Queuing for Auckland Food Service Automation: Contextual photographs

I have set my operations management students a deceptively simple queuing problem. The images here establish some context for the problem. As you will see, the number of kiosks that vendors choose to locate temporarily at The Cloud depends on their expectation of crowd numbers. Crowd numbers vary considerably from day to day, event to event.

The problem

Auckland Food Service Automation wishes to establish how many automated food service machines should be installed at The Cloud, an events facility in downtown Auckland.

The client requires that:

no customer waits longer than 150 seconds and

no more than 4 people in the queue at any time.

People are expected to arrive at the food service area at rate of 40 people per hour. The machines process requests at 25 per hour.

Given the above data determine whether the client’s expectation can be met with no more than three machines.

With the number of machines you establish are required to meet the client’s expectation, determine:

the average time a person spends in the system

the average time a person spends in the queue

the probability that there is no person in the system.

the average number of people in the system at any time.

Sensitivity analysis:

For each of the preceding measurements, what is the impact if the rate of people arriving per hour increases to 50 people per hour?

For each of the preceding measurements, what is the impact if the machines process requests at 30 per hour?

A visit to The Cloud

References

Barlow, J. (2006). Models for Production Operations: Queueing Models. In Excel Models for Business and Operations Management (2nd ed., pp. 353–369). John Wiley & Sons.

Begin, T., & Brandwajn, A. (2012). A tool for solving Ph/M/c and Ph/M/c/N queues. In Proceedings of the 9th ACM International Conference on Quantitative Evaluation of SysTems, QEST12. Retrieved from http://perso.ens-lyon.fr/thomas.begin/Publis/QEST12.pdf