Red Cross also accepts donations via text message. Text “redcross” to 90999, and you can make a $10 donation to the organization.
• You can also keep up with updates from the nonprofit by following its Twitter feed or watching its Facebook updates.
• The Salvation Army, which has had a presence in Japan since 1895, is sending a team to Sendai to assess damages and hopes to provide food, water and other necessities. In Tokyo, the Salvation Army opened its main building to house and feed commuters who were unable to reach their homes.
• The aid organization is issuing updates via its blog, its Twitter feed andFacebook page. It also is accepting donations via text message. Those interested in contributing $10 to the group’s efforts can text “japan” or “quake” to 80888.
• The group AmeriCares, which is also accepting donations, said it is mobilizing resources and sending an emergency response manager to the region.
• Save the Children is sending an emergency team to assist its staff in Japan.Donations to the group’s Children’s Emergency Fund will help preserve the welfare of young ones, who “are always the most vulnerable in a disaster,” Eiichi Sadamatsu in central Tokyo said in a statement.
• Globalgiving.com, another aid organization that allows donors to select the causes they want to support – including climate change, economic development, human rights and natural disaster relief – has established a Japan Earthquake and Tsunami Relief Fund. GlobalGiving will disburse funds to organizations providing aid and relief to victims.
• Among those organizations are Save the Children and the International Medical Corps. The Corps, which focuses on “health care training and relief and development programs,” is speaking with local partners to determine Japan’s most pressing needs. It will send supplies or emergency teams based on those communications.
• To contribute to the Corps’ efforts, visit its website or text “med” to 80888 to send the group a $10 donation.

Japan Earthquake and Tsunami 2011, a Queue, Crowd Control, How You Can Help is a post from: Lean Six Sigma Consulting

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Given just a few properties of a queue, we can answer some questions about waiting times without knowing anything other than the average line length and the average customer arrival rate.

For example, If a customer joins the line just after a customer begins to be served, then intuitively one would expect the newly arriving customer to wait (Line Length) x (Cycle Time). Let’s use numbers to make this point more concrete. Assume a Queue at Starbucks Coffee is:

(8 customers) x (1 min/customer) = 8 minutes

If the line length is doubled to 16 people, then the waiting time should be

(16 customers)(1 min/customer) = 16 minutes

Similarly, doubling the cycle time to 2 minutes should also raise the waiting time to 16 minutes. This last point on Cycle Time is critical, because this often becomes the most controllable variable available to the firm – in other words, line length, demand fluctuations or arrival rate are often not controlled by the firm, but the Cycle Time it takes to serve a customer is controllable and so becomes a critical variable to focus on.

The above example all points us to Little’s Law, but before I show Little’s Law, here are some definitions:

• Lq: The average number of people in a line awaiting service.
• Wq: The average length of time a customer waits before being served.
• Throughput: Mean Outflow (average numbers of items leaving a system, not entering it)

Little’s Law

Now, let’s generalize the example above and arrive at Little’s Law:

W_q = L_q / Throughput

Little’s Law and can be applied in any system in which the mean waiting time, mean line length (or inventory size), and mean throughput (outflow) remain constant. To some extent this is an arbitrary decision, but in most real-world situations, measuring the outflow of a queue is easier than measuring its inflow.

Another interesting point is the generality of this formula. For one thing, this relation will hold no matter what the distribution of inter-arrival times or processing times is. Even more amazingly, Little’s law is not restricted to simple systems with one line and a number of servers. It will hold no matter what the internal structure of a system is.

Little’s Law Example: Patient Flow in Hospital

To illustrate the use of Little’s Law, let’s use an example of Queueing in Healthcare. What if we wanted to know the following:

What the average time in the system for a patient at a hospital?

This includes all the multiple phases, disease states, surgery procedures, etc.

Suppose we know the following:

• Lq: The average number of patients is 102.5
• Wq: [This is the unknown]
• Throughput: Average discharge rate is 67.5 patients per day.

In other words,

W = L/Throughput => Average Time in Hospital = Average # of Patients / Average Discharge Rate = 102.5 patients /67.2 patients per day = 1.53 Days

Knowing that a patient in this hospital can expect to stay an average of 1.53 days can help the hospital administrators plan for care, staffing, budgeting, and other internal items that will help the hospital’s level of service.

Weaknesses of Little’s Law

While Little’s Law is convenient to use and gets us a decent approximation to most queueing questions, it’s clearly not perfect. For example, process utilization must be less than 100% or else the line will grow to infinity (this is otherwise known as WIP Explosion).

Little’s Law Applications

Other ways in which Little’s Law can be used are the following:

• Estimate Waiting Times: [W = Average Number of Customers / Average Throughput] (as the patient flow example above)
• Planned Inventory Time: Suppose a product is scheduled so that we expect it to wait for 2 days in finished goods inventory before shipping to the customer. This two days is called planned inventory time and is sometimes used as protection against system variability to ensure high delivery service. Using Little’s Law, the total amount of inventory in finished goods can be computed as [FGI = Throughput x Planned Inventory Time]
• WIP Reduction: Reducing WIP in a process without making any other changes will also reduce throughput. So, simply reducing inventory is not enough to achieve a Lean Manufacturing system. An integral part of any Lean Manufacturing implementation is an effort to reduce variability (often the domain of Six Sigma), to enable a line to achieve the same (or greater) throughput with less WIP.
• Little’s Law for Product Development
• Little’s Law and Multi-tasking

Little’s Law is a post from: Lean Six Sigma Consulting

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Below are common Queueing Theory definitions, glossary, and vocabulary. Knowing the concepts below will help you better understand these articles on Queues.

• Queue: A line (or buffer or inventory) feeding a number of servers
• Server: An operation fed by a queue.
• Arrival rate (λ): Mean number of arrivals per unit time (usually per hour or day).
• Service rate (μ): Mean number of customers that can be served at 100% utilization by each individual server per unit time (usually per hour or day). At the individual workstation level, the service rate will equal capacity.
• Channels (M): The number of parallel operations connected to an individual queue. For example, if each queue has 2 operations then it will have two channels.
• Utilization (u): A measure of how “busy” the system is. It is generally defined as the ratio of throughput to capacity. Note that u = λ/(Μμ) if λ < Μμ, i.e. the utilization is less than 100%. (Also, note that while the Greek letter μ— or mu— looks a bit like u, they are in fact two different variables.)
• Phase: A queue and its connected servers, or routes to a server.

