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.