# Times of arrivals and wait times in a queuing system

Let's consider the following problem: a post office has one counter that it recognizes is insufficient for its customer volume. It is considering putting in another counter and wishes to model the effect on the maximum number in a queue at any one time, which is considered to be a measure of the quality of its service. The post office is open from 9am to 5pm each working day. Past data show that when the doors open at 9am there will be the following number of people waiting to come in:

People

Probability

0

0.6

1

0.2

2

0.1

3

0.05

4

0.035

5

0.015

People arrive throughout the day at an average rate of 1 every 12 minutes. The amount of time it takes to serve each person is Lognorm(29,23) minutes. What is the maximum queue size in a day?

This problem requires that one simulates a day, monitors the maximum queue size during the day, and then repeats the simulation. One thus builds up a distribution of the maximum number in a queue. The solution is provided in the Queue Model. It uses a looping Visual Basic Macro called "Main Program' at each iteration of the model.

The link to the Queue Model software specific model is provided here:

This is an advanced technique and, although this problem is very simple, one can see how it can be greatly extended. For example, one could change the rate of arrival of the customers to be a function of the time of day; one could add more counters, and one could monitor other statistical parameters aside from the maximum queue size, like the maximum amount of time any one person waits or the amount of free time the people working behind the counter have.

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