# Question

Jake’s Machine Shop contains a grinder for sharpening the machine cutting tools. A decision must now be made on the speed at which to set the grinder.

The grinding time required by a machine operator to sharpen the cutting tool has an exponential distribution, where the mean 1/μ can be set at 0.5 minute, 1 minute, or 1.5 minutes, depending upon the speed of the grinder. The running and maintenance costs go up rapidly with the speed of the grinder, so the estimated cost per minute is $1.60 for providing a mean of 0.5 minute, $0.40 for a mean of 1.0 minute, and $0.20 for a mean of 1.5 minutes.

The machine operators arrive randomly to sharpen their tools at a mean rate of 1 every 2 minutes. The estimated cost of an operator being away from his or her machine to the grinder is $0.80 per minute.

(a) Obtain the various measures of performance for this queueing system for each of the three alternative speeds for the grinder. (Set t = 5 minutes in the Excel template for the waiting time probabilities.)

(b) Use the cost figures to determine which grinder speed minimizes the expected total cost per minute.

The grinding time required by a machine operator to sharpen the cutting tool has an exponential distribution, where the mean 1/μ can be set at 0.5 minute, 1 minute, or 1.5 minutes, depending upon the speed of the grinder. The running and maintenance costs go up rapidly with the speed of the grinder, so the estimated cost per minute is $1.60 for providing a mean of 0.5 minute, $0.40 for a mean of 1.0 minute, and $0.20 for a mean of 1.5 minutes.

The machine operators arrive randomly to sharpen their tools at a mean rate of 1 every 2 minutes. The estimated cost of an operator being away from his or her machine to the grinder is $0.80 per minute.

(a) Obtain the various measures of performance for this queueing system for each of the three alternative speeds for the grinder. (Set t = 5 minutes in the Excel template for the waiting time probabilities.)

(b) Use the cost figures to determine which grinder speed minimizes the expected total cost per minute.

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