Choose M/M/1, M/M/c or M/D/1 to calculate utilization, wait times and lengths from arrival and service rates. Little’s Law separately links average number in system, throughput rate and time in system.

Steady-state averages, not guaranteed individual waits. Queue models assume Poisson arrivals, independent service and unlimited waiting space. M/M models use exponential service; M/D/1 uses fixed service time.

Illustrative examples

Queuing Theory Formulas

The following formulas are for an M/D/1 queue: Poisson arrivals, independent fixed service times, one server and unlimited waiting space. They assume a stable system with p < 1 (arrival rate less than service rate). Choose M/D/1 to use these formulas; M/M/1 and M/M/c instead assume exponential service times. The selected rate units are converted automatically.

AQL = (p²) / (2(1 - p))
ATT = (2 - p) / (2u(1 - p))
AWT = (p) / (2u(1 - p))
  • Where p is the arrival rate divided by the service rate (utilization, p = λ/μ).
  • u is the service rate (μ).
  • AQL is the average queue length (average number waiting, Lq).
  • ATT is the average total time in the system (waiting + service, W).
  • AWT is the average waiting time in the queue (Wq).

Queuing Theory Definition

Queuing theory is the mathematical study of waiting lines (queues) and how systems behave under congestion. It is used to estimate measures such as average queue length, average time in the system, and average waiting time.

How does queuing theory work?

Queuing theory works by using mathematical models (often probabilistic) to represent arrivals, service processes, and the number of servers. Many formulas are derived for idealized queuing situations with specific assumptions.

Why is queuing theory important?

Queuing theory is important because it can allow a business to understand how lines and queues will form even before customers enter the store. The business can then make adjustments to change the wait and service times.

Queuing Theory Example

How to calculate average wait time of a queue?

  1. First, determine the arrival rate.

    Calculate or measure the rate of arrival (people per time).

  2. Next, determine the service rate.

    Calculate the average service rate (people per time).

  3. Finally, calculate the average wait time.

    Using the formula above, calculate the average wait time.

FAQ

What is a queue?

A queue is typically referred to as a line of people waiting for something, usually a service.