Calculate the utilization the cpu proportion time cpu busy

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COMP SCI/SFWR ENG 4/6E03 - Sample Questions for Test 2

(b) Jobs arrive to a system at average rate 15 per hour. On observing the system, the average number occupying it is 10 jobs. What is the mean response time for a job? (c) Name the model: Arrivals follow a Poisson process and form a single queue. There are 3 identical processors, each with exponentially distributed processing times. Processing is FCFS and any jobs arriving when there are more than 7 jobs in the system are rejected.

3. Jobs arrive at a system according to a Poisson process with rate 10 per hour. The service requirement of each job has an exponential distribution with mean 5 minutes. For use of the system, each arriving job is charged one dollar, with a 20 cent rebate if the job has to wait before entering service (the jobs wait in one queue). A server costs 1 dollar per hour to operate (this is a fixed cost, and does not depend on the server utilization). To maximize expected revenue for the system, should one or two servers be used?

5. A call centre is using the following staffing rule: the average time in queue (before service commences) of a customer must be less than 5 minutes. The minimum staff level should be kept. The time spent with a customer (service time) is exponentially distributed with mean 4 minutes and interarrival times are exponentially distributed.

(a) For what range of arrival rates can one staff person be used?

(b) Now assume that µ = 5. The server gets a bonus of $1 per job that does not have to wait in the queue before entering service. Calculate the expected bonus per hour that the server makes.

7. Arrivals to a single server follow a Poisson process with rate 30 per hour. Service times are exponentially distributed with mean 60/35 minutes.

(b) Is the value calculated in (a) an increasing or decreasing function of T? Explain your answer.

9. An existing system consists of a single server. Requests arrive according to a Poisson process with rate three per hour. The processing times are exponentially distributed with mean 18 minutes. You are asked whether it would be beneficial to add a second (identical) processor. The two processors would share a common queue. The costs are as follows: operating costs of 25 dollars per hour per processor and cost of delay of five dollars per hour per request.

the processing times being exponentially distributed with mean 40 seconds. If there are three jobs in the system, two servers process the jobs, each server having processing times exponentially distributed with mean 40 seconds. Finally, an arriving job that sees three jobs in the system is rejected. In steady-state, calculate the expected number of jobs in the system.

13. Arrivals occur to a system according to a Poisson process with rate one per minute. The blocking probability for the system is measured to be 0.1. The expected number of jobs in the system is 2.5. What is the expected response time?

(ii) If the queue only has low priority jobs, an arriving high priority job will instantly displace the low priority job at the processor and take its place.

(a) Let the arrival rate be 5 jobs per millisecond for each class and the processing rate be 20 jobs (of either class) per millisecond. Calculate the average number of high priority jobs in the system and the average number of jobs of both classes in the system. Use these two answers to deduce the average number of low priority jobs in the system. (Hint: Think carefully how high priority jobs are processed.)
(b) Try to repeat as much of (a) as possible, if the processing times (in milliseconds) of high priority jobs are uniformly distributed between 0.03 and 0.07. The low priority jobs remain exponentially distributed. If it is not possible to calculate any of the quantities in question, indicate why.

18. A designer is trying to decide how many processors to have in a system. Each processor works at a rate of 100 jobs per hour and the system is such that a processor upon becoming idle chooses the next waiting job. The system is being designed for an arrival rate of 150 jobs per hour. Assume all underlying distributions (interarrival and service times) are exponential. If the design criterion is that the probability that an arriving job has to wait is no larger than .5, what is the smallest number of processors that can be chosen?

19. Consider two M/M/1 systems with arrival rates λ1 = 0.5 per second and processing rates µ1 = 1 per second and an M/M/2 system with arrival rate λ2 = 2λ1 = 1 per second and processing rate µ2 = µ1 = 1 per second for each server. Compare the mean waiting time times for both systems. Discuss your result, giving a physical explanation for why one system is better.

22. (a) Consider an M/M/1/4 system. The observed mean number in system is 2, the mean waiting time (from arrival to departure) is 4 seconds. The arrival rate, λ (without taking the losses into account) is λ = 0.75 jobs per second. What is the loss (blocking) proba-bility?

(b) Consider an M/M/c system with arrival rate 5 per minute. Each processor has a mean processing time of 20 seconds. How many processors are required so that the system is stable (i.e. the steady-state probabilities pn exist)?

(b) If the processing times and interarrival times are both deterministic (the values are the same as the means in (a)), would your recommendation change?

24. Consider a system with four nodes. External arrivals occur to the first node according to a Poisson process with rate 9 per hour. From node one, with probability .17, jobs go to node four, with probability .47 they go to node two, otherwise they go to node three. From node three, all jobs go to node four. From node two, with probability .95 a job goes to node four, otherwise it returns to node one. There is a single processor at each node. The processing times at each node are exponentially distributed, with the following means: node one - 6 minutes, node two - 12 minutes, node three - 16 minutes, and node four - 6 minutes.

(b) You wish to improve the expected time in system for jobs that originally arrive to node 2. To do this, you are allowed to increase the processing rate of one node by ten percent. Which processor would you improve?

26. We consider a system that allows a constant level of N programs (jobs) sharing main memory. It has 2 servers, server 1 being the CPU and server 2 being the I/O. The mean service time for a job at the CPU is 0.02 seconds, the mean service time for a job at the I/O subsystem is 0.04 seconds (assume all times are exponentially distributed). After a job completes at the I/O subsystem, it always goes to the CPU. After completion of a job at the CPU, a job exits the system with probability 0.05, otherwise it goes to I/O. When a job exits a system, another immediately enters at the CPU, thus there are always exactly N jobs in the system. For N = 2, calculate the utilization of the CPU (proportion of time CPU is busy).

