Problem $2$: Spreadsheet Simulation $$ Variant of Single Server Queue Problem

For this problem, you should modify the single server queue spreadsheet simulation from Section $2\mathrm{.}7\mathrm{.}2$ of the textbook in a number of ways, and then compute some performance measures of interest.

Change the spreadsheet options so that observed values of the inter$-$arrival and service times are resampled only when you hit the F$9$ key.

The inter$-$arrival times should be sampled from an exponential distribution with a mean of $2\mathrm{.}4$ minutes.

The service times should be sampled from a probability density function that has the form:

f$\_$X $($x$)=\{($Cx$^3,$ if $0<=$x$<=2\mathrm{.}5$ min@$0,$ if x$<0$ or x$>2\mathrm{.}5$ min$)$

To sample from this distribution, you should use the method of inversion. First find the value of the normalizing constant C that makes a proper density function; and then compute and invert the corresponding cumulative distribution function:

Incorporate your scheme for sampling from this service$-$time distribution into the revised spreadsheet simulation.

Compute a $95\%$ confidence interval $($CI$)$ on the average time in the queue for the first $150$ customers based on $10$ replications of the simulation. There are places to enter the mean, standard deviation, t$-$value, CI center, CI half$-$length, CI lower limit$,$ and CI upper limit$.$ Compute the steady$-$state time in the queue using the Pollaczek$-$Khinchin formula. This will require you first to compute the expected value service time X$,$

the variance of the service time X$,$

and the coefficient of variation of the service time X$,$

Recall that the formula for the expected value of the steady$-$state average waiting time in the single$-$server queue with a constant customer arrival rate is:

This is the Pollaczek$-$Khinchin formula as given in the book and the Coursepack.

Use the file Homework$4\_$Template.xlsx $$ Problem $2$ tab. You should fill in the highlighted cells on the left side. You must fill in answer and the appropriate units for each of the following:

Mean interarrival time $(1/\backslash$lambda $)$

Minimum Service Time $($a$)$

Maximum Service Time $($b$)$

Mean Service Time $($X$)$

Variance of Service Time $()$

Standard Deviation of Service Time $(\backslash$sigma X$)$

Server Utilization $($Traffic Intensity$)$ Factor $(\backslash$rho $)$

Coefficient of Variation $($CV$($X$))(=\backslash$sigma X $/$X$)$

Steady$-$State Expected Mean Waiting Time in the Queue

Answer the following questions clearly.

Does your $95\%$ CI always cover the steady$-$state waiting time in the queue?

Do you think your estimate of the average waiting time in the queue for the first $150$ customers and $10$ replications is a good estimate of long$-$run average waiting time in the queue?

What could we change in order to ensure the $95\%$ CI covers the steady$-$state value almost every time?

You should use formulas for the answers in your spreadsheet so that even when you hit F$9,$ all of the cells are filled in appropriately. For the question of whether or not the $95\%$ CI covers the steady$-$state value, you can use the Excel IF function to display $$yes$$ and $$no$$ under the appropriate conditions.

For this problem, in addition to the Excel file, you should turn in a word or PDF file with your computations of the normalizing constant C$,$ the CDF the inverse CDF $,$ the expected value and variance of service time, coefficient of variation and steady state waiting time in the queue.