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)
Now, let’s generalize the example above and arrive at Little’s Law:
Wq = Lq / Throughput
Littles 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, Littles 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