(a) Find the expected number of customers at node 2.

(b) If you could increase the service rate at one node, which node would be your last choice to do so?

31. Jobs arrive to node one of a network according to a Poisson process with rate two per minute. With probability 0.4, jobs exiting node one go to node two, with probability 0.5, jobs exiting node one go to node three, otherwise they exit the system. Jobs exiting node two go to node three or node one with equal probability. Jobs exiting node three go to node two or node one with equal probability. Processing times at nodes one, two

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(b) Which node is the bottleneck?

34. Consider two M/M/1 systems with arrival rates λ1 = 0.5 per second and processing rates µ1 = 1 per second and an M/M/2 system with arrival rate λ2 = 2λ1 = 1 per second and processing rate µ2 = µ1 = 1 per second for each server. Compare the mean waiting time times for both systems. Discuss your result, giving a physical explanation for why one system is better.

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(b) Using the value of γ from (a), if the bottleneck processor has its processing rate doubled, calculate the mean number of jobs in the network.

40. Short answer questions. Explain your answers for each.

(a) True or False. MVA may be used to calculate the probability that there are zero jobs at a node.

41. Parts (a) and (b) are unrelated.

(a) A single server system has arrivals occurring according to a Poisson process with rate one per minute. You have two design choices. The first has processing times that are exponentially distributed with mean two per minute, the second has processing times that are uniformly distributed between 20 seconds and 40 seconds. If the mean waiting time is the performance measure of interest, which would you choose?

(b) BCMP networks include those with state dependent routing. (c) Resource utilization can be calculated using the MVA algorithm.

43. Users are connected to a database server through a network. There are two users, each having a think time that is exponentially distributed with mean 20 seconds. A user request goes to the database server through a network, where the network processing time is exponentially distributed with mean 0.1 seconds. The database server takes a period of time that is exponentially distributed with mean five seconds to process a request. Requests returning from the database server to the users must go via the network, where the processing time is exponentially distributed with mean one second. The network is modelled as a PS node, while the database server is FCFS. Calculate the utilization of the database server.

46. You are asked to design a system for which you receive 1 dollar in revenue for every arriving job that is served without waiting before processing and 50 cents for every job that is processed but has to wait before processing. Arrivals occur according to a Poisson process with rate 54 per hour and the processing times are exponentially distributed with mean 1 minute. You are considering the following two choices:

(a) A single processor, admit all arrivals, processing in FIFO order.

distributed with the following rates:

(a) What is the system bottleneck?

(b) Calculate the throughput at node 1.

(b) If you were to improve the processing rate at exactly one node, which would it be?

52. As a first cut at dimensioning a database server, we assume that the server has arrivals occurring as a Poisson process with rate 2 per minute and that the processing times are

53. A system has room for at most two in system (i.e. arrivals finding two in the system are lost). Arrivals follow a Poisson process with rate 1 per minute. The processing times are exponentially distributed with mean 40 seconds if there is one job in the system and 20 seconds if there are two jobs in the system. Find the expected number of lost arrivals in a one hour interval.

54. True or False.

(b) Removing the constraint that there are 2 jobs circulating, what is the maximum possible throughput at node 1?

56. You are setting up a server system that has a single queue. Service is FCFS and you need to decide the number of servers. For cost reasons, you would like to choose the smallest number of servers. Arrivals follow a Poisson process with rate 20 per hour. Processing times are exponentially distributed with mean 2 minutes. There is a requirement that there is at most one job in the system 75 percent of the time. How many servers would you choose?

(b) Calculate the throughput at node 3.

59. An administrator is trying to calculate how many (identical) servers are required for a database server system. Suppose arrivals occur according to a Poisson process with rate 1 per second. The administrator estimates that the processing time of a request is exponentially distributed with mean 0.8 seconds. For the questions below, use appropriate models to justify your answers.

(a) Calculate the steady-state probability that the server is idle.

(b) Suppose that the last arrival occurred twenty seconds ago. How long would you expect to wait until the next arrival occurs?

63. The performance of a system with fault detection is evaluated as follows. When operating, faults occur at a rate of λ = 1 per hour, with fault times being exponentially distributed. With probability 0.99, a fault is detected and recovery takes an exponentially distributed period of time with mean 1 second. If the fault is not detected, recovery takes an ex-ponentially distributed period of time with mean 5 minutes. Calculate the steady-state probability that the system is operating.

64. Consider the following open network. External arrivals occur to node 1 according to a Poisson process with rate 2 per second. External arrivals occur to node 2 according to a Poisson process with rate 1 per second. After processing at node 1, jobs immediately return to node 1 with probability 0.25, otherwise they go to node 3. After processing at node 2, jobs immediately return to node 2 with probability 0.2, otherwise they go to node 3. After processing at node 3, jobs leave the system. The processing times at each node are exponentially distributed with rate µ per second, 2 per second and 5 per second, respectively. There is one processor at each node.

(a) What fraction of the arrival rate given in the question yields half the expected number in system? Explain why this fraction is more or less than one half.

(b) It is now 9:00. The last job arrived at 8:57. What is the probability that at least one job arrives between now and 9:02?

(b) If the processing times and interarrival times are both deterministic (the values are the same as the means in (a)), would your recommendation change?

68. A closed network operates as follows. It has 4 nodes, the processing rates being 5, 10, 10, 1 at the nodes, respectively. After finishing processing at node 1, jobs proceed to either node 2 or 3 with equal probability. After processing at either nodes 2 or 3, with probability 1/10 a job visits node 4, otherwise it returns to node 1. After processing at node 4, jobs return to node 1.