So, welcome to the review session for Module four. It's the same format as in the past. We going to work through a couple of questions. This time it's going to be five questions. And we'll do this in the format that I'll present the question to you, You pause the video, And then I'll present the answer to you once you're ready to check your results. Okay. So, the first question is about this firm my-law.com. And the way my-law works is basically they have a lawyer, and he's answering e-mails. and e-mails come in from eight to six:00 p.m. And there are ten e-mails arriving per hour. There's only one lawyer on call and it takes the lawyer five minutes to write a response e-mail with a standard deviation of four minutes. Alright, a couple of questions are coming up. Pause me now and take a shot at these questions yourself. Alright. This is obviously a waiting time problem. The magic word here, as you see, wait. How long does the customer have to wait? And we want to be careful with this question, because it is asking you to consider not just the wait questions, the time, and the queue. But you have to also, if you want to wait for the e-mail, the customer has to wait for the agent or the lawyer to get to the e-mail and then the customer also has to wait for the five minutes that it takes to respond to the e-mail. So, we're also looking for the processing time. Now, the hardest thing is really coming up with the wait time. And so just remember, time in the q is P times U divided by one minus u times CVa square plus CVp square divided by two. Let me emphasize it's one lawyer so m equals one. For bigger m, as we will see in a moment, for, on the next question, for bigger m, we need the other, the bigger formula. So, this is a simple case for m equals one. Alright, ten e-mails per hour that means there's an e-mail coming in every six minutes so the inter-arrival time is six. The processing time, we notice, P is five minutes and so if you look, then it's the utilization. Utilization is flow rate divided by capacity. Flow rate as e-mails flow in the system every six minutes. One over five is our capacity, and so the utilization is really five over six. That means here I have P equals to five times U five over six divided by one minus five over six times, okay, CVa square coefficient variation of the inter-arrival time is one and so that is simply a one, One squared stays a one, Plus CVp squared Well, so standard deviation here is four minutes and the mean is five minutes. And so, we are dealing here with 0.8, four divided by five, 0.8 and we're going to square this. And then we're going to divide both of these by two, Alright? So, that is basically giving you here, we have one minus five-sixths. That is one over six, five over six by one over six. That is going to be five, Five times five is 25. And the other ratio here, This is 0.64 plus one is 1.64, divided by two is 0.82. And that is giving you 20.5 minutes. Again, this is just the time in the, the queue. If you want to add up to that, The actual writing time of the e-mail then you have to add another five minutes and that would give you then a total of 25.5 minutes. That's the answer to the question. Now, B and C are actually, I don't know what I did when I wrote these questions, but these are really friendly question in many ways. Just over ten hour days. Remember, ten e-mails are coming in per hour, and so really we have nothing but a hundred e-mails here that are answered. And then finally, the last question asks, is the time the lawyer can use to pursue these kind of large settlement cases, and for that, we have to look at the idle time which is one minus the utilization, times the ten hours in the day we know that the idle time is one-sixth because the utilization is five-sixths and so that is simply ten over six hours per day. Okay. The next question, A small computer store here with five employees that is doing repairs for people like Jim, who has his computer broken. And so, this computer store, PC Fixers, or PF for short, is going to get customers walk in every ten minutes and they take 40 minutes to fix the computers. And there are five computer technicians to do the work. Alright. Same business as usual. I shut up for a while. You work on the problem and then press on play again so that you hear my answers. Okay. So, again, we are dealing with a waiting time problem. But this time, because of the multiple technicians, we are dealing with the more complicated waiting time formula that has Tq as a function of m, multiple servers expressed. So let's just start writing down the formula from the lecture. So we're looking at the expected wait time, P divided m times u raised to the square root 2m+1.1. Now, very careful, the square root ends after the one, minus one. The minus one is outside the square root. One minus u times CVa square plus CVp squared divided by two. Alright, so now, let's look for what's easy and what is hot. We clearly see that there's a customer walking in every ten minutes, so that's the inter-arrival time a. We have a service time P of 40 minutes. And so, we can first compute the utilization. As of flow rate, which is one over a divided by the capacity, which is m over p and so that simplifies to p divided by a times m, and that is, in this case where we have a P of 40, divided by ten times five so that is 40 by 50 or 0.8 utilization. Now, note that in this example here, CVa is equal to one, because both the mean inter-arrival time and the standard deviation is equal to ten, so that cancels out. And CVp is also equal to one, because 40 is both the mean and the standard deviation. So, this whole thing here at the end is all going to be equals to one. That simplifies the math because now we have 40 minutes divided by five servers times 0.8, the utilization is raised to a power of twelve, which is 2m+1,1. Five plus one is six, Two times six is twelve. So, square root of twelve minus one divided by 0.2 times one. And you plug that into your calculator, and at least I get a 23.08. Now, the next question asks for the length of the line or how many customers will be waiting for their computer to be fixed. This is ultimately something that we can determine with Little's Law, I equals R times T. Remember, already we know that there is a customer coming in every ten minutes. So, one over ten customers per minute is going to be our flow rate. And our flow time, well that's a 23.08 here, here in the queue plus the 40 minutes that they spent for the, waiting for the actual service time. And so, this is then going to be 63.08 minutes. And that gives you simply 6.308 customers, and that is the inventory of customers in the store. Now, Recompute is a company that offers super computers to users via the internet. And they get jobs every four minutes. Standard deviation of the inter-arrival time is also four minutes. It takes ten minutes to execute these jobs. And here's a trick that if the super computers are busy, the job actually gets rerouted to another super computer of another I mean, or a supplier of ours, or a partner of ours. In which case, we have to pay these people actually $40 per job. So, you see the question here test your self and then put me back on play. Alright, the key to this question is that