• Balking: When a person, who would otherwise have entered a line, decides not to enter it.
• Reneging: When a person, who has entered a line, later decides to leave it without being served.
• Interarrival Time: The time between when one customer arrives at a queue and when the next customer arrives.
• Service Time: The time it takes for one particular server to complete a customer’s service. The average service time will be the same as the cycle time.
• CV: The coefficient of variation. This is a measure of a random variable’s variability. For a random variable x, CVx is defined as Standard Deviation (x) x mean (x) CV = .
• CVIAT: The coefficient of variation of the interarrival time. The greater the CVIAT, the “lumpier” the arrival rate.
• CVST: The coefficient of variation of the service time. The smaller the CVST, the more “consistent” a server is.
• Lq: The average number of people in a line awaiting service.
• Wq: The average length of time a customer waits before being served.

After you have become familiar with the terms above, go ahead and read the articles on Queueing Theory.

Queueing Theory Definition is a post from: Lean Six Sigma Consulting

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Eid ul-Adha is an Islamic holiday marking the last day of the Hajj, the annual pilgrimage to the holy city of Mecca. The Hajj is the largest organized movement of people in the world. One can only imagine the logistical nightmare for such a massive event – not to mention all the queueing properties that are present when you’ve got a confluence of people in tight constraints of both time and space.

Hajj Congestion

Congestion comes in many forms with such a large gathering:

• Congestion to perform holy rites or holy rituals
• Congestion to use the restroom or other basic services such as food purchasing and water
• Congestion to travel from one holy city to another or one holy place to another

Given the logistical challenges, the Saudi government is attempting to modernize the Hajj and recently completed a massive transit system connecting the holy city of Medina to Mecca and to surrounding cities (source: http://goo.gl/CAD8i, emphasis are mine):

Well, they’ve actually completed a mass transit system. And that will shuttle people between these key locations. And the idea is to take 72,000 people an hour in each direction to relieve some of the congestion on the road, because you’ve got two and a half, maybe three million people. There’s no concept of personal space. There’s no concept of privacy. You probably won’t eat properly either or sleep properly.

As for the Saudi government, you’re right. They are doing what they can to modernize the Hajj. The mass transit system is one of them. They’re also building a 440 kilometer rail link between the two holy cities, Mecca and Medina. Now, currently, that journey takes four or five hours. They want to reduce it to 30 minutes.

Imagine that type of improvement – moving 72,000 people an hour and reducing travel time from five hours to 30 minutes – an amazing technical accomplishment and one that allows ease of worship for millions of those of the Muslim faith.

Below are some pictures of the Hajj and also a few videos of the massive transit system that was recently completed.

Hajj – Kaaba: source, Wikipedia

Hajj 2010 – Kaaba: source, Wikipedia

Hajj – Transit System over Medina

Hajj: Eid ul-Adha and Queueing is a post from: Lean Six Sigma Consulting

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The books I list here are books focused on Queueing Theory or have a major component of Queueing Theory.

For those interested in the Theory of Queues, the books listed here might be a bit advanced for the beginner; for those just starting out, I suggest you grab an introductory book on operations research or industrial engineering.

The books I list below are in no particular order of preference and I do not endorse any book. I simply list them here for your consideration.

Queueing is such an obscure topic, with few people really interested in it, but its applications are wide and far-reaching: from restaurant operations to airports; from amusement park management to sporting events and crowd management; from emergency room waiting times to fulfillment and logistics; from evacuation and terrorism preparation to tax preparation; from hotel reservation systems to the department of motor vehicles (DMV). And more.

I hope this list of books on Queueing is helpful to you.

Advances in Queueing: Theory, Methods, and Open Problems by Jewgeni H. Dshalalow. The progress of science and technology has placed Queueing Theory among the most popular disciplines in applied mathematics, operations research, and engineering. Although queueing has been on the scientific market since the beginning of this century, it is still rapidly expanding by capturing new areas in technology. Advances in Queueing provides a comprehensive overview of problems in this enormous area of science and focuses on the most significant methods recently developed.Written by a team of 24 eminent scientists, the book examines stochastic, analytic, and generic methods such as approximations, estimates and bounds, and simulation. The first chapter presents an overview of classical queueing methods from the birth of queues to the seventies. It also contains the most comprehensive bibliography of books on queueing and telecommunications to date. Each of the following chapters surveys recent methods applied to classes of queueing systems and networks followed by a discussion of open problems and future research directions.Advances in Queueing is a practical reference that allows the reader quick access to the latest methods.

Applied Probability and Queues by Soeren Asmussen. This book is a highly recommendable survey of mathematical tools and results in applied probability with special emphasis on queueing theory . . . The second edition at hand is a thoroughly updated and considerably expended version of the first edition . . . This book and the way the various topics are balanced are a welcome addition to the literature. It is an indispensable source of information for both advanced graduate students and researchers.

Fundamentals of Queueing Theory, Solutions Manual by Donald Gross, John F. Shortle, James M. Thompson, and Carl M. Harris. With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Big Queues, Lecture Notes in Mathematics by Ayalvadi Ganesh, Neil O’Connell, Damon J. Wischik. Big Queues aims to give a simple and elegant account of how large deviations theory can be applied to queueing problems. Large deviations theory is a collection of powerful results and general techniques for studying rare events, and has been applied to queueing problems in a variety of ways. The strengths of large deviations theory are these: it is powerful enough that one can answer many questions which are hard to answer otherwise, and it is general enough that one can draw broad conclusions without relying on special case calculations.

The Disney Queue Line Survival Guidebook by Kimberly Button. Do you dread the hour-long waits for theme park attractions at the Walt Disney World® Resort? Wondering how to keep everyone entertained and still enjoy a stress-free vacation? Then you need The Disney Queue Line Survival Guidebook, the only guidebook with scavenger hunts, trivia questions, word searches, hidden Mickey locations and more, uniquely tailored to every attraction and show within Disney’s four Florida theme parks. With hours of fun-filled activities as well as essential attraction details, The Disney Queue Line Survival Guidebook is the only book you need to guarantee a magical time at Disney’s theme parks.

Does This Line Ever Move?: Everyday Applications of Operations Research by Kenneth Chelst and Thomas Edwards. Explore Real-World Mathematics! When students solve real-world problems, they see the usefulness of mathematics, they understand concepts better, and they gain insights into related careers. Authors Kenneth R. Chelst and Thomas G. Edwards, professors of engineering and education, respectively, have collaborated to convert real problems from the field of operations research into 10 easy-to-use teaching modules.

Relevant to Student’s Lives. Students will recognize and appreciate the contexts: queuing theory as applied to long lines for concert tickets, multi-attribute decision making for chosing a college, and maximizing profit and minimizing waste by developing a system for ordering supplies for a concession stand.

More Curriculum Choices. Teachers of advanced algebra, precalculus, or general survey courses can use these modules to motivate or review topics. As they work, students will develop skill in using mathematics to model and solve real-world problems.

Egress Design Solutions: A Guide to Evacuation and Crowd Management Planning by Jeffrey Tubbs and Brian Meacham. Egress systems, in concert with evacuation plans and crowd management, form the fundamental basis of good life-safety design for emergency events, yet there are fewcomplete and comprehensive reference books covering the broad nature of this subject.

