time distribution not just has an average but also a standard deviation.

We denote by the coefficient of variation of the inter-arrival time, CVa, as the

ratio between the standard deviation and the mean.

We call the idea of the coefficient of variation from the module of customer

choice and variety. The idea here is that the standard

deviation by itself is not a good measure of variability.

Is a standard deviation of ten minutes a lot or a little?

Well, it really depends on the underlying demand process.

You might have heard concepts such as exponential distribution or Poisson

distributions. For those cases, the CVa is equal to one, and the assumption is

really that at any given moment in time, the likelihood of a new patient arriving

in the next unit of time is constant. I'll talk about seasonal arrival times a

little later on in this module. For now, I'll just assume that the

likelihood of a patient coming in is random, but is reasonably constant over

time. Variability however, is not limited to the

demand process. We also have variability in what we do

internally. In other words, we have variability in our

processing time. Let me use P to define the average

processing time, Keeping in mind that the processing time

will vary from one customer to the other. Just like we used the coefficient of

variation for the inter-arrival times, it also defines a coefficient of variation

for the processing time, as a standard deviation of the processing times divided

by the average processing time. The coefficient of variation of the

processing time really measures how standardized the work is.

This varies a lot by application. In an assembly line forever, you might see

situations where this coefficient of variation of the processing time might be

close to zero. In other settings, if you think about an

operating room, you think about developing a piece of software, there might be a fair

bit of variability in the processing time. Hence, CVps can be one or bigger.

Again, this is something that you would have to measure as you go into the

analysis that we're about to do. You just collect a bunch of processing

times, put them in Excel, And just compute the standard deviation

and the mean. Equipped with all of these definitions, we are now ready to compute

the time in the queue. Notice that the time in the queue will

vary for each individual customer. So, the best we can do, is we can compute

the expected or the average time in the queue.

Now, the following formula will allow us to compute this time in the queue.

The time in the queue is given as a product of the average processing time P

for the average activity time, I use those interchangeably here,

Times the utilization divided by one minus the utilization times the coefficients of

variations added up and then divided by two.

Now, let me comment on this and go to each of these three factors step by step.

The first one is simply the average processing time.

In other words, we see that the time in the que grows linearly here with the

processing time P. Whenever in math you see a formula that

has the shape u divided by one minus u, you know that it gets ugly as you're

approaching one, then we just plug in some numbers. At an 80% utilization,