Tandem queue with rework: John Mirage of New York has a regional service center in RTP that operates 8 hours each day. Suppose faulty parts arrive to the service center at a rate of 20/day. The diagnosis desk is equipped with ny technicians. The diagnostic tests require the full attendance of one technician and takes an average of 30 minutes. Since testing is prone to measurement error, about one in every four items (irrespective of how many rediagnoses they had) are put to the back of the queue to be re-diagnosed. On the other hand, about one in every ten diagnosed parts are found to be irreparable (irrespective of how many rediagnoses they had) and a returns/scrap/warranty process is opened. Others are sent to the repair center. The repair center has n2 repair stations, where each station is capable of repairing one part at a time. Repairing requires about 2 hours. A quick input analysis shows that the exponential distribution is a reasonable fit for the diagnosis and repair times: they vary widely due to an intractable number of factors. re-diagnosis Diagnosis Repair faulty parts repaired parts scrap (a) What is the minimum number of technicians and repair stations to ensure that John Mirage can keep up with its demand for repair services? (b) Suppose the center chooses to run with n1 = 2 technicians and n2 = 6 repair stations. Approximate the average time spent in the diagnosis center waiting for a technician assuming that all external arrival streams to each unit of the service center are Poisson processes, all routings are independent, and all service times in the service center are exponentially distributed. (C) Suppose the center chooses to run with n1 = 2 technicians and n2 = 6 repair stations. Compute the probability that the center will simultaneously have at least one part in diagnosis and at least one part in repair. Assume that all external arrival streams to each unit of the service center are Poisson processes, all routings are independent, and all service times in the service center are exponentially distributed. Tandem queue with rework: John Mirage of New York has a regional service center in RTP that operates 8 hours each day. Suppose faulty parts arrive to the service center at a rate of 20/day. The diagnosis desk is equipped with ny technicians. The diagnostic tests require the full attendance of one technician and takes an average of 30 minutes. Since testing is prone to measurement error, about one in every four items (irrespective of how many rediagnoses they had) are put to the back of the queue to be re-diagnosed. On the other hand, about one in every ten diagnosed parts are found to be irreparable (irrespective of how many rediagnoses they had) and a returns/scrap/warranty process is opened. Others are sent to the repair center. The repair center has n2 repair stations, where each station is capable of repairing one part at a time. Repairing requires about 2 hours. A quick input analysis shows that the exponential distribution is a reasonable fit for the diagnosis and repair times: they vary widely due to an intractable number of factors. re-diagnosis Diagnosis Repair faulty parts repaired parts scrap (a) What is the minimum number of technicians and repair stations to ensure that John Mirage can keep up with its demand for repair services? (b) Suppose the center chooses to run with n1 = 2 technicians and n2 = 6 repair stations. Approximate the average time spent in the diagnosis center waiting for a technician assuming that all external arrival streams to each unit of the service center are Poisson processes, all routings are independent, and all service times in the service center are exponentially distributed. (C) Suppose the center chooses to run with n1 = 2 technicians and n2 = 6 repair stations. Compute the probability that the center will simultaneously have at least one part in diagnosis and at least one part in repair. Assume that all external arrival streams to each unit of the service center are Poisson processes, all routings are independent, and all service times in the service center are exponentially distributed