This book fills this gap by providing a comprehensive review of egress design and analysis, covering egress fundamentals and strategies, performance solutions, human behavior, evacuation modeling, evacuation planning, and crowd management. The authors, both recognized experts in the field, present egress concepts that are both prescriptive and performance-based, accounting for human behavior in emergency conditions as well as a range of expected hazards.

Hardwiring Flow: Systems and Processes for Seamless Patient Care by Dr. Thom Mayer and Dr. Kirk Jensen. You know you have great healthcare providers. But are your systems and processes letting them maximize the time they spend with patients? It’s a deeply important question. Even the best, most talented, most compassionate healthcare providers are only as good as the context in which they practice. Organizations must set them up to do their best possible work. Only when patients flow smoothly through the care process can physicians, nurses, and other care providers execute their tasks efficiently and effectively.

This book delves into one of the most critical issues facing healthcare leaders today. Patient flow. Essentially, it means patients spend exactly the right amount of time at every juncture in their journey through an organization-just enough time to maximize their clinical outcomes in the most cost-effective manner.

Readers will learn: Why patient flow helps organizations maximize the Three E’s: Efficiency, Effectiveness, and Execution How to implement a proven methodology for improving patient flow Insights for improving flow’s teammates;-patient safety, customer service, risk management, and more Why it’s important to engage physicians in the flow process (and how to do so) How to apply the principles of better patient flow to emergency departments, inpatient experiences, and surgical processes Of course, better clinical outcomes must be at front and center of all change. But leaders cannot ignore financial impact, either (especially these days). And as the authors of this book assert, organizations that master flow not only provide better care and reduce the likelihood of litigation, they also attract and retain the best possible talent. In short, they enjoy a decided competitive advantage in the healthcare market.

Health Operations Management: Patient Flow Logistics in Healthcare by Jan Vissers and Roger Beech. Focusing on the mannagement of patient flows and resources in and between healthcare organizations, this book will include both a theoretical framework and case studies for practical use by students. Addressing one of the key challenges associated with the healthcare industry, this textbook leads readers across geographical boundaries and through the logical steps taken in health operations management, and its development.

Introduction to Discrete Event Simulation and Agent Based Modeling: Voting Systems, Health Care, Military, and Manufacturing. Discrete event simulation and agent-based modeling are increasingly recognized as critical for diagnosing and solving process issues in complex systems. Introduction to Discrete Event Simulation and Agent-based Modeling covers the techniques needed for success in all phases of simulation projects. These include: • Definition – The reader will learn how to plan a project and communicate using a charter.

Input analysis – The reader will discover how to determine defensible sample sizes for all needed data collections. They will also learn how to fit distributions to that data.

Simulation – The reader will understand how simulation controllers work, the Monte Carlo (MC) theory behind them, modern verification and validation, and ways to speed up simulation using variation reduction techniques and other methods.

Output analysis – The reader will be able to establish simultaneous intervals on key responses and apply selection and ranking, design of experiments (DOE), and black box optimization to develop defensible improvement recommendations.

Decision support – Methods to inspire creative alternatives are presented, including lean production. Also, over one hundred solved problems are provided and two full case studies, including one on voting machines that received international attention. Introduction to Discrete Event Simulation and Agent-based Modeling demonstrates how simulation can facilitate improvements on the job and in local communities. It allows readers to competently apply technology considered key in many industries and branches of government. It is suitable for undergraduate and graduate students, as well as researchers and other professionals.

An Introduction to Management Science: Quantitative Approaches to Decision Making by David Anderson, Dennis Sweeney, and Thomas Williams. Learn today’s management science concepts and techniques from a leader in the field. AN INTRODUCTION TO MANAGEMENT SCIENCE provides a sound conceptual understanding of the role that management science plays in the decision-making process. AN INTRODUCTION TO MANAGEMENT SCIENCE is applications-oriented and continues to use the problem-scenario approach that has been a hallmark feature of each edition in which a problem is described in conjunction with the management science model that’s introduced. The model is then solved to generate a solution and recommendation to management.

Introduction to Queueing Systems by Sanjay Bose. This volume accomplishes the unique task of providing the reader with the analytical fundamentals for both single queues and queueing networks while keeping the description simple enough so that the results may be directly used for modeling and analysis. A very wide range of single-queue models has been covered, while networks are analyzed through a very comprehensive set of approximation algorithms ready to apply in modeling. The text was honed through years of research and teaching, and is an excellent tool for engineers and students who wish to apply queueing methods to study the performance of systems.

Introduction to Queueing Theory: Modeling and Analysis in Applications by U. Narayan Bhat. An introductory chapter including a historical account of the growth of queueing theory in the last 100 years. A modeling-based approach with emphasis on identification of models using topics such as collection of data and tests for stationarity and independence of observations. Rigorous treatment of the foundations of basic models commonly used in applications with appropriate references for advanced topics. A chapter on modeling and analysis using computational tools. A comprehensive treatment of statistical inference for queueing systems. A discussion of operational and decision problems. Modeling exercises as a motivational tool, and review exercises covering background material on statistical distributions.

This book may be used as a textbook by first-year graduate students in fields such as computer science, operations research, industrial and systems engineering, as well as related fields such as manufacturing and communications engineering. Upper-level undergraduate students in mathematics, statistics, and engineering may also use the book in an elective introductory course on queueing theory. With its rigorous coverage of basic material and extensive bibliography of the queueing literature, the work may also be useful to applied scientists and practitioners as a self-study reference for applications and further research.

Lean Manufacturing: A Plant Floor Guide. Internationally renowned editors John Allen, Charles Robinson, and David Stewart take readers on a comprehensive, ‘street-level’ journey through lean implementation, from the seven wastes and flow processes to developing a business case, using lean tools, and applying readers’ newfound knowledge at greenfield and brownfield sites. Specific chapters on mapping the value stream, policy deployment, the five-phase implementation process, and problem-solving crystallize concepts with a pragmatic treatment. In addition, the brownfield implementation chapter is a must.

Mastering Patient Flow: Using Lean Thinking to Improve Your Practice Operations by Elizabeth Woodcock. Using lean thinking principles to structure workflow improvements, this third edition of the best-selling primer on medical practice operations delivers the newest trends for practice efficiency. Centering workflow around the patient, Mastering Patient Flow, 3rd edition shows you how to take efficiency to a new level. This MGMA classic is a completely updated comprehensive manual on practice operations. This easily readable book delivers sound and timely techniques for reducing patient cycle time, streamlining scheduling methods, managing telephones, maximizing space capacity and utilization and controlling costs. Add proven tools for benchmarking, creating action plans and self-assessment, along with worksheets, tips and case studies, and you have a one-stop “power punch” for a most profitable medical practice!