this is not a waiting time question but a loss question. If you hear words in the questions, see words in the questions, such that you'll redirect or the job you know, a probability that the job can or cannot be executed. We're not dealing with waiting time questions, We're dealing with loss questions. For loss question, we have the LN loss function, and so, remember, we start by looking at the ratio of P divided by a. So, we have a processing time here that is ten minutes, and we have a inter-arrival time that is four minutes. So, that ration here is 2.5. We have m equals four. We then look at the table that I had mentioned in the lecture, but it is posted in the, in the wiki for the current week. Look at the table, and there you will find that the probability that we have a loss where is R equals 2.5 and M equals four is equals to 14.99%,. So 0.4, 1.499. So, notice a question here is asking what's that probability which, with which an incoming job can be executed, and so we have to take one minus that probability as the answer to the question. And so, the answer to this question is that it is 85 or one%.. The second part of the question looks at the and sometimes, sometimes looks at the overflow and it's really interesting to keep in mind or important to keep in mind that we have fifteen jobs coming in per hour, Right, That is just 60 minutes divided by the excuse me, by the inter-arrival time of four minutes here. So, fifteen jobs come per hour and we know from the previous question that jobs will be turned down or turned away to the partner company on computer that happens with the probability of 14.99%.. So, in that case, simply, we have to turn 2.2485 expected jobs away. And if we multiply that with the penalty of $40 per job, We're going to get a hourly payment, an expected hourly payment to our partner company of $89.94, so this is dollars per hour. Okay, next question. We have a contractor here that is doing lots of renovation and building work. And the contractor has six projects lined up. You'll see their processing times listed here. And he wants to minimize the average wait time that projects are waiting for. And so, that's how he sequences his work. And the question is what will he be doing in 30 days from now. Alright ready? You go first and then, I'll give you the solution. Alright. The key thing here that these not working first come first serve, but he is using the shortest processing time rule. The shortest processing time rule will minimize the average wait time of these projects. Now, notice that regardless in which sequence first come first serve, shortest processing time rule, or whatever else, the total amount of time it takes is simply just the sum of these various tasks. But, by doing the shortest processing time rule, he can reduce the average wait time. It simply makes more sense that we start with the short jobs so that the long job will waits for the short job, as opposed to the other way around. So, if he follows the shortest processing time rule, he will do this job first. This job second, third, Fourth, fifth, and, this one's here's going to be last. And so, now I know the sequence in which he's going to do the work, and so the, the first job is going to be over after two days. The second job he's going to touch is going to be over of the two plus six equals to eight days. The next job is going to be over of the sixteen days. Then, he's going to take the job number four that takes ten days, so that's hoing to get him to 26 days. So, after 26 days, he has the first four jobs completed, and then he's going to start the fifth job in his, in his sequence, and that is going to be the new kitchen here at, at Rosemont. And so, with that in mind, it is the new kitchen in Rosemont that answers a question on what he will be working 30 days from now. Alright. The last question is about a call center. The call center, right now, has a constant staff available all over the day, and there is some seasonality in the demand, in particular, there is lots of demand in the morning hours, and very little demand in the afternoon hours. So, utilization of the call center is high in the morning and low in the afternoon. Now, which of the following action will decrease the average waiting time in the call center? Take some time for yourself, and then I share my thoughts with you. Alright so, this is a qualitative question, there is really no number here that you could crunch into the waiting time formula. Nevertheless, I think it's helpful to remind ourselves of the waiting time formula. Tq is equals to P divided by m. Utilization raise to the 2m+1 minus one divided by one minus u times CVa squared plus CVp squared divided by two. Alright, now let's look at the suggestions that I put up here of adding more servers. So, adding more servers is a good idear, right? You notice a larger m here will decrease Tq. Now I want to be careful here in reminding you that the utilization, which we defined as flow rate divided by capacity, and as one over the inter-arrival time times m over P and then P divided by a times m, I want to just emphasize that the utilization is also a function of m and so if you're taking the sensitivity analysis derivative of Tq with respect to m, you cannot just look at this term. Also u is changing as you're adding servers, but, either way, adding servers is a good thing. It will shorten the waiting time, Decreases service coefficient of variation, That one is fairly straightforward. You see the marginal impact here. Cvp shows up just linearly in here, and everything that you do to CVp will be reducing the wait time, so this one also makes sense. Decreasing the average service time. You see this will help through P here in the formula. That same comment is on mP, is also sitting in u, and will also have an x3 effect by allowing you to operate at a lower level of utilization. And so, that one is certainly true as well. Now, the last one, option D, is a little tricky here. Leveling the demand. Why would that matter, and how do you think about that? For that place, remember that if you plot the time and the cube as a function of the utilization u, you have this very steep, nonlinear effect. And so right now, in the morning hours, we have a very high level of utilization. So, say, say, we are, we are here, high level of utilization. And in the afternoon hours, we have a low level of utilization. Now, conceptually, you see that if we could now bring this together, and level the demand, So we move some of the demand from the morning into the afternoon hours, Well, the folks who had previously a super short wait time, well, they will lose a little. They wait a little longer here, right? So, you see, there's a little increase here in the waiting time, for these guys. But the morning people with really, really benefit, right? And so, you notice that, on average, it's nice to actually have a constant level of utilization as opposed to having half of the customers get really high level of utilization, and half of the customers having a really low level of utilization. So, leveling the demand because of these convexities, of steep increase of the waiting time as a function of u will be a good thing. And thus, we can check option E, all, all of the above.
