# Question

At the Forrester Manufacturing Company, one repair technician has been assigned the responsibility of maintaining three machines. For each machine, the probability distribution of the running time before a breakdown is exponential, with a mean of 9 hours. The repair time also has an exponential distribution, with a mean of 2 hours.

(a) Which queueing model fits this queueing system?

(b) Use this queueing model to find the probability distribution of the number of machines not running, and the mean of this distribution.

(c) Use this mean to calculate the expected time between a machine breakdown and the completion of the repair of that machine.

(d) What is the expected fraction of time that the repair technician will be busy?

(e) As a crude approximation, assume that the calling population is infinite and that machine breakdowns occur randomly at a mean rate of 3 every 9 hours. Compare the result from part (b) with that obtained by making this approximation while using (i) the M/M/s model and (ii) the finite queue variation of the M/M/s model with K = 3.

(f) Repeat part (b) when a second repair technician is made available to repair a second machine whenever more than one of these three machines require repair.

(a) Which queueing model fits this queueing system?

(b) Use this queueing model to find the probability distribution of the number of machines not running, and the mean of this distribution.

(c) Use this mean to calculate the expected time between a machine breakdown and the completion of the repair of that machine.

(d) What is the expected fraction of time that the repair technician will be busy?

(e) As a crude approximation, assume that the calling population is infinite and that machine breakdowns occur randomly at a mean rate of 3 every 9 hours. Compare the result from part (b) with that obtained by making this approximation while using (i) the M/M/s model and (ii) the finite queue variation of the M/M/s model with K = 3.

(f) Repeat part (b) when a second repair technician is made available to repair a second machine whenever more than one of these three machines require repair.

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