Multiaccess, Reservations, and Queues by Dee Denteneer, J.S.H Van Leeuwaarden. Reservation procedures constitute the core of many popular data transmission protocols. They consist of two steps: A request phase in which a station reserves the communication channel and a transmission phase in which the actual data transmission takes place. Such procedures are often applied in communication networks that are characterised by a shared communication channel with large round-trip times.

In this book, we propose queuing models for situations that require a reservation procedure and validate their applicability in the context of cable networks.

We offer various mathematical models to better understand the performance of these reservation procedures. The book covers four key performance models, and modifications to these: Contention trees, the repairman model, the bulk service queue, and tandem queues.

The relevance of this book is not limited to reservation procedures and cable networks, and performance analysts from a variety of areas may benefit, as all models have found application in other fields as well.

Traffic Theory by Denos C. Gazis. Traffic Theory describes and illustrates the key models of traffic flow and associated traffic phenomena such as conflicts in traffic, traffic generation and assignment, and traffic control. The use of these various models are explored both in terms of how they have improved traffic systems over the years and how better implementation of these models can accelerate the successful deployment of Intelligent Transportation Systems (ITS). Furthermore, the book outlines opportunities for development of additional models needed for continued improvement of ITS.

The book is intended as a textbook for a college Transportation Science curriculum, and as a reference book for researchers in Transportation Science. Dr. Gazis has concentrated in the book’s presentation on the fundamental concepts and methods in the various areas of traffic theory.

Retrial Queueing Systems: A Computational Approach by Jess Artalejo and Antonio Gomez-Corral. The application of auto-repeat facilities in telephone systems, as well as the use of random access protocols in computer networks, have led to growing interest in retrial queueing models. Since much of the theory of retrial queues is complex from an analytical viewpoint, with this book the authors give a comprehensive and updated text focusing on approximate techniques and algorithmic methods for solving the analytically intractable models.

Retrial Queueing Systems: A Computational Approach also

• Presents motivating examples in telephone and computer networks.
• Establishes a comparative analysis of the retrial queues versus standard queues with waiting lines and queues with losses.
• Integrates a wide range of techniques applied to the main M/G/1 and M/M/c retrial queues, and variants with general retrial times, finite population and the discrete-time case.
• Surveys basic results of the matrix-analytic formalism and emphasizes the related tools employed in retrial queues.
• Discusses a few selected retrial queues with QBD, GI/M/1 and M/G/1 structures.
• Features an abundance of numerical examples, and updates the existing literature.

Heavy Traffic Limits for Multiphase Queues by FI Karpelevich and Ya Kreinin. This book analyzes several types of queueing systems arising in network theory and communication theory. Karpelevich and Kreinin use numerous methods and results from the theory of stochastic processes. The main emphasis is on problems of diffusion approximation of stochastic processes in queueing systems and on results based on applications of the hydrodynamic limit method. The book will be useful to researchers working in the theory and applications of queueing theory and stochastic processes.

Operations Research in the Airline Industry. The field of operations research has had a tremendous impact on the management of today’s air transportation. Driven by enormous demand from management to gain a competitive advantage in the market, airlines are turning to advanced optimization techniques to develop mission-critical decision support systems for management and control of airline operations. The purpose of Operations Research in the Airlines Industry is to show some recent advances in optimization techniques and decision support systems applications in air transportation. It covers a wide variety of operations research topics in the air transportation industry including: + Demand forecasting + Network design + Revenue management + Route planning + Airline schedule planning + Irregular operations aircraft routing + Integrated scheduling + Real-time crew management + Crew pairing optimization + Air traffic flow management + Crew rostering + Airport traffic simulation and control + Coping with the FAA’s traffic control program.

Optimal Design of Queueing Systems by Shaler Stidham Jr. Focusing on the underlying structure of a system, Optimal Design of Queueing Systems explores how to set the parameters of a queueing system, such as arrival and service rates, before putting it into operation. It considers various objectives, comparing individually optimal (Nash equilibrium), socially optimal, class optimal, and facility optimal flow allocations.

After an introduction to basic design models, the book covers the optimal arrival rate model for a single-facility, single-class queue as well as dynamic algorithms for finding individually or socially optimal arrival rates and prices. It then examines several special cases of multiclass queues, presents models in which the service rate is a decision variable, and extends models and techniques to multifacility queueing systems. Focusing on networks of queues, the final chapters emphasize the qualitative properties of optimal solutions.

Written by a long-time, recognized researcher on models for the optimal design and control of queues and networks of queues, this book frames the issues in the general setting of a queueing system. It shows how design models can control flow to achieve a variety of objectives.

Optimizing Emergency Department Throughput by John Shiver and David Eitel. Across the country ambulances are turned away from emergency departments (EDs) and patients are waiting hours and sometimes days to be admitted to a hospital room. Hospitals are finding it hard to get specialist physicians to come to treat emergency patients. Our EDs demand a new way of thinking. They are not at a tipping point; they are at a breaking point. Under current loads and trends they are going to begin to break and these breakdowns will be painful and ultimately dangerous to society.

Stop hemorrhaging limited health care funding through your emergency department …

Rethink, Reorganize, Reprioritize, Retool.

Recognizing that the ideal in health care is presently beyond our immediate grasp, this book instead focuses on providing health care leaders with the tools they can employ to optimize the performance of EDs and thereby improve service to patients, employees, and communities.

Written by 20 of the most progressive and successful health care reformers in the country, the approaches described can be utilized to quantify improvements, enhance predictability of workflow, and improve staff scheduling. The data derived using these techniques can serve as powerful evidence in support of change. While a common discussion among ED professionals is the perception that many patients are not really emergency patients and could be treated in another setting at another time, that argument is not germane until we as a nation elect to reform the way we chose to deliver healthcare to the underserviced.

In the meantime this book provides invalauable information to help individual hospitals to retool their ED’s. It offers new approaches that think outside of the box for all stakeholders It also provides the statistical evidence that administrators need to make their cases for changes and added resources. It will help you forecast the demand for services and give your center an approach that will allow the ED to become a source of income rather than one that continues to hemorrhage needed limited health care funding.

Patient Flow: Reducing Delay in Healthcare Delivery by Operations Research and Management. This book is dedicated to improving healthcare through reducing the delays experienced by patients. It is the first book treatment to have reduction in patient delay as its sole focus, and therefore, provides the foundation by which hospitals can implement change. In short, the book provides “hands-on” discussion and methods for solving a variety of problems, and is a guide to motivate change in Health Care Systems around the world.

