Two different queuing systems

Introduction

This report presents the modeling of two different queuing systems in a typical bank environment using the arena software. The confidence intervals for both the systems are constructed based on the simulation results. The systems are then compared to find out which queuing system performs better.

Assumptions

For both systems, no real data was collected. Both the interarrival times and service times were taken from known probability distributions. Other assumptions also include no balking, reneging and queue jumping. Each replication had the same initial conditions and terminating events. Lastly, both systems are assumed to be stable, have infinite calling population and no limit on system capacity.

Modeling of the systems

In this section of the report, the actual modelings of both the systems using the arena software are discussed. Configuration of the models and steps to run the system are also highlighted. Firstly, system 1 is explained, followed by system 2.

System 1 modeling

System 1 has a separate queue for each individual bank teller. Based on Kendall’s notation, system 1 is an M/M/4 system. It is a Poisson process and disallows batch arrivals. The table below summarizes the categorization of the system based on the parameters of the system.

In this system, customers arrive and choose to join the shortest queue. The highlighted mean values in the table represent the exponential mean value ?. For the interarrival time, 100 customers arrive in 1 hour. Hence,

β= 1/ (100/60) = 0.6

1. Firstly, create the customer arrival portion by clicking and dropping the create button. Next configure it by doubling clicking the diagram. The Figure shows the dialog box to configure the entity.
2. Type the parameter as shown in Figure 2 above for this system. The configuration can also be shown in the figure below.
3. Create the four individual processes for each of the Bank Tellers by using the process button. Configure the process as shown below.
4. Since the customers can choose the shortest queue to join upon arrival, create a decision box by using the decide button. Configure the decision box as follows:
5. Click on the Add button to include the conditions for the branching conditions. Select Expression and right click and select expression builder to construct the expressions.
6. Finally, create the customer departure by using the Dispose button. Double click on the button to configure by naming it.
7. Lastly, connect all the components together to model the system 1.

System 2 modeling

System 2 has only a single queue for all the arriving customers. When a bank teller becomes available, the customer will be served by that bank teller. Based on Kendall’s notation, system 2 is an M/M/1 system. The table below shows the categorization of the system 2 based on Kendall’s notation.

Running the Simulation

Once the models of both the system are constructed, simulation runs are conducted to evaluate the performance of the systems. The steps in running the simulation are as follows:

1. Click on the Run tab and select Setup.
2. Click on the Replication Parameters tab. Input number of replications as 15 and replication length as 480 change all the units to minutes. This is shown in the Figure below.
3. Click on Run tab and select Go to run the simulation.

Simulation Results

This section of the report summarizes the results produced by both the queuing systems. The performance measure parameter is the average time the customer spends in the bank. The results for each individual system are evaluated and the following confidence interval is constructed. Then the two systems are compared by constructing another confidence interval.

System 1 Results

The system 1 results are based on the average time a customer spends in the system as its performance measure. The average time for each replication is summarized in the table below.

Firstly, the mean is computed using

(n) = 4.8121

Variance is also computed using

(n) = 1.103800987

Hence the 95% confidence interval (? = 0.05, t14, 0.975 = 2.145) for system1 is computed using

Confidence interval: [4.2302, 5.3940]

System 2 Results

The system 2 results are also measuring the average time the customer spends in the system. The results are summarized in the table below.

By using the same formulas, the mean, variance and confidence interval are as follows:

(n) = 3.804533333

(n) = 2.231921051

Confidence interval: [2.9771, 4.6319]

Comparison between Two Systems

From previous results, the confidence intervals of both the systems overlap each other. Therefore, it is hard to determine which system performs better. Hence, paired- t confidence interval is used to compare the two systems. It is important to note that the number of replications for each system must be the same for this type of comparison. The table below summarizes the results of this comparison.

The mean, variance and the confidence interval is computed and the results are as follows:

(n) = 1.007566667

(n) = 3.578001252

Confidence interval: [0.5192, 1.4960]

Since the confidence interval does not contain zero, there is strong evidence to conclude that system 1’s average time customer spends in the system is larger than that of system 2. Hence, system 2 performs better than system 1.

Conclusion

This report presents the models of two different queuing systems in a bank environment. Through the simulation results, it is found that system 2 performs better than system 1. In order to get more accurate results, the number of simulation runs must be increased and other performance measure parameters can be tested to further gauge the performance of both the systems.