From the balance equations given in Table 12.2: a) give the details to verify the solution given in Table 12.3 and b) give the details to verify the associated performance metrics shown in Table 12.4. Consider the following "system." Eight students are always in the weight room at the Rec Center. As soon as one student exits, another enters. Upon entering, a student goes to treadmill, then to the stationary bike, and then to the rowing machine. There is a single treadmill, stationary bike, and rowing machine. A student exercises for an average of 5 minutes on the treadmill, 8 minutes on the stationary bike, and 4 minutes on the rowing machine. A typical student makes two cycles through the three pieces of equipment before leaving. If a particular piece of equipment happens to be busy when a student wants to use it, he/she patiently waits until it becomes free. Use MVA to find the average number of students leaving the weight room per hour and also the average amount of time that each student stays in the weight room. Plot the average amount of time that each student stays in the weight room as a function of the number of students allowed in the weight room. Vary the number of students allowed in the weight room from 1 to 15. Suppose that it is desired to place a cap on the maximum utilization that any piece of equipment should be used (i.e., to allow for equipment maintenance time). If this cap is set at 80%, what is the maximum number of students that should be allowed in the room? Which piece of equipment is the system bottleneck? If a certain percentage of students were allowed to bypass this particular piece of equipment, what percentage of students should be allowed to bypass it so that it is no longer the system bottleneck? If an additional stationary bike were purchased and students were randomly assigned (i.e., equally likely) to use either bike, by how much would the average time be reduced that a student spends in the weight room?
In Section 12.4, the database server example was balanced by moving files from the slow disk to the fast disk. Another way to achieve balance would be to speed up both the slow disk and the CPU. By how much would the speed of these devices need to be improved to achieve a balanced system? How much would these upgrades improve overall system performance (e.g., throughput and response time)? In the original database server example with two customers, the files are evenly distributed over the two disks. That is, after leaving the CPU, customers are equally likely to visit either disk. Use MVA and vary the proportion of time that the fast disk is visited as opposed to the slow disk. (This models the movement of files from one disk to the other.) Find the optimal proportion of files that should to stored on the fast disk in order to maximize overall system throughput. Repeat the previous problem six times, varying the number of customers in the system to be 1, 2, 3, 5, 10, and 15. Plot the results. Provide an hypothesis of why this optimal proportion changes as a function of the number of customers in the system. Hypothesize what the optimal proportion would be if the number of customers in the system grows toward infinity. Justify your hypotheses. A Web server has one CPU and one disk and was monitored during one hour. The utilization of the CPU was measured at 30%. During this period, 10,800 HTTP requests were processed. Each request requires, on average, 3 I/Os on the server's disk. The average service time at the disk is 20 msec. What are the service demands of an HTTP request at the CPU and at the disk? Find the throughput, X_{0}(n), of the Web server for n = 0, 1, 2, and 3, where n is the number of concurrent HTTP requests in execution at the Web server. Assume that the Web server receives requests at a rate of l = 5 requests/sec. At most three HTTP requests can be in execution at any point in time. Requests that arrive and find 3 requests being processed will be placed in a processing queue, which is assumed to have an infinite size. Find the average response time of an HTTP request. This time includes the time spent by a request in the processing queue plus the time required to process the request. [Hint: use the Generalized BirthDeath theorem of Chapter 10 together with the results of the previous item.]