Performance Analysis and Optimization of Inbound Call Centers. The focus of this book is on the management of inbound call centers. Based on technical performance measures this book develops economic performance measures for different classes of telephone service numbers. Both the numbers of agents and the number of offered phones lines are decision variables in the operational personnel planning process. Since call arrivals as well as call-handling times are random in inbound call centers, this book concentrates on performance analysis and optimization using queueing models. These models may differ with respect to several features, for example, the number of customer classes, the number of differently trained agent groups, the limitation of the waiting room, or the customer’s impatience. This book describes mathematical methods and algorithms to relate these decision variables to technical as well as economic performance measures.

Point Processes and Queues by P. Bremaud. From the Introduction: The emphasis has been placed on topics of interest in systems science at large…The level of exposition and the inclusion of a large number of exercises with complete detailed solutions make this book usable as a text for graduate students in applied probability, electrical engineering, computer science, and operations research. The prerequisites in probability and random processes are recalled in the Appendices.

Probability, Markov Chains, queues, and Simulation: The Mathematical Basis of Performance Modeling by William J. Stewart. Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics.

The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation.

Frontiers in Queuing Models and Applications in Science and Engineering. Queueing systems and networks are being applied to many areas of technology today, including telecommunications, computers, satellite systems, and traffic processes. This timely book, written by 26 of the most respected and influential researchers in the field, provides an overview of fundamental queueing systems and networks as applied to these technologies.Frontiers in Queueing: Models and Applications in Science and Engineering was written with more of an engineering slant than its predecessor, Advances in Queueing: Theory, Methods, and Open Problems. The earlier book was primarily concerned with methods, and was more theoretically oriented.

This new volume, meant to be a sequel to the first book, was written by scientists and queueing theorists whose expertise is in technology and engineering, allowing readers to answer questions regarding the technicalities of related methods from the earlier book.Each chapter in the book surveys the classes of queueing models and networks, or the applied methods in queueing, and is followed by a discussion of open problems and future research directions. The discussion of these future trends is especially important to novice researchers, students, and even their advisors, as it provides the perspectives of eminent scientists in each area, thus showing where research efforts should be focused. Frontiers in Queueing: Models and Applications in Science and Engineering also includes applications to vital areas of engineering and technology, specifically, telecommunications, computers and computer networks, satellite systems, traffic processes, and more applied methods such as simulation, statistics, and numerical methods. All researchers, from students to advanced professionals, can benefit from the sound advice and perspective of the contributors represented in this book.

Queueing Theory for Telecommunications: Discrete Time Modelling of a Single Node System by Attahiru Sule Alfa. Queueing theory applications can be discovered in many walks of life including; transportation, manufacturing, telecommunications, computer systems and more. However, the most prevalent applications of queueing theory are in the telecommunications field.

Queueing Theory for Telecommunications: Discrete Time Modelling of a Single Node System focuses on discrete time modeling and illustrates that most queueing systems encountered in real life can be set up as a Markov chain. This feature is very unique because the models are set in such a way that matrix-analytic methods are used to analyze them.

Queueing Theory for Telecommunications: Discrete Time Modelling of a Single Node System is the most relevant book available on queueing models designed for applications to telecommunications. This book presents clear concise theories behind how to model and analyze key single node queues in discrete time using special tools that were presented in the second chapter. The text also delves into the types of single node queues that are very frequently encountered in telecommunication systems modeling, and provides simple methods for analyzing them. Where appropriate, alternative analysis methods are also presented.

This book is for advanced-level students and researchers concentrating on engineering, computer science and mathematics as a secondary text or reference book. Professionals who work in the related industries of telecommunications, industrial engineering and communications engineering will find this book useful as well.

Queueing Modelling Fundamentals: With Applications in Communication Networks by Chee-Hock Ng and Soong Boon-Hee. Queueing analysis is a vital tool used in the evaluation of system performance. Applications of queueing analysis cover a wide spectrum from bank automated teller machines to transportation and communications data networks.

Fully revised, this second edition of a popular book contains the significant addition of a new chapter on Flow & Congestion Control and a section on Network Calculus among other new sections that have been added to remaining chapters. An introductory text, Queueing Modelling Fundamentals focuses on queueing modelling techniques and applications of data networks, examining the underlying principles of isolated queueing systems. This book introduces the complex queueing theory in simple language/proofs to enable the reader to quickly pick up an overview to queueing theory without utilizing the diverse necessary mathematical tools. It incorporates a rich set of worked examples on its applications to communication networks.

Queues, Inventories, and Maintenance: The Analysis of Operational systems with Variable Demand and Supply by Philip Morse. A seminal work by one of the founders of the science of operations research, this text for upper-level undergraduates and graduate students examines general queuing problems, discusses the effect of changes of arrival and service distributions on queuing results, and describes the application of queuing theory to maintenance and inventory problems.

Analysis of Waiting Time Data in Health Services Research by Boris Sobolev and Lisa Kuramoto. Why some patients wait longer than others remains an important question. This book is a reference for health services researchers looking for statistical tools with which to study waiting times. The book offers detailed coverage of statistical concepts and methods for the analysis and interpretation of waiting-time data. It provides analysis from health services research perspective, rather than operations management, and contains a collection of examples.

To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems by Rafael Hassin and Moshe Haviv. Focuses on the interesting, practical viewpoint of customer behavior and its effect on the performance of the queueing system.

Leadership for Smooth Patient Flow by Kirk Jensen. The payoff for improving patient flow goes far beyond shorter wait times. When patients flow smoothly through the care process, nurses and physicians have the time they need to provide safe and compassionate care. Treatment is provided in the optimal setting, which reduces risk and improves outcomes. Patients feel satisfied and recommend your services to others, and your bottom line improves as more patients are treated. This book provides the inspiration, information, and ideas you need to lead patient flow improvement efforts at your organization. You will learn from the successes and failures of the authors healthcare leaders who have played pivotal roles in patient flow improvement projects. The book begins by explaining the fundamentals of patient flow and providing a solid business case for pursuing improvement efforts. It uses real-life examples to explain common patient flow theories and improvement methods.

Queueing Theory Books is a post from: Lean Six Sigma Consulting

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For those who study the theory of lines and queues, holiday shopping (next to airport dynamics) is a time full of learning opportunities: a real-world lab to study and learn how people respond and behave within the dynamics of lines.

But, this Black Friday 2010 and CyberMonday 2010, consider avoiding the lines all-together and just shop online. You’ll physically avoid waiting in lines, but you will have to wait for your package – in the comfort of your own home. All the work of pick, pack, and ship is handled quite well by Amazon’s highly efficient fulfillment operations.

So, consider ordering online instead, it’ll be much easier to find that Lego Minifigure you’ve always wanted to get your kid.

Disclosure: When you purchase on Amazon from shmula.com, it allows me to continue writing and the money goes to support the site. Thanks.

