If you care about the total time in the system, we simply have to add the expected

processing time P. So that gets us from entering the

physician's practice to leaving it. Notice that by Little's Law, I can also

compute the inventory. Since I hold the flow rate constant,

That's the rate of demand, I can simply apply Little's Law.

I have the flow time, I have the flow rate, and I can compute Little's Law.

This is summarized over here. The inventory in the queue is the flow

rate through the queue times the time in the queue.

Notice that the inventory that is currently in process, meaning the number

of patients that currently see the doctor, can be computed as the utilization times

the number of servers. Finally, I can compute the total

inventory. We see inventory in the waiting room, plus

the inventory with the servers. So far, we've assumed that the demand is

really constant throughout the day. While it is variable from a perspective

uncertainty, but it's at every minute and every hour,

It's the same underlying distribution from which they enter arrival times, and the

processing times are drawn. Now, in practice, oftentimes, you see

situations when this assumption is not fulfilled.

You see situations where you have spikes in demand at certain busy hours.

In this example here, you see a call center where we have a spike of demand in

the morning hours, and another spike in the early afternoon.

It would be misleading to simply ignore this effect and just assume that the inter

arrival times are drawn from the same distribution for every hour in the day.

What you do in situations like this, This is very similar to what we did in the

Subway analysis a little while ago, You slice the data into 30 minutes or hour

long time intervals. And then, you behave as if the arrivals are constant within

each of these intervals. I do believe that it's an imperfect

approximation, but it's certainly better than ignoring it altogether.

This is quite an important consideration when you are putting together a staffing

plan. Let me illustrate this in Excel.

In this Excel spreadsheet, I've summarized the calculations of our earlier example of

the online retailing. Recall we had a situation where the

processing time was on average four minutes, the inter-arrival time was two

minutes, three employees, and so on. You notice here, the utilization is P

divided by a times m. M, for the down here, you see the argued

waiting time formula this time in excel. Now, typically, the question here, assumed

that we have a given staffing level and given parameters.

Instead of taking the staffing plan as given, you might ask the question a

different way. You might ask, how many employees would it

take to get the average waiting time to under a minute?

You can then keep on adding employees to the point where this constraint is

honored. So, this is quite an easy way to find a

staffing plan. You can do this for one time slot in, in

isolation, but also consider the situation where you have seasonal demand as

illustrated on the earlier slide. Seasonal demand simply means that the

inter-arrival times are actually changing over the course of the day.

There's 60 minutes in an hour, so if I have 30 customers arrive in an hour, I

have an inter-arrival time of two. When demand gets busier,

I have a shorter inter arrival time. So, say for a sake of argument, I have

some times in the day when there are not 30 customers arriving, but there are 50

customers arriving. The new inter-arrival time, in this case,

would simply be 60 minutes in an hour divided by 50 customers in an hour, which

means there's a customer coming in every 1.2 minutes.

Notice that this blows up our waiting time formula.

At that point, actually, Our implied utilization is bigger than one

and our formula does not apply. I have to keep on adding employees to make

the staffing feasible. If I add from three to four employees, I

have an average waiting time of 2.5 minutes.

If I have a goal, as articulated earlier on of having a response time under a

2-minute waiting time. Well, let's see if five minutes do the

job, five employees do the job. And you notice that I can just increment

my m to the point where the constraint is fulfilled.

This gives us a staffing plan if I do this for every hour in the day.

In this session, we extended the waiting time formula from the previous session to

the case of a general m, a general number of resources,

It was not pretty. I suggested if you want to, to impress a

coworker, fellow student, or somebody in the family, just go memorize this formula

and recite it at dinner. It would make for quite an impression.

But it's a powerful formula, ugly or not. It is powerful because it can let you

drive a staffing decision. We saw that often times, the demand

changes over the course of the day or over the season an effect that we refer to as

seasonality. We then followed an approach to this

somewhat similar to what we did in the, the productivity analysis of the Subway

case in Module Two. We said that we would level the demand,

Fpr example, in 30 minutes time brackets, and we would then choose an appropriate

staffing level for each of these time brackets.

This allowed us to match supply with demand and thereby, balance the

conflicting objectives from the services provider's perspectives.

The desire of obtaining a high utilization, and from the customer's

perspective of obtaining a short response time to their order.