In the last session, I introduced the concept of the waiting time formula. One assumption that we made in this formula is that we are dealing with a case where we have just one single resource doing all the work. We looked at the case where we had m equals one. In this session we will generalize this formula to the case where the resource is staffed by m equals to multiple people. Most interesting problems, I would argue, have this flavor. If you think about a physician’s office, it's often times not just one physician doing the work, but maybe three, four, or five. In a call center, you don't have just one operator picking up the phone, but you might have hundreds of seats with busy operators. So the purpose of this session is to generalize the waiting time formula to this general m. This will be a little bit more technical. I have to alert you to this. But as a byproduct or as a reward for this extra work. We'll be able to derive a staffing time. We'll be looking at a specific target weight time and then think about how many operators we'll have to put to work to actually be able to meet that level of responsiveness. All right. I warned you. This is not going to be pretty. Now let's look at the process flow diagram before we turn our eyes to the math. And this process flow diagrams I've realized now, that I have m parallel servers. Think of m for simply standing for multiple. I've multiple servers, but know further I have a common queue. So the assumption here is that the process flow diagram still consists out of this basic picture. I have a common queue for everyone and whoever's available next, is going to serve the next customer. Once again I want to find the time it takes a customer to wait in the queue. It turns out that this expected time in the queue can be written in a very similar way as we saw it before. Here we go. We have to look again at the activity time or the processing time, this time divided by m, times the utilization. And here's the really ugly part. Look at the square root here. So, the utilization is raised to the exponent of the square root of 2m plus 1 minus one, divided by one minus the utilization. The third factor is unchanged to the previous case. Now as we apply this formula, keep in mind that the utilization all throughout, remain the flow rate divided the the capacity. Remember further that in all of our discussion around variability we assume that the flow rate was constrained by demand. Again, if you have a demand that exceeds the capacity, waiting is not driven by variability, it's just driven by insufficient capacity. Now, if the flow rate is driven by demand, I simply have a customer arrive every a units of time, where a was the inter-arrival time. And I can write my capacity just like in every other module in this course is the number of resources divided by the processing time. So you can see here, that I can simplify the utilization to p divided by a times m. So when you interpret this formula, be careful in the sense that the utilization is also a function of m. So do not just look at this equation as a whole. That makes sense. M is driving down the time into queue. M is also sitting in the utilization over here. And over here. I think you be a lot more comfortable with this formula. If we crunch through another example. So let's consider the following situation. We have an online retailer that is staffing a help desk with three employees. These three employees are getting an email every two minutes on average. Standard deviation of these inter-arrival times is two minutes. It takes four minutes to write a response email with a standard deviation of two minutes. What's the waiting time for the customers? Well again as before I suggest we start writing down the formula. P divided by m times the utilization, square root two. M plus one minus one divided by one minus u. Be careful with the minus one. That's outside the square root here. The square root ends but still in the exponent. Times cva squared plus cvp squared divided by two. All right now lets look at the ingredients here to the question. We have a fairly obvious p equals to four minutes. We have three employees so m equals three. The utilization as I've said before we've got a computer utilization as flow rate divided by the capacity which we said equates to p divided by a times m. So p is four, a is two, the emails come in every two minutes, times three employees. So that is 66%. And then here is where things get ugly with the exponent. So we have here the square root of n plus one, that's three plus one is four times two is eight. Square root of eight minus one, divided by 0.333. Times the cva squared, that is one squared. Plus the cvp, that is the standard deviation of the service time, two minutes divided by the average four minutes. So that is 0.5 squared divided by two. That's all there is to it. When I plug this into my calculator I see an expected wait time in the queue of 1.19 minutes. Again as before in an example we have to debate whether what matters here is the time in the queue or the time to respond. If you care about the time to the response to the customer you would have to add two minutes here of the actual service time. On this slide I've summarized all the calculations you need to be able to do in the context of waiting times. The most important building blocks for the calculations is the utilization. Remember the utilization is the flow rate divided by the capacity, which simplifies to p divided by a times m. Once we are done with the utilization, we can use the waiting time formula to get the time of the queue. Which is the time from entering the system to the time when you going to start your service. If we care about the total time of 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 that holds the flow rate constant, that's the rate of demand, I can simply apply Little's Law. I have, it's 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 of 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 a utilization times the number of services. Finally I can compute the total inventory as the inventory in the waiting room plus the inventory with the service. So far, we have 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. With 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 way. 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. Then you behave as if the arrivals are constant within each of these intervals. Arguably they're just 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 have summarized our calculations of the earlier example of the online retailer. Recall, we had a situation where the processing time was on average four minutes, the arrival time was two minutes, three employees and so on. You notice here the utilization is p divided by eight times m. And further down here you see the ugly 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 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 are 60 minutes in an hour, and 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 the sake of argument, I have some times in the day when there are not 30 customers arriving. But, there are 50 customers arriving. When you enter 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 this staffing feasible. If I add from three to four employees I have an average waiting time of two point five minutes. If I have a goal as articulated earlier as having a response time under two minute waiting time. Well let's see if five minutes was a job, five employees to 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 section to the case of a general m, a general number of resources. It was not pretty. I suggested if you want to impress a coworker, fellow student, or somebody in the family. Just go memorize this formula and recite it at dinner. It will 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 oftentimes, the demand changes over the course of the day or over the season. An effect that referred to as seasonality. We then followed an approach that is somewhat similar to what we did in the productivity analysis of the Subway case in module two. We said we would level the demand for example in 30 minute 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/providers perspective, the desire of obtaining a high utilization, and from the customer's perspective of obtaining a short response time to their order.
