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 pushe