CyberMonday: Avoid the Lines is a post from: Lean Six Sigma Consulting

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In the United States, Thanksgiving Holiday is always on the last Thursday of November. The Friday immediately after Thanksgiving is traditionally called Black Friday (I have no idea why they call this day “Black Friday”). Black Friday is usually the biggest shopping day of the year and there are very big discounts and sales and deals on items and the newspaper is usually full of Black Friday ads.

Usually, stores open very early in the morning – 4AM and people are waiting in line – sometimes people have been camping out in front of the stores so they are the first one inside to take advantage of the sale and discounts on items.

This Black Friday 2010, I learned a good lesson on Queueing and Mob Mentality and I’ll be shopping online next time.

Black Friday 2010

This Black Friday, my wife and I decided to be courageous and act crazy like everyone else. Below is a picture I took at Wal-Mart.

Black Friday: A Lesson in Queues

We also went to Toys ‘r’ Us, which had really good discounts on items and a big after Thanksgiving sale. When we arrived, there was a long line outside and a long line inside. We decided not to wait in the line, but Toys ‘r’ Us applied a really good Queueing principle:

No customers entered Toys ‘r’ Us until a customer exited.

On the one hand, this approach prevents crowd-related violence and stampedes. On the other hand, it creates a very long line of customers waiting in the cold – but, as you can see from the fuzzy picture below – these are customer willing to wait in the cold, so Toys ‘r’ Us can take the risk and make customers wait.

It was 17 Degrees Fahrenheit outside and there were about 75 people waiting outside and several hundred customers waiting inside Toys ‘r’ Us. And, yes, that’s a police car by the entrance just in case a stampede starts over toys.

For your enjoyment, here’s a picture of a festive lady with a crazy hat at Wal-Mart.

Below are a few videos of Black Friday 2009 after Thanksgiving Sale – in one video, a stampede at Wal-Mart caused the death of a Wal-Mart employee.

Black Friday at Wal-Mart

Black Friday at Circuit City

Black Friday at Target

Black Friday Fight over XBox

Black Friday Fight over PS3

Black Friday at Best Buy

Black Friday After Thanksgiving Day Sale is a post from: Lean Six Sigma Consulting

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This article looks at the tradeoff between Service Level and Costs, within the context of Waiting Line Management.

Before we begin, let me present two common assumptions:

1. Quicker is always better
2. Quicker service cost more than slow service

But, are those assumptions true?

In general, the answer to (1) is “yes”. But, with lean thinking and process improvement as a backdrop, we know that faster service doesn’t have to be a cost burden to the firm. In other words, most people and organizations assume (2) is correct, but we’ll refute that claim here.

There is one thing we know for sure:

Customers’ encounters with waiting lines (which are often their initial encounter with the company or its service) can significantly affect their overall level of satisfaction with the organization.

So, the question becomes:

How can we offer sufficiently fast service, within cost constraints?

In other words,

So, if we’ve captured the tradeoff between speed and service level within the context of queues, then the above cost function gives us a glimpse as to how to answer the question:

How can we offer sufficiently fast service within cost constraints?

Why the Need for Speed?

In general, the items below are considered reasons for why faster service is desired over slower service:

1. Time is a commodity and is very valuable.
2. Faster service and convenience is considered a premium feature and can demand a higher price.
3. As products get more commoditized, few things can create brand or service differentiation – speed is a feature that can create differentiation.

Tradeoff Between Cost and Service in Queues

The traditional view is that the faster the service, the higher the cost it is to provide that service. An alternative view is pictured below:

The Tradeoff Curve above tells us a few things:

1. When the wait is short, the cost is low.
2. When the wait is long, the cost is high.
3. In some operations, the cost to serve increases as the wait time increases.
4. In most cases, for the customer, the cost to wait increases as the wait times are longer.

How Can Cost Decrease With Shorter Wait Times?

A few simple examples are instructive:

A Call Center Example

• When there is a problem with a product, a customer will call the customer service center. The longer the customer is on the telephone with the customer service agent, the higher the cost of that call or contact.
• In contrast, through process improvement, efforts to improve product quality will likely lead to contact avoidance – that is, prevent customer contacts (or failure demand).

A Restaurant Example

• When the wait time is long to receive service, the cost for the customer increases, but the cost for the restaurant might not – except in the form of customer loyalty, but not in hard net present costs.

The Role of Continuous Improvement

We know from the Theory of Constraints that if we improve a non bottleneck, then no material improvement will have been made to the service or the process. What is critical in the context of queues is to identify the constraint and then apply the principles of lean or six sigma to improve quality, reduce defects.

Practically, this approach might mean the following, assuming we have identified the constraint in the system:

1. Reduce set-up times.
2. Reduce changeover time through SMED.
3. Reduce steps, defects, rework.

And, when the above are accomplished at the system bottleneck, then we will have reduced over wait time and within reasonable cost constraints.

Waiting Line Management: Costs and Service Level is a post from: Lean Six Sigma Consulting

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There are many crowd related disasters and crowd related violence throughout history. Here is just one story:

Jan 14th 2010 – HARIDWAR, 6 dead, 12 injured. India: Hundreds of thousands of bathed in the icy waters of the Ganges river yesterday as a months-long Hindu festival expected to attract more than 10 million people kicked off in one of India’s holiest cities. The festival was marred by tragedy elsewhere in the country when pilgrims jostling to get to a river temple stampeded, crushing six elderly women and a child and injuring 12 pilgrims, police said. The stampede occurred as thousands of pilgrims tried to board boats to the island temple nearly 100km south of Calcutta, the capital of West Bengal state, said Surajit Kar Purakayastha, an inspector-general of police. Along the river banks, hundreds of thousands of men, women and children entered the fast-moving waters in a ritual that is part of the Kumbh Mela, touted as the largest religious gathering in the world. Braving rain and cold weather, devotees from all over India in cars, buses, trains and tractor-driven carriages have been pouring into Haridwar, a temple-filled town at the foothills of the Himalayas. One pilgrim, Shyam Lal, 55, said, “It is Hindus’ belief that bathing in the Ganges river on the occasion purifies one’s soul. It also helps in controlling one’s desires in a materialistic world,” he said. Hundreds of tents have been spread out over a vast 130km2 area where more than 15 000 makeshift toilets have been erected and 10 000 people employed to keep the tent city clean, a festival organiser said. – Sapa-AP. Published on the web by Mercury on January 14, 2010. ^{1 2 3 4}

Lessons from Queueing Theory can inform our perspective on crowds and how to prevent disasters that can come from crowd-related violence such as a stampede, mob violence, evacuation, and crowd management and planning.

A Few Concepts

A few terms relevant to our discussion:

• Focal Route Analysis: A study of the primary routes to the destination, including routes to the turnstiles, kiosks, front desk, and the entry or exit door.
• Crowd Craze: This a term used to describe a deliberate attempt to create a large group, usually for an event

Crowd Craze and Queueing

In General, crowd violence follows a similar model in Queueing typically called WIP Explosion – that is, when the capacity of a system has reached its near maximum. The difference in crowd control is not so much the capacity, but how the units in the system are behaving – in the case of crowds, it’s how people are behaving and how the crowd is behaving en masse.

