Problem description

The problem at hand relates to the management of queues in a typical bank. In a bid to improve the level of service at the tellers, a bank manager seeks to find strategies to employ for better service offering that will improve customer satisfaction. The manager is prepared to employ up to $5$ cashiers, but not less than $1.$ The main management objectives are

$($i$)$ To have an average customer to wait less than $2$ minutes for service, and

$($ii$)$ the average length of the queue to be $2$ persons or fewer.

The manager tests several models to help in the decision process.

Model $1$

Suppose the manager makes the following assumptions:

$-$ The distribution of the length of time it takes the cashiers to carry out a service is exponential, with mean $2$ minutes and standard deviation of $1\mathrm{.}25$ minutes;

$-$ There is no limit to the queue length during business hours as the bank has a large floor area;

$-$ Customers arrive at an average rate of $25$ per hour

$-$ Service is done on a first come first serve basis.

Model $2$

Suppose the manager uses empirical data collected from past observations, from which it is estimated that the bank serves $150$ customers per day, and existing arrival and service times are as given in the tables below:

Table $1$: Inter$-$arrival times

Table $2$: Service times

Model $3$

Suppose management needs to adjust the service strategies in order to comply to the health restrictions imposed for the use of public space during a disease outbreak. A typical requirement that banks need to implement is to have a limited number of customers within the service area and provide limited sitting outside the service area.

Your Assignment

As a consultant you want to model the different scenarios so that you can make recommendations to management, stating the criteria used in your modelling and subsequent findings for your recommendations.

You are first required to describe the problem using the terminology of continuous time stochastic processes used in the study of this module, stating the processes and their distributions underlying the modelling.

To meet management expectation, you decide to formulate specific questions to base your decision analysis. The questions address various aspects of the queue operation, in terms of certain metrics common in the given models, that you then compare to come up with your recommendations in line with management goals.

Your output should consist of a report stating the approach you have undertaken for the task at hand. The models used should be included in your report. Your recommendations should be accompanied by relevant explanations of the questions you sought to address and$/$or justification of these recommendations based on your findings.