# Simulation of Queuing System

In this section, we provide examples of how simulation models can be used for Queuing System.

## Example 1: Simulation of Queuing System

People arrive at the New Delhi Railway station to buy tickets according to the following distribution.

Inter-arrival Time (Min.) Frequency
2 10
3 20
4 40
5 20
6 10

The service time is 5 minutes and there is only one ticket counter. The Railway station incharge is interested in predicting the operating characteristics of this counter during a typical operating day from 10.00 a.m. to 11.00 a.m. Use simulation to determine the average waiting time before service and average time a person spends in the system.

Solution.

From the given distribution of arrivals, the random numbers can be assigned to the arrival times as shown in table 1.

Table 1

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Inter-arrival Time
(Min.)
Frequency Probability Cumulative
Probability
R.No.
2 10 0.10 0.10 0 - 09
3 20 0.20 0.30 10 - 29
4 40 0.40 0.70 30 - 69
5 20 0.20 0.90 70 - 89
6 10 0.10 1.00 90 - 99

The first random number generated is 17, which corresponds to the inter-arrival time of 3 minutes. This implies that the first person arrives 3 minutes after the service window opens, as shown in table 2. Since the first person arrives at 10.03 a.m., therefore, the server has to wait for 3 minutes. The server takes 5 minutes and thus, the first person leaves the system at 10.08 a.m. (10.03 + .05). Similarly, other values can be calculated.

Table 2

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S.No. R.No. Inter- Arrival
Time
Arrival
Time
Service Starts Service
Ends
Waiting Time
Server Person
1 17 3 10.03 10.03 10.08 3 -
2 86 5 10.08 10.08 10.13 - -
3 84 5 10.13 10.13 10.18 - -
4 79 5 10.18 10.18 10.23 - -
5 33 4 10.22 10.23 10.28 - 1
6 55 4 10.26 10.28 10.33 - 2
7 6 2 10.28 10.33 10.38 - 5
8 42 4 10.32 10.38 10.43 - 6
9 93 6 10.38 10.43 10.48 - 5
10 38 4 10.42 10.48 10.53 - 6
11 58 4 10.46 10.53 10.58 - 7
12 71 5 10.51 10.58 11.03 - 7
Total             39

Average waiting time before service.
= Total waiting time (person)/Total no. of arrivals
= 39/12 = 3.25 minutes.

Average time a person spends in the system.
= Service time + Average waiting time before service
= 5 + 3.25 = 8.25 minutes.