# Simulating a Multiple Server Queue

Simulating a Multiple Server Queue

In queuing theory, the simplest model is called the M/M/1 or M/M/c model (Markovian arrivals, Markovian service, and 1 or servers). Customers arrive at a facility and either get served immediately by a free server or join a queue that waits for a server to become available. Standard notation is:

c

λ

1/λ

μ

1/μ

c

In the Demonstration graphic, the number of people in the queue is indicated by the height of the red bars at the bottom; the horizontal green lines indicate customers being served; red lines indicate customers waiting. The table below the graphic shows the expected and observed values of several values associated with such a process. Note that the basic units are hours, but some of the parameters are given in minutes.

In order to look at a typical window of the queue situation, a certain percentage of the initial customers is ignored. This is controlled by the choice of the initial deletion ratio. If this is set to 0 then one sees a simulation that starts with the first customer that walks in the door. In that case, the number in the queue or the system will generally be less than the expected value, since the queue is smaller than the limiting value at the start. If the deletion ratio is set to, say, 0.5, then the graphic shown will start with the customer in the order where is the total number of customers. Regardless of deletions, the time starts at 0.

N/2

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