TASK $1(20\%)$ Basics of Queueing Theory

A:

Problem $2\mathrm{.}6\mathrm{.}1$

For an M$/$M$/1$ queue with a mean interarrival time of $1\mathrm{.}25$ minutes and mean service time

of $1$ minute, find all five of Wq$,$ W$,$ Lq$,$ L$,$ and $\backslash$rho $.$ For each, interpret in words. Be sure to

state all your units $($always$!),$ and the relevant time frame of operation. Hint: You might

want to calculate manually or use $$mmc$.$exe$($e$.$g$.$ mmc$\_$calculator, mmc$\_$solver$\_$python$).$

These tools are included in the exam files, also available in Canvas$$s module.

Problem $2\mathrm{.}6\mathrm{.}4$

In Problem $2\mathrm{.}6\mathrm{.}1,$ suppose that we$$d like to see what would happen if the arrival rate were to

increase in small steps; maybe a single$-$server barbershop would like to increase business by

some advertising or coupons. Create a spreadsheet or use a computer program $($if you haven$$t

already done so to solve those problems$),$ and re$-$evaluate all five of Wq$,$ W$,$ Lq$,$ L$,$ and $\backslash$rho $,$

except increasing the arrival rate by $1\%$ over its original value, then by $2\%$ over its original

value, then by $3\%$ over its original value, and so on until the system becomes unstable $(\backslash$rho $>=1).$

Make plots of each of the five metrics as functions of the percent increase in the arrival rate.

Discuss your findings.

B:

Problem $2\mathrm{.}6\mathrm{.}8$

In the urgent$-$care clinic of Figure $2\mathrm{.}6\mathrm{.}8\mathrm{.}1,$ suppose that the patients arrive from outside into

the clinic $($coming from the upper right corner of the figure and always into the Sigh In

station$)$ with interarrival times that are exponentially distributed with mean $6$ minutes. The

number of individual servers at each station and the branching probabilities are all as shown

in Figure $2\mathrm{.}6\mathrm{.}8\mathrm{.}1.$ The service times at each node are exponentially distributed with means $($all

in minutes$)$ of $3$ for Sign In$,5$ for Registration, $90$ for Trauma Rooms, $16$ for Exam Rooms,

and $15$ for Treatment Rooms. For each of the five stations, compute the $$local$$ patient traffic

intensity $\backslash$rho $\_$station there. Will this clinic $$work$,$ i$.$e$.,$ be able to handle the external patient

load? Why or why not? If you could add a single server to the system, and add it to any of the

five stations, where would you add it$?$ Why? Hint: Use $$mmc$.$exe$$ program $($available in

Canvas$)$ unless you like using your calculator or a spreadsheet to calculate values manually.

Figure $2\mathrm{.}6\mathrm{.}8\mathrm{.}1$ A queueing system represen$$ng and urgent$-$care clinic

$3$

Problem $2\mathrm{.}6\mathrm{.}9$

In Problem $2\mathrm{.}6\mathrm{.}8,$ for each of the five stations, compute each c$,$ and interpret in words. Would

you still make the same decision about where to add that extra single resource unit that you

did in Problem $2\mathrm{.}6\mathrm{.}8?$ Why and why not? $($Remember these are sick people, some of them

seriously ill, not widgets being pushed through a factory.$)$

TASK $2(20\%)$ Basic Modeling with Simio

A:

Problem $4\mathrm{.}11\mathrm{.}13$

Start with a problem formulation of Problem $4\mathrm{.}11\mathrm{.}1.$ Run the model and report the results.

Skip the animation step, but do the following instead:

Create the experiment as described in Problem $4\mathrm{.}11\mathrm{.}3$

Modify the original model from Problem $4\mathrm{.}11\mathrm{.}1$ assuming you are modeling a manufacturing

process that involves drilling holes in a steel plate. The drilling machine has the capacity for

up to $3$ parts at a time $($c $=3$ in queueing terms$).$ The arrival rate should be $120$ parts per hour

and the processing rate should be $50$ parts per hour.

Add a simple Animation:

Use Trimble $3$D Warehouse to find and import appropriate symbols for the entities $($steel

plates$)$ and the server $($a drill press or other hole$-$making device$).$ Adjust physical dimensions

of imported symbols so they look realistic. Add a label to your animation to show how many

parts are being processed as the model runs.

B:

Problem $4\mathrm{.}11\mathrm{.}15$

Build a Simio model to confirm and cross$-$check the steady$-$state queueing$-$theoretic results

from your solutions to the M$/$D$/1$ queue of previous Problem $2\mathrm{.}6\mathrm{.}9.$ Remember that your

Simio model is initialized empty and idle, and that it produces results that are subject to

statistical variation, so design and run a Simio Experiment to deal with both of these issues;

make your own decisions about things like run length, number of replications, and Warm$-$up

Period, possibly after some trial and error.

For each of the five steady$-$state queueing metrics, first Compute numerical values for the

queueing$-$theoretic steady$-$state output performance metrics of Wq$,$ W$,$ Lq$,$ L$,$ and $\backslash$rho from

your solutions to Problem $2\mathrm{.}6\mathrm{.}9.$ Compare these with your simulation estimates and

confidence intervals. Discuss results. All time units are in minutes, and use minutes as well

throughout your Simio model. Take the arrival rate to be $\backslash$lambda $=1=1$ per minute, and the

service rate to be $\backslash$mu $=1/0\mathrm{.}9=1/0\mathrm{.}9$ per minute.