Visually, the chart below shows WIP Explosion, in general:

Crowd Control and Queueing Applications

Consider the picture below:

What do you see? Is there an ordered line? Can you tell where the people are supposed to self-organize?

If there is no focal route, that is a signal of risk that a crowd-related disaster might happen. What might be done to the line above to encourage and enforce order and thereby preventing potential violence?

Now consider the picture below?

A small hole in the fence such as the above picture shows draws focus away from the focal route and brings the focus to this very small hole in the fence. As with most things, a small hole such as the above can quickly create mob mentality and crowd violence – imagine hundreds of people, within seconds, trying to get through a small hole in the fence – yes, that is a recipe for crowd craze. Often, it can happen in seconds.

There are many opportunities for Queueing Theorists in the “real world”. Our discussion on Crowd Control is just one.

Crowd Control, Queueing, and Violence is a post from: Lean Six Sigma Consulting

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Most organizations use the average or the mean for their metrics. This is common, convenient, and easy. But, the average doesn’t describe the experience of the individual customer.

Suppose you run a call center or contact center. Assume the following measures for Average Speed of Answer (ASA), which  is a common call center metric that is supposed to measure how long a customer has to wait on the telephone before a customer service representative answer the customer’s phone call:

• Target: 60 seconds
• Average ASA: 57 seconds

Most managers would say that this call center is doing just fine. But is it?

Here are the other numbers:

• Shortest Wait or Fastest ASA: 31 seconds
• Longest Wait or Slowest ASA: 85 seconds
• Standard Deviation: 11 seconds

Given the above measures, we know the following according to the axioms of the normal distribution:

1. 68.3% of calls will be answered within 57 seconds (+/-) 11 seconds
2. 95.5% of calls will be answered within 57 seconds (+/-) 22 seconds
3. 99.7% of calls will be answered within 57 seconds (+/-) 33 seconds

Metrics, The Mean: Applications

Beyond the call center, where else might customers wait for service or for product? Think about the specific processes that you are involved in, whether they be software engineering, project management, in a hospital – wherever there are queues or waiting, the example above can be applied.

What The Customer will Feel and Remember

We must keep in mind that the customer will not remember “an average experience”; the customer will remember usually the BEST and the WORST experiences he or she has had with a company, product, or a service provider. The average or the mean, as a measure of customer experience, is inadequate by itself. We must also include the standard deviation, for that will give us a better picture for what the individual customer might be feeling.

Averages Do Not Describe Individual Experience is a post from: Lean Six Sigma Consulting

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My kids have been in school now for about a two weeks. So, the whole back-to-school shopping thing is still all-too fresh in my mind. Which is why this recent article on back-to-school and Queueing was so relevant.

According to a recent study, back-to-school retailers are losing up to 39% of revenues due to long lines and in-line frustration. With overall K-12 and college spending combined expected to reach a whopping $55 billion this year, back-to-school shopping is the second largest event for retailers, right behind the winter holidays.

According to the study, $21 billion of that revenue is lost to poor customer queue management. That is an outstanding number – and tremendous opportunity.

Waiting Line Survey Data

According to the survey results,

In the nationwide study of 500 parents with school-aged children, the vast majority, 75% of respondents say long lines are the worst part of the back-to-school shopping experience. Nearly four in ten (39%) parents, leave stores while back-to-school shopping without purchasing a single item because of long lines. Long lines are cited by 35%, 1 in 3 parents, as the primary reason for leaving the store — worse than high prices, lack of supplies or inexperienced sales personnel.

According to the researchers,

“Time is money — reducing ‘walkaways’ and optimizing queue time is the end game. By deploying a smart queue management strategy, retailers can take control of and ideally manage the customer journey, from when they enter the store, at the retail point-of-sale, in-line at checkout and once they walk out the door.”

Convert Wait Time to Shopping Time

This is a tremendous opportunity for the retailer – by converting Queueing Time or Wait Time to extended Shopping Time. Potentially, retailers can cost-effectively convert waiting time into extended shopping time, boost impulse sales, improve retail efficiencies and eliminate customer frustration and employee stress during the back-to-school shopping season, enriching the overall customer experience and ultimately increase margins.

Lean Thinking Applied to Queues

How might Lean Thinking be applied?

The Theory of Constraints teaches us that the overall output of any process is determined by the bottleneck. So it’s critical that we do the following at the bottleneck:

• eliminate or reduce steps at the bottleneck
• eliminate or reduce waste (muda) at the bottleneck
• create more capacity at the bottleneck

For a retailer, the point-of-sale or the cashier or the cash registers is the bottleneck step.

Ideally, we would want to design a pull system, such as as one customer leaves the system, another enters the system. But, in a point-of-sale queue, such as we see in retail, we do not see a pull system – we see a push system: customers line-up and wait, regardless of customers leaving the system or regardless of exit rate in the queue.

In general, If a customer is already waiting in line, one can eliminate waste at the server (or point of sale). Doing this could potentially reduce wait time, increase flow, which means revenue recognition much sooner.

Here are a few specific suggestions:

1. Make sure the cashier is only (a) scanning and (b) taking payment. There should be baggers to offload unnecessary steps that the cashier should not be doing.
2. Make sure payment systems and scanning systems are functioning perfectly, are easy to use for the cashier; that the payment system is clear and easy to use for the customer.
3. Make sure the customer is ready to pay, removing the need for the “search, reach, grab, pay” steps during the payment steps. If the customer is ready when the cashier if finished scanning, that could eliminate a few seconds of wait time for the entire queue.

It’s Your Turn

If you were to provide advice to Wal-Mart or some other large brick-and-mortar retailer, how might you suggest they manage their queues in order to convert wait time to shopping time?

source: http://www.lawrencemetal.com/about/news.aspx

Back to School, Revenue Loss, Waiting Line Frustration is a post from: Lean Six Sigma Consulting

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Yes, I feel like a manchild as I write this post about a vampire love story, but there is some relevance to the long-time readers of shmula.com:

why would people wait for 3 days to see The Twilight Saga Eclipse?

The answer can be found in the Psychology of Queueing (Queing) or Psychology of Waiting Lines.

Psychology of Waiting Lines

As a review,

1. Unoccupied time feels longer than occupied time.
2. Process waits feel longer than in-process waits.
3. Anxiety makes waits seem longer.
4. Uncertain waits seem longer than known, finite waits.
5. Unfair waits are longer than equitable waits.
6. The more valuable the service, the longer the customer is willing to wait.
7. Solo waits feel longer than group waits.

