Only after the last tree has been cut down,
Only after the last river has been poisoned,
Only after the last fish has been caught,
Only then you will find out that money cannot be eaten.

Cree Proverb, and today’s Fresh Tracks, Inc.™ AM Fuel


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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


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

Brandwajn, A., & Begin, T. (2011). Numerical Solutions for Queueing Systems. Retrieved from

Heizer, J., & Render, B. (2014). Queuing Models (Part 4 - Module D). In Operations Management: Sustainability and Supply Chain Management [Global Edition] (11th ed., pp. 771–800). Pearson Education.

Hillier, F., Hillier, M., & Lieberman, G. (2000). Introduction to Management Science: A Modeling & Case Studies Approach. McGraw-Hill/Irwin.

Multiple Channel Queueing Model Solved Examples Help for Queueing Models, Management, Homework Help. (n.d.). Retrieved November 1, 2012, from

Perros, H. (n.d.). Queueing Theory - A Primer.

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From supply chain to value chain.
Excellent illustrated lecturer using cartoon animations by Prof Andrew Ferne, Kent Business School, UK.

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