Everyday queueing theory problems come down to one relationship between arrival rate (λ), arrival rate (λ), probability of queue with, probability of queue with, number of the servers (s), service rate (μ), service rate (μ) and one way. This calculator turns those numbers into probability of zero customers in the queue (p0), number of customers in the queue (lq) and average time spent in the system (w), with the working shown so you can check the result rather than simply trust it.
What Is the Queueing Theory Calculator?
The queueing theory calculator works through arrival rate (λ), arrival rate (λ), probability of queue with, probability of queue with, number of the servers (s), service rate (μ), service rate (μ) and one way to give you probability of zero customers in the queue (p0), number of customers in the queue (lq) and average time spent in the system (w).
It is aimed at the case where you know the method but not the arithmetic: schoolwork, a quick check on a worked example, or a figure that needs to be right first time.
How the Queueing Theory Works
Each step is applied in order and depends only on the values before it, which makes it possible to follow the calculation rather than take it on trust.
If you are working through the same problem by hand afterwards, compare the intermediate values. That is where a mistake usually shows itself.
Checking Your Own Working
Round sensibly as you go and keep the full precision in the calculator. Rounding too early is the most common way of getting a right method to give a wrong answer.
Where a result should be a whole number, such as a count, check it against the constraint that produced it. A fraction of a person is a sign that something upstream needs revisiting.
A Worked Example
Working through an example, with arrival rate (λ) set to 10, arrival rate (λ) set to 5, probability of queue with set to 2, the calculator returns traffic intensity (ρ) of 2, and average number of customers (l) of -2, and number of customers in the queue (lq) of -4. Substitute your own figures and the same steps produce the same kind of answer.
Frequently Asked Questions
What do I need to work out probability of zero customers in the queue (p0)?
Enter arrival rate (λ), arrival rate (λ), probability of queue with, probability of queue with, number of the servers (s), service rate (μ), service rate (μ) and one way. The calculator then works through the relationship and reports probability of zero customers in the queue (p0), number of customers in the queue (lq) and average time spent in the system (w).
What does the queueing theory actually calculate?
It takes arrival rate (λ), arrival rate (λ), probability of queue with, probability of queue with, number of the servers (s), service rate (μ), service rate (μ) and one way and converts them into probability of zero customers in the queue (p0), number of customers in the queue (lq) and average time spent in the system (w) using the standard relationship for this problem. Each step is applied in turn, so every output is traceable back to the numbers you typed in.
How should I enter arrival rate (λ)?
Use the value you actually have, in the unit shown next to the field. Enter every value in the same unit system. The calculator assumes the figures you supply are consistent, so it cannot correct a mistake that happens before the calculation starts.
Why does a small change in one value move the answer so much?
That happens when an output depends on a ratio, because dividing by a small number magnifies any change in it. Check the units first, then check whether the value you entered is in the range the relationship was designed for.
Can I use this for a quick estimate as well as a precise answer?
Yes. The calculator is built for quick checks, so round your inputs to a sensible number of significant figures and the result will be just as useful. If you need more precision, keep the same method and add digits to the inputs rather than adjusting the method itself.