As you can see from the picture below, on the whole, people aren’t really complaining, there’s a community of like-minded and interested (crazy twilight fans) people together. They obviously value the end result and because they are all waiting together, it doesn’t feel long.

As a reference point, these fans are our modern day “dead heads” – fans used to follow around the Grateful Dead in my day.

Team Edward or Team Jacob?

I don’t care. My wife loved the movie – that’s all that matters, but I’m comfortable enough with my manhood to say that it was pretty good. What is instructive for us is that people are CRAZY about this movie and are willing to wait for days to see it. Do Apple Fan Boys wait for the iPhone 4 like Twilight fans wait for the movie?

More about Twilight Saga Eclipse below, if you’re interested.

Twilight Saga: Eclipse 2010

Bella once again finds herself surrounded by danger as Seattle is ravaged by a string of mysterious killings and a malicious vampire continues her quest for revenge. In the midst of it all, she is forced to choose between her love for Edward and her friendship with Jacob — knowing that her decision has the potential to ignite the struggle between vampire and werewolf. With her graduation quickly approaching, Bella is confronted with the most important decision of her life. Written by Summit Entertainment Isabella Swan was a normal teenage girl, in a normal world. Until she met Edward Cullen and Jacob Black.

Since then she has been up against a Sadistic Vampire, The Volturi and an angry werewolf, Paul. But what lies ahead is something only Edward’s ‘brother’, Jasper Hale, can help the Cullens with. A Newborn Vampire Army created by a Revenge ridden Victoria for what Edward did to her mate, James. ‘Mate for Mate’ as Victoria sees it. But what she doesn’t know is the Cullens have a secret on their side. With Graduation coming fast, Bella gets more and more worried about becoming immortal and the fight. But the real question is: Who will she choose? Written by Jemma From England Bella and Edward have been reunited, but their forbidden relationship is threatened to be torn apart again with an evil vampire still seeking her revenge. And Bella is forced to choose between her true love for Edward or her friendship with Jacob Black as the struggles between vampires and werewolves continues.

But there is still another choice for Bella to make, mortality or immortality? Written by Mel from the United Kingdom. After the vote about her becoming immortal, Bella realizes graduation isn’t too far away and starts to panic. But she has bigger problems arising, her feelings for Jacob, his feelings for her and Victoria. The vampire who is trying to take revenge on her for Edward killing her James.

photo credit: Reuters

The Twilight Saga: Eclipse, A Queueing Perspective is a post from: Lean Six Sigma Consulting

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Little’s Law is an incredibly helpful principle for business. Unfortunately, it is not used enough, or it is poorly understood.

As review,

Little’s Law: For a Queueing (Queuing) System in steady state, the average length L of the queue equals the average arrival rate λ times the average waiting time W.

Or,

L = λW

Put another way,

Total Cycle Time = Number of Things in Process / Average Completion Rate

For example, let’s assume 8 feature request are what a team can consider per month.  If 16 features were in the backlog, then it will take the team 60 days to complete.

In other words,

60 Days = 16 features / 8 feature capacity

But, assume that 4 feature request were released, then it will take the team an average of 14 days to complete those features.

So, what does Little’s Law teach us about speed of delivery?

To deliver faster, we can do two things:

1. Reduce the size of work in process (things in process)
2. Increase the average completion rate

That’s it.  When reduced to the physics of Queueing, those are the two variables that are drivers of speed.

Other Applications of Little’s Law

How else might you apply Little’s Law?  Would Little’s Law be helpful in your work?  How?

Software Development is Queue Management is a post from: Lean Six Sigma Consulting

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A Queueing (Queuing) System is a structure where things arrive (anything, including people), they join a line, wait for service.  After being served, they exit the system.

One result of Queueing Theory is Little’s Law:

Little’s Law: For a Queueing (Queuing) System in steady state, the average length L of the queue equals the average arrival rate λ times the average waiting time W.

Or,

L = λW

Given this definition, a warehouse (distribution center, fulfillment center) is a Queue in which Stock Keeping Units (SKU) are the customers that arrive at the receiving dock, where they join a queue (a line) to be processed and then wait for service (shipped).  The processing in this case is the Pick, Pack, Ship part of the internal warehouse process.

Here’s a simple example of how to use Little’s Law to expose information that is not obvious.

Imagine a warehouse with 10,000 pallets, and the company has an average of 4 inventory turns per year.  How much labor is needed to support this activity?  Using Little’s Law, we get:

10,000 pallets = λ(1/4 year)

so,

λ = 40,000 pallets / year

Let’s assume an 8 hour shift and 250 working days per year and 2,000 working hours per year.  Then, we get:

λ = 20 pallets / hour

Of course, most systems aren’t conveniently in a steady state, but the principle still applies.  With the little information we had, we estimated the labor required to support the activity of this warehouse.

Other Applications of Little’s Law

Little’s Law can be applied to many areas of business.  Here are scenarios where Queueing Theory can be helpful:

• how to staff an emergency room (ER)
• how to staff a call center
• how many features a software engineering team can develop
• how many projects can be completed

There are more.  Think about it – I bet you can find many applications of Queueing Theory.  Think about your areas of responsibility – how might you apply Little’s Law?

A Fulfillment Center is a Queueing System is a post from: Lean Six Sigma Consulting

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Does the shape of the queue matter?  What determines how waiting lines are organized?

A provocative young artist recently published a set of infographics that compare the differences between the West and the East.  In one of those, she portrays the differences in waiting between the West and the East.

In that infographic, you get a sense of the organization, shape, and cultural biases ¹.

What is the reason for this cultural difference in wait line experience?

It turns out there’s a simple explanation.

Waiting with Teenagers

If you’ve every waited in a line with many teenagers in it – like opening night for a heart throb movie or something – what do you notice?  The line is more like a cluster.  Why?  It turns out the line shape becomes a cluster shape because of socialization: queue occupiers find people they know, start talking, and the shape of the line changes.

In the East, apparently, there is a higher likelihood of ancestral or familial relationships.  In other words, people know each other because of family relationships.  So, the line shape looks difference and queueing dynamics changes because of socialization.  Makes sense.

The Psychology of Queueing

If what I claim is true, how does socialization in queues impact the psychology of queueing?

1. Unoccupied time feels longer than occupied time.
2. Process-waits feel longer than in-process waits.
3. Anxiety makes waits seem longer.
4. Uncertain waits seem longer than known, finite waits.
5. Unfair waits are longer than equitable waits.
6. The more valuable the service, the longer the customer is willing to wait.
7. Solo waits feel longer than group waits.

It’s Your Turn

Does socialization in queues make waiting “feel shorter”?  What about other queueing dynamics, such as the physics of wait time, number of servers – are queues managed differently because of this cultural phenomenon?

Queue Shapes, Culturally Influenced is a post from: Lean Six Sigma Consulting