New Exercise Bank Content Qu$.13-16$

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 $\frac{1}{}$ can be set at $1$ minute, or $1\mathrm{.}5$ minutes, or $2$ 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\mathrm{.}60$ for providing a mean of $1$

minute, and $$0\mathrm{.}90$ for a mean of $1\mathrm{.}5$ minutes, and $$0\mathrm{.}40$ for a mean of $2$ minutes. The machine operators arrive randomly to sharpen

their tools at a mean rate of one every two minutes. The estimated cost of an operator being away from his or her machine to the

grinder is $$0\mathrm{.}80$ per minute.

Click here for the Excel Data File

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.$)$

a$-1.$ Determine the performance for this queueing system for $1$ minute.

Note: Round your answers to $1$ decimal place. 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 $\frac{1}{}$ can be set at $1$ minute, or $1\mathrm{.}5$ minutes, or $2$ 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\mathrm{.}60$ for providing a mean of $1$

minute, and $$0\mathrm{.}90$ for a mean of $1\mathrm{.}5$ minutes, and $$0\mathrm{.}40$ for a mean of $2$ minutes. The machine operators arrive randomly to sharpen

their tools at a mean rate of one every two minutes. The estimated cost of an operator being away from his or her machine to the

grinder is $$0\mathrm{.}80$ per minute.

b$-1.$ Determine which grinder speed minimizes the expected total cost for $1$ minute.

Note: Round your answer to $2$ decimal places.

Total cost for $1$ minute

b$-2.$ Determine which grinder speed minimizes the expected total cost for $1\mathrm{.}5$ minute.Template for the M$/$M$/$s Queueing Model

Note: Round your answer to $2$ decimal places.

Total cost for $1\mathrm{.}5$ minutes