Consider the following two process layouts.

In the first example, option number one, I have two waiting lines for two doctors or

for two servers. People that come to this server have to be

served by this server. And people who come to the other server

have to be served by the other server. Contrast this with the example of a pooled

service. In the pooled service, no matter from

where you come in, you're going to get served by whoever is available next.

Which one will have the shorter waiting times.

Most of us I think have an intuition that this process here will have a better

waiting time. I would argue the intuition behind it,

goes something like follows. In the example up here, you might end up

in a situation where we have a lot of people waiting in the upper case, and you

have a doctor idle here in the lower case. As a result of that, that cannot be

efficient, and that's what this pooling addresses in this system down here.

Now, let's understand how this relates to our waiting time formula.

First and foremost, let's see how it influences utilization.

Recall that utilization is P divided by a times m.

You know, example here, we have, again, customers come in every four minutes, and

the inter-arrival time is five minutes, with m equals one, we simply get four

divided five equals to 80%. Now, how does it look down here if I pool

two such systems? I have no impact on the processing time,

however, I have now a shorter inter-arrival time.

I here had twelve customers arrive per hour.

If I pool, I have to have 24 customers per hour showing up.

So, pooling at least two similar systems, I move from twelve to 24 customers per

hour. Meaning an enter arrival time from five

minutes to 2.5. And notice often times students get

confused on this point. A shorter enter arrival time means more

demand. Finally, I have an m2, equals two, m2,

equaks two, p4, equals four, a2.5. Equaks 2.5, if I plot this into the

formula, p divided by a times m. I'm going to get a utilization of four

divided by two times 2.5 and voila, it's the same 80% as I had before.

So, pulling in it by itself, actually does not change the utilization but it will

impact the waiting time. Let's try this out on our Excel

spreadsheet. I'm using the same Excel spreadsheet that

I used in the last session, where you have P, as the processing time, inter-arrival

time, number of servers, CVas CVps, the utilization as defined by P divided by a

times m, and finally, the actual waiting time.

Alright, now let's take a look. Processing time stays unchanged.

Inter-arrival times, if I basically pool two similar servers with the same demand

wait together, I cut the inter-arrival time in half.

However, the server capacity doubles because I'm moving from m equals one to m

equals two. And everything else is a matter of

calculation. Notice as we saw on my calculations on the

previous slides, the utilization does not change, yet the time in the queue does.

The average time in the queue goes down from sixteen minutes to 7.2 minutes.

This is the power of pooling applied in queing.

Let me point out though that when you pool, you don't necessarily have to assume

that the inter-arrival times are going to be exactly identical, and so that the new

inter-arrival times are simply cut in half, or generally, the way that you're

going to figure out new enter arrival times is by just adding up the demand

rates and then dividing 60 minutes by the new demand rate in customers per hour and

that gives you the new enter arrival time. Early on this session, I discussed how the

utilization drives up the waiting time. We notice that as the utilization

approaches 100%,, the waiting time really goes through the roof.

I've taken the waiting time formula and I plug it in Excel and then I've entered

various level of utilization. So, this is the more exact graph that I

should have showed you early on the previous slide.

Now, what you notice here is that, as you're increasing the utilization,

Indeed, the waiting time goes up very steeply.

However, you notice that while for an m equals one already an 80% utilization is

quite a steep increase for every additional unit of utilization added.

At an m equals ten you can safely keep on adding.

Additional units of utilization. So, this is really the effect of pooling

at work. Pooled systems are much more able to be

loaded to very high volume without really experiencing a steep increase in the

waiting time. One nice way of illustrating this is

thinking about the tradeoff here when designing the service.

We have to balance the forces of responsiveness and efficiency.

Now, responsiveness means the service is responsive, as the time in the queue is

short. High time in the queue,

We would say the service is not responsive.

Efficiency we can proxy by utilization. High utilization is typically a more

efficient process than a low utilization. As we have seen previously by changing the

staffing plan by any more workers, I can move from to the lower right to the upper

left. So, a change in staffing extra m, number

of workers, will increase the responsiveness, but it comes at the

expense of the efficiency. Now, notice that when we pool the system

however, we keep the utilization constant and we're moving to a new frontier.