In the previous session we saw the waiting time can occur even if the recourse utilization is below 100 percent. This is driven by the variability in the process floor. Now unfortunately, the type of tools that we've introduced so far in this course can really not explain that waiting time. We so far naively believe that if the utilization is less than 100%, there wouldn't be any waiting. The reason for this is our previous analysis was always built on averages. We have simply ignored the concept of variability so far. The purpose of this session is to help us describe the variability in this process, and start putting together a set of tools that help us predict the amount of waiting, even if the utilization is less than 100%. Now we first need to describe the variability. We need some form of measuring it. We see with the example of the doctors office, this is pretty representative for most service operations. I have some waiting area, and I have a resource. Notice that in this section now, we're going to look at the case where the flow rate is constrained by demand. If we have more demand than capacity, our implied utilization is more than 100% and we don't need fancy variability models to tell us that there's waiting. This was my first doctor's office in the previous session. We further assume for the remainder of the session that for everybody who arrives to the practice will wait in line until they're served, and leave the patients after having seen the doctor. Okay, now let's first describe variability and demand. We notice that the demand process was somewhat random. When we said random, what we meant, is if were customers were not lined up, Toyota Camrys at the end of an assembly line, they came when they wanted. So the formal way of capturing this, is we define the arrival times as a time when patients arrive to the practice, and then we define the inter-arrival time as the time between two subsequent arrivals. When we say that the arrivals are random, what we really mean is that these inter-arrival times are drawn from some underlying statistical distribution. If we were to note with A, the average inter-arrival time. Like any distribution the inter-arrival time distribution not just has an average, but also a standard deviation. We denote by the coefficient of variation of the inter-arrival times, CVA, as a ratio between the standard deviation and the mean. We call the idea of the coefficient of variation from the model of customer choice and variety. The idea here is that the standard deviation, by itself, is not a good measure of variability. Is the 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 personal distributions. So 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 talk about seasonal arrival times a little later on in this module. For now, just assume that the likelihood of a patient coming in is random, but it's 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 the 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. And in the assembly line forever, you might see a situations where this coefficient of variation of 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. Then CVPs can be one or bigger. Again, this is something that you would have to measure as you go into the analysis that we are 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, or the average activity time, I use this 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 end goal 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 queue grows linearly here with the processing time, p. Whenever in math you see a formula that has the shape u divided by 1 minus u, you know that it gets ugly as you're approaching one. Let me just plug in some numbers. At an 80% utilization, this ratio is 0.8 divided by 0.2, which is simply 4. At a 90% utilization, the same ratio is 0.9 divided by 0.1, which is already 9. You notice that a simple 10% increase in utilization can more than double the waiting time. You see this on the graph here on the right that the utilization grows very steeply as we approach 100% utilization. This is practically very important. The reason for that is that managers in service operations are incurring very big fixed costs. So it is in their interest to squeeze more and more customers through the process. However, you notice that those last customers, which from a profitability perspective are really interesting of course, because their revenue goes right into the bottom line after all of the fixed costs are already paid for. They look very profitable, but they create havoc to the system. Again, as these last customers get added to the process, our waiting times get up through the roof. Finally, look at the variability here. I'm squaring the coefficient of variation for both the arrival process, as well as for the processing times. The more variability there is in the process, the longer is the time in the queue. If you look at a case where you have a coefficient of variation of a exponential distribution, you notice that these two fellows here are going to be equal to one, and this whole loss factor will degenerate to one. Let's practice our new equation with a quick example. Again, let's look at the doctor's office. Let's assume that patients come every 30 minutes with a standard deviation of 30 minutes, and then consultations last 15 minutes with a standard deviation of 15 minutes. The first thing that I'd encourage you to do for these types of problems is simply write down the waiting time formula. The time of the queue is p divided by u divided by 1- u, times the coefficients of variations added up, after they got squared individually. Now which one of these ingredients here in the formula is easy? Well, we see that the processing time is simply 15 minutes. What else? The coefficient of variation of the inter-arrival time is 30 minutes of the standard deviation divided by 30 minute average. So this means it's one, which gets squared, but that still stays one. Same for the processing times. The standard deviation here is 15 minutes, the average is 15 minutes, so I have another one squared, and I divide those two by two. So this old fellow here at the end is simply going to be equal to one. Now utilization is not immediately visible in this question, and so, we have to remind ourselves that utilization is the flow rate divided by the capacity. Flow rate, in this practice is one patient every 30 minutes and constrained by demand. I divide this by capacity which is one divided by 15, there'd be utilization of 50%. In other words then, if I plug this in, this middle factor here is simply 50% divided by one minus 50%, which is equals to one. So, the total wait time, thus, is 15 x 1 x 1 = 15 minutes. Now, I want to be a little hair-splitting here, because the question actually doesn't ask for the waiting time. But asks for the time when our friend, Newt, can walk out of the practice again. That includes the time in the queue, which we said was 15 minutes, but also the time in the practice when he is in service. And so this is the time of the queue plus the processing time all together, 30 minutes. And so at 10:30, he can expect to be out of the office again. We can predict the average time in the queue based on the processing time, the utilization, and the amount of variability in this system. We saw on the waiting time formula that as the utilization goes up and it approaches 100%, the utilization would drive the waiting time through the roof. So once a system becomes more congested in a 90, 95% utilization, it becomes very sensitive to every additional customer walking in. Please only use this waiting time formula for situations where the utilization is less than 100%, where we have more capacity than we have demand. If demand exceeds capacity, first of all, remember we speak of an implied utilization. Utilization by itself can, at the maximum, be 100%, but if the amount exceeds capacity, we really don't have a variability-induced waiting time. We just have the problem of a doctor who is taking six patients in, and can only serve three. To predict that this waiting room will fill up over time doesn't require some fancy math. It's just a simple matter of understanding that we are serving three fewer customers than we have demand. Also notice that once we have computed the time in the queue, we can compute the total flow time, the time of the system, by simply adding the time of the queue plus the processing time, p. We can also compute inventory using little thought. So once you have computed the time in the queue, every other measure that we talked about in this class can be computed. That is why this waiting time formula is so important.