Math Forum / Mathematics / Operations Research / July 2006

Schaum's Outline of Operations Research would be the analog to a Cliff
Notes for OR.  You can find it listed on Amazon.

For more detail, snag any introductory OR text (Hillier & Liebermann,
Taha, ...) and look for the chapter(s) on queueing theory.

You need more detail to finish your problem (distribution of
interarrival times, distribution of service times, whether the cashiers
have mustard/ketchup on their fingers when they return, ...).

/Paul

The specific Queueing models that you are looking for are called G/G/k
models.  While you will find M/M/k models in some of the introductory
text, the G/G/k model is more complicated. However, the M/M/k models do
provide great insight into the system with minimal data.  I think
Hillier & Liebermann has some of the curves you may be looking for.

If you really do need the G/G/k models, one of the seminal papers in
approximations for this case is available at
http://www.columbia.edu/~ww2040/ApproxGIGm1993.pdf.

If you would not like to look at the math but just want the numbers,
you can download software free from http://www.okstate.edu/cocim/raqs/.

Here are some of my thoughts.....

The question you have posed seems to deal with a sudden change in
operation for the queue (not a steady state).  Most queueing analysis
however assumes steady state.  So, Instead of trying to use queueing,
you can use discrete event simulation.

Typically most customer service situations are analyzed on max waiting
times (or more accurately, 95% percentile of the waiting times). A
simulation can help you build a policy table that reads
something like ..if number of people in queue is X, then you need Y
active cashiers to keep the max waiting time under 30 (or 40 or 50..)
seconds.