Queuing Theory Calculator

Enter the arrival rate and service rate of a queue into the calculator to determine the average queue length, average total time, and average waiting time.

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Queuing Theory Formulas

The following formulas are for an M/D/1 queue: random (Poisson) arrivals, deterministic service times, and one server. They assume a stable system with p < 1 (arrival rate less than service rate).

AQL = \frac{p^2}{2(1-p)}

ATT = \frac{2-p}{2u(1-p)}

AWT = \frac{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, L_q).
• ATT is the average total time in the system (waiting + service, W).
• AWT is the average waiting time in the queue (W_q).

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

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