$1.(2$ pts$)$ Arrivals to a gate for entry into a stadium are Poisson with an average interarrival time of $3\mathrm{.}2$ seconds. There are $6$ servers, each taking $17$ seconds on average to conduct a security check and scan a ticket. Service times are exponentially distributed.

$($a$)(0\mathrm{.}5$ pts$)$ Assume each server has a separate line with arrivals equally distributed among servers. That is$,$ we have six M$/$M$/1$ queues and the arrival rate to each queue is $1/6$ of the arrival rate to the gate. For each server, what is the average utilization, the average cycle time for a fan, the average time in line for a fan, the average number of fans at each server $($including queue$)$ and the average queue length?

$($b$)(1$ pt$)$ Now assume a single line for all servers. That is$,$ consider an M$/$M$/6$ queue. What is the average utilization of a server, the average cycle time for a fan, the average time in line for a fan, the average number of fans at each server $($including queue$)$ and the average queue length?

$($c$)(0\mathrm{.}5$ pts$)$ Management is considering merging the arrivals to two identical gates. Each gate faced an arrival rate of one per $3\mathrm{.}2$ seconds as described in $($b$).$ After merging, the fan arrivals would double to each merged gate. What is the smallest number of servers that are needed at the merged gate? $[$Hint: we look for the minimum that results in $.]$

$.2.(1\mathrm{.}5$ pts$)$ An unmanned space probe is being designed. It has a single$-$processor computer that will receive instructions from a space station orbiting Earth and processes them for execution. All times are in seconds unless otherwise stated. The natural process time is $2$ seconds with a CV of $1\mathrm{.}25.$ The instructions will arrive at an average rate of one every $2\mathrm{.}5$ seconds with an arrival time CV of $1\mathrm{.}5.$ Unprocessed instructions are stored in a memory buffer for future processing. The processor experiences glitches every $30$ hours on average, and these take $10$ minutes to repair on average with a CV of $0\mathrm{.}8.$

$($a$)(0\mathrm{.}5$ pts$)$ What is the effective processing time $()$ and its CV $()?$

$($b$)(0\mathrm{.}5$ pts$)$ What is the average time for an instruction in the memory buffer? $($

c$)(0\mathrm{.}5$ pts$)$ The chief engineer tells you that the average time for an instruction in the memory buffer can be at most $25$ seconds. Otherwise, the risk of collision or some other disaster is too high. It is your job to change the processor design to accommodate this requirement. You decide to reprogram the processor repair function to reduce the variance of repair time. What is the new requirement for maximum CV of repair time? $($This is called a derived requirement in engineering design.$)[$Hint: First solve for the new squared CV of effective process time $()$ by setting to $24\mathrm{.}99$ seconds. Then solve for based on the new $.]$

$3.(2\mathrm{.}5$ pts$)$ A single$-$server machine has a natural process time of $15$ minutes with of $5.$ It has preventive maintenance done after every $75$ jobs. The maintenance lasts $2$ hours with a CV of $0\mathrm{.}75.$

$($a$)(0\mathrm{.}5$ pts$)$ What is the effective processing time and its CV$?$

$($b$)(0\mathrm{.}5$ pts$)$ An alternate maintenance program is proposed. Preventive maintenance is done after every $25$ jobs, but it lasts only $1\mathrm{.}5$ hours. It has the same CV as the first maintenance program. What is the effective processing time and its CV$?$

$($c$)(1$ pt$)$ Compare the two alternatives based on their capacity $($i$.$e$.,)$ and cycle time $().$ For cycle time, use the input stream of average interarrival time of $20$ and interarrival CV of $1$ for both alternatives.$($d$)(0\mathrm{.}5$ pts$)$ Which of the alternatives is preferred to maximize sales? Which is preferred to maximize individual customer satisfaction?

$3.(2\mathrm{.}5$ pts$)$ Patients are screened for diabetes at the nurse station in a small clinic. This process takes $10$ minutes on average with a CV of $0\mathrm{.}3.$ There are $3$ nurses. Then the patients see a physicians assistant to receive advice based on their screening result. This takes $12$ minutes on average with a CV of $1\mathrm{.}2.$ There are $4$ physicians assistants. Patients arrive at a rate of $1$ every $4$ minutes on average with CV$=1.$

$($a$)(1$ pt$)$ What are the utilizations of each station and the arrival process parameters $($rate and arrival time CV$)$ for the second station?

$($b$)(1\mathrm{.}5$ pts$)$ For the entire line, what is the average time in queue, average time in system, average number in queue and average number in system?

$4.(1\mathrm{.}5$ pts$)$ An assembly station produces electronic devices, and a testing station determines whether the devices function correctly. There is a conveyor between the two stations with room for $6$ devices. Each station has one server. The assembly station takes $15$ minutes on average, and the testing station takes $12$ minutes on average. The assembly station is never starved and has infinite buffer space. Assume Poisson arrivals and exponential service times. Use the M$/$M$/1/$b queue to model the behavior of the testing station. $($a$)(1$ pt$)$ What is the average WIP, TH and CT of the testing station?

$($b$)(0\mathrm{.}5$ pts$)$ Explain, intuitively, why is smaller than the bottleneck rate $?$

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