IE 1082: Practice Problems
Spring 2022
运筹学考试代做 1.Consider a single machine where jobs arrive according to a Poisson process with a rate of 10 jobs per hour. The processing times are exponentially
1.
Consider a single machine where jobs arrive according to a Poisson process with a rate of 10 jobs per hour. The processing times are exponentially distributed, but it costs more to run the machine faster. More specifically, the processing cost is $16µ2per hour when the machine is running at rate µ (and zero when it is idle). The waiting cost is $20 per job per hour in the system.
(a) What type of queueing system is this?
(b) Calculate the average number of jobs in the system if the processing rate is set at 10.8 jobs per hour. (12.5 jobs)
(c) Calculate the average time in system if the processing rate is set at 10.8 jobs per hour. (1.25 hours)
(d) Calculate the long run average cost rate if the processing rate is set at 10.8 jobs per hour. ($1978/hr)
(e) Determine the processing speed µ that minimizes the long run average cost per hour. (11.12/hr)
2. 运筹学考试代做
A computer consists of three processors. Their main task is to execute jobs from users, which arrive according to a Poisson process with rate 15 jobs per minute and have execution times that are exponentially distributed with mean 10 seconds. When a processor completes a user’s job and there are no other users’ jobs waiting to be executed, the processor starts to execute maintenance jobs. Maintenance jobs are always available and take an exponentially distributed amount of time with mean 5 seconds, however, as soon as a job from a user arrives, the processor interrupts the execution of the maintenance job and starts to execute the new job; the execution of the maintenance job will be resumed later at the point where it was interrupted.
(a) What type of queueing system is this for the jobs from users?
(b) What is the mean number of processors busy with executing jobs from users? (2.5 processors)
(c) What is the probability that a job from a user has to wait? (0.2978)
(d) Determine the mean waiting time of a job from a user. (3.51 min)
(e) What is the mean number of processors busy with executing maintenance jobs? (0.5 processors)
(f) On average, how many maintenance jobs are completed per minute? (6/min)
3. 运筹学考试代做
In a robotic dairy barn, cows are automatically milked by a robot. When a cow is in the robot, the robot first detects whether the cow has to be milked. If so, then the cow will be milked; otherwise, the cow can leave the robot and walk to the feeder. A visit to the robot with milking takes an exponential time with a mean of 6 minutes; otherwise, the visit takes an exponential time with a mean of 3 minutes. Cows arrive at the robot according to a Poisson process with a rate of 10 cows per hour, and 25% of cows need to be milked.
(a) What type of queueing system is this?
(b) What is the average service time? (3.75 min)
(c) What is the variance of the service time? (17.4375 min2 )
(d) Determine the average number of cows waiting for the robot. (1.167 cows)
4. 运筹学考试代做
Consider a four server system with exponential interarrival and service times; no room for a queue; and a controller that adjusts the arrival rate depending on how busy the system is. More specifically, when i (i = 0, 1, 2, 3) servers are busy, the controller sets the arrival rate to 15 − i. Moreover, the servers distract each other when more than one of them are busy, which causes their service rate to slow down. More specifically, when i (i = 1, 2, 3, 4) servers are busy, each busy server’s rate is 12/i.
(a) Model this process as a birth-death process by drawing the transition diagram.
(b) Calculate the long run fraction of time that the system is empty. (0.1456)
(c) Calculate the long run fraction of time that the system is full. (0.23)
(d) Calculate the long run average rate at which customers enter the system. (10.25/hr)
(e) If dividers are installed to eliminate distractions so that each server always works at rate 12, how many more customers can be served per unit time? (3.36/hr)
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