Homework $5$

Problem $1(25$ points$)$

The time between incoming phone calls to customer service is Exponential with a mean of $14$

minutes. Each call typically takes min$=5,$ max$=20,$ and most likely $=10$ minutes, $($Triangularly

distributed$).$ Simulate the system based on the following scenarios $($run each scenario for $6$

hours, with $50$ replications$).$

a$)$ There are two operators, and the system has only a single queue. Report the histogram

and mean for the average waiting time$1$

$.$

b$)$ There are two operators, and each operator has its own queue. Phone calls will be

directed to the random queue $($policy $=$random$).$ Report the histogram and mean for

the average waiting time.

c$)$ Compare the average waiting time of parts a and b$.$ Is there any meaningful difference?

Explain.

Problem $2(25$ points$)$

This problem is the revised version of problem $1.$ Suppose the time between incoming phone

calls to customer service is Exponential with a mean of $5$ minutes. And, suppose there are $3$

operators and each operator has its own queue. Phone calls will be directed to queues

randomly. Operators $1$ and $2$ respond to calls typically between min$=5$ to max$=20$ minutes

$($Uniformly distributed$).$ The third operator is chattier, and her service takes more time, which is

Normally distributed with a mean of $15$ and a standard deviation of $3$ minutes. Run your model

for $6$ hours, with $50$ replications.

a$)$ Report a utilization plot for each operator. Discuss what you learned from the plot.

b$)$ Report a waiting time for each operator. Discuss your results.

c$)$ Find a $95\%$ confidence interval for the average waiting time for each operator.

Problem $3(25$ points$)$

Consider a manufacturing system comprising two different machines and two operators.

$$ Each operator is assigned to run a single machine.

$$ Parts arrive with a Truncated$-$Normally distributed interarrival time with a min of $0\mathrm{.}5$

minutes, mean of $4$ minutes, and stdev of $2$ minutes. The arriving parts are one of two

types.

$$ Sixty percent of the arriving parts are Type $1$ and are processed on Machine $1.$ These

parts require the assigned operator for a $1-$minute setup operation.

$$ The remaining $40\%$ of the parts are Type $2$ and are processed on Machine $2.$ These parts

require the assigned operator for a $1\mathrm{.}5-$minute setup operation.

$1$ You might have NAs in your waiting time, because some people start the service but do not finish it before the

end of simulation clock. Therefore, in R$,$ you can use mean$($waitTime$,$ na$.$rm$=$T$).$

$$ The service times $($excluding the setup time$)$ are Normally distributed with a mean of $4\mathrm{.}5$

minutes and a standard deviation of $1$ minute for Type $1$ parts and a mean of $7\mathrm{.}5$

minutes and a standard deviation of $1\mathrm{.}5$ minutes for Type $2$ parts.

Run your model for $2,000$ minutes, with $50$ replications.

a$)$ Plot the utilization, usage, flow$\_$time, and waiting$\_$time. Discuss your findings.

b$)$ Report the average total time spent in the system $($flow time$)$ for each type of part.

Problem $4(25$ points$)$

Consider an Urgent Care $($UC$)$ comprising three doctors and one nurse to serve patients during

the day.

$$ On a typical day, the interarrival time is Exponentially distributed with a mean of $6$

minutes.

$25\%$ of patients are high$-$priority, and the remaining are low$-$priority.

$$ Upon arrival at UC$,$ the patients are triaged by a nurse into one of the two types of

patients. The service time for triage is distributed by Triangular distribution with min $=$

$3,$ max $=10,$ and the most likely value $=5$ minutes.

$$ Then, the patients wait in the waiting room and get called to visit doctors on a firstcome$-$first$-$served basis. If more than $10$ people are waiting for service, an arriving

patient will exit before being triaged.

$$ Finally, low$-$priority patients may depart if they have to wait longer than$20\backslash$pm $5$ minutes

$($Uniformly distributed$)$ after triage.

$$ The doctor service time distributions are given as follows.

Priority Service Time Distribution $($in Minutes$)$

High Normal$($mean $=40,$ sd $=5)$

Low Gamma$($shape $=15,$ rate $=1)$

Assuming that the UC opens at $8$ hours, simulate the process for $50$ replications. The UC would

like to estimate the following:

$($a$)$ the average flow time of each type of patient.

$($b$)$ the probability that low$-$priority patients balk