Once again, we will focus on process flow diagrams that are very simple and just

include one resource, potentially those with multiple servers.

Now, on the very left of the slide, we see the situation that we've analyze so far in

this module. We have a resource and we have a waiting

line. Now, let's move from the very left to the

right. You can imagine that customers are

impatient and after a while, get tired of waiting and they abandon the waiting line.

Examples of this would be an emergency room where patients after a couple of

hours waiting might just quit or a call center where customers hang up if you have

had them wait too long. Further on the right, we can imagine that

some customers might not even enter the system when the line is very busy.

Think about a drive thru restaurant. Imagine, you're pulling in at a

McDonald's, where there are three parking lots at a busy road.

If all these three parking lots are full, well, pulling in will not work and you

have to keep on driving on. So, the customer gets lost from the

perspective of the server, if there's not enough space to store them here in

inventory. In the very extreme case, waiting time is

just not an option. If you cannot serve the customer

immediately, the customer will be lost. Notice all these cases are related, as we

move from, for example, to the very left to the right here.

We are just adding some impatience of the customer as we move from here to here.

We have basically just said that the size of this inventory equals to zero.

We have focused the previous session on this very left case.

We will focus this session on the right case.

Now, there are mathematical models to also analyze the cases in between, but they're

a little bit more tedious than what I would like to do in this first course on

operations management. Consider yet another health care example.

We previously talked about emergency rooms.

Big inventories in front of a resource. Next, consider the case of trauma care.

Patients that have to be moved to trauma care, they're so acutely sick that waiting

is not an option. What happens if all the trauma bays and

all the trauma capacity in a hospital is busy?

The hospital goes in what is called diversion.

It tells the regional fire and rescue to stop sending ambulances or helicopters.

Simply put, the demand rate is shut down. Now, let's analyze the situation.

As before, we have a random demand process.

Say, for the sake of argument, a patient comes in every three hours.

We will enforce in this session here that the CVa is equals to one, in that we're

dealing with exponential inter-arrival times.

On the service process, we find that in this trauma center, patients stay on

average in a trauma bay for two hours. This is nothing else but our good old

friend, the processing time. Again, it is naive to believe that Dr.

Toyota like, the patients would stay exactly two hours in the hospital buthe

can have many distribution. So, it turns out that the standard

deviation of this distribution will not matter for our analysis.

The magic number that we want to compute is a probability with which an incoming

customer will not be served. This is an important number.

Once I know this probability, I can multiply this with the demand rate.

And I can compute the number of customers that get lost, as well as the number of

customers that get served. So, how do we find that probablity?

It turns out that there's a big mathematical formula that computes a

probability as a function of the numbers of servers, the processing time, and the

inter-arrival time. I will show you that formula just in a

moment, but you will thank me one day that you will not have to use it.

Here's how I want you to find out the probability.

The first thing I want you to do is compute the ratio between the processing

time and the inter-arrival time. I would prefer that you don't try to

interpret that number just call it r for ratio.

Next, you look at how many resources do you have or how many services do you have

in that resource. And so, in this case we have three trauma

bays, m3. = three.

And then, the last thing that you need to do is you need to go into a big table that

I will provide you. And in this table, you're going to go in

to row r and you're going to go into column m.

And where row and column meet, you will have the diversion probability.

In this case, 2.55%.. Again, once you have this number, you are

in good shape. If you want to figure what percentage of

the time and the date the hospital is on diversion, just multiply 24 hours with

2.55%.. If you want to figure out how many

patients that you're going to lose because of diversion, remember that we have a

patient arriving every a units of time, Which is one every three hours so eight

patients per day. And if you multiply this with 2.55%,

you'll figure out how many patients we lose.

Similarly, if you're going to take one minus that probability, we can figure out

how many patients that we have served. Again, figuring out this probability is at

the heart of dealing with the subs of loss problems.