IND $2249060-$ Stochastic Models Midterm Exam INSTRUCTIONS$1.$ You have from $10$:$30$ to $12$:$30$ to complete this exam$\mathrm{.}2.$ The exam consists of $4$ questions$\mathrm{.}3.$ You have to write your answers by hand, sign each page, place your IDcard on each of your exam paper, and then take the photo of each of your paper, and then submit your answers to Microsoft Teams as a single PDF file.Q$3(35$ pts$)$ Duration: $40$ minutesAn average of $40$ students per hour arrive at the computing lab. The average student uses a computer for $10$ minutes. Assume exponential interarrival and service times. Use the following M$/$M$/$s Excel templates to answer the questions:Re sul tsn Pn L$=23\mathrm{.}88939328$ Lq $=17\mathrm{.}22272661$ W$=0\mathrm{.}597234832$ Wq $=0\mathrm{.}430568165$ r$=0\mathrm{.}952380952$ l$=$m$=$s$=$Pr$($W $>$ t$)=0\mathrm{.}916851$Da ta $4067($mean arrival rate$)($mean service rate$)($# servers$)$ when t $=0\mathrm{.}083$ Prob$($Wq $>$ t$)=0\mathrm{.}729422$ when t $=0\mathrm{.}08300\mathrm{.}0003531210\mathrm{.}00235413120\mathrm{.}0078471053140\mathrm{.}0174380110\mathrm{.}02906335150\mathrm{.}038751135$ l$=$m$=$s$=$l$=$m$=$s$=$s$=$Pr$($W $>$ t$)=0\mathrm{.}638991$when t $=0\mathrm{.}083$Da taDa ta$4069$Da ta$($mean arrival rate$)($mean service rate$)($# servers$)$nRe sul ts $4068($mean arrival rate$)($mean service rate$)($# servers$)$L$=$Lq $=$ W$=9\mathrm{.}3300995532\mathrm{.}6634328860\mathrm{.}233252489$ Pr$($W $>$ t$)=0\mathrm{.}756296$ Prob$($Wq $>$ t$)=0\mathrm{.}274221$Wq $=0\mathrm{.}066585822$ when t $=0\mathrm{.}083$r$=0\mathrm{.}833333333$ when t $=0\mathrm{.}083$Pn $040\mathrm{.}00091742610\mathrm{.}006116174230\mathrm{.}0203872470\mathrm{.}0453049930\mathrm{.}07550832250\mathrm{.}100677763$ Pr$($W $>$ t$)=0\mathrm{.}677026$Re sul tsn PnRe sul tsL$=7\mathrm{.}561682847$ Lq $=0\mathrm{.}89501618$ W$=0\mathrm{.}189042071$ Wq $=0\mathrm{.}022375405$ r$=0\mathrm{.}740740741$ when t $=0\mathrm{.}083$ Prob$($Wq $>$ t$)=0\mathrm{.}098005$ when t $=0\mathrm{.}08300\mathrm{.}00113296810\mathrm{.}00755311923450\mathrm{.}0251770620\mathrm{.}0559490280\mathrm{.}093248380\mathrm{.}124331173$ l$=$ m$=40610($mean arrival rate$)($mean service rate$)($# servers$)$L$=$Lq $=$ W$=$ r$=7\mathrm{.}0158305140\mathrm{.}3491638470\mathrm{.}175395763$ Wq $=0\mathrm{.}0087290960\mathrm{.}666666667$ Prob$($Wq $>$ t$)=0\mathrm{.}033195$ when t $=0\mathrm{.}083$nPn $00\mathrm{.}001217737123$ a$)$ If we want the average time a student waits for a computer to be at most $11$ minutes, how many computers should the lab have? Indicate how you used the template to arrive at this decision. $(15$ pts$)240\mathrm{.}0081182490\mathrm{.}0270608310\mathrm{.}060135180\mathrm{.}100225350\mathrm{.}133633734$ b$)$ If we want $90\%$ of all students to spend $5$ minutes or less waiting for a computer, how many computers should the lab have? Indicate how you used the template to arrive at this decision. $(10$ pts$)$c$)$ Suppose each computer costs the lab $$3$ per hour and each hour a student spends in the lab costs $$0\mathrm{.}5.$ What is the total expected cost $($i$.$e$,$ expected server cost $+$ expected cost of waiting time$)$ for s$=7$ and s$=8?(10$ pts$)3$