Consider a single machine that has access to unlimited supply of raw material. The machine works continuously to process raw material one unit at a time at a rate of $40$ units per hour. $($a$)$ What is the average processing time per unit? $($What is the average utilization of themachine? What is the average throughput $($output rate$)$ of the machine? $($b$)$ Does the processing time distribution affect any of your answers to the above questions? Explain why. $($c$)$ Assume from here on that the processing time at this machine $($let$$s call it machine $1)$ has the exponential distribution. Machine $1$ feeds into a single machine $($machine $2)$ where the material gets processed further. What is the arrival rate to this machine $($assuming there is ample waiting room at machine $2)?($d$)$ Machine $2$ can process units at a rate of $50$ units per hour. What is the throughput of thesystem $($machine $1+$ machine $2)?$ What is the average utilization of machine $2?($e$)$ Assume processing time at machine $2$ is also exponentially distributed, what is expected total time a unit spends in stage $2$ of the process $($queueing and getting processed at machine $2)?$What is the average queue size at machine $2?($f$)$ What is the probability of having less than $2$ units in the queue at machine $2?$ What is theprobability of having exactly four units in stage $2($queueing or in process at machine $2)?($g$)$ Assume now that there is room for at most three units to queue at machine $2.$ Once queue sizeat machine $2$ reaches $3,$ units are diverted away to a different facility. What is the probability thatan arriving unit is diverted? What is the throughput of the system $($i$.$e$.,$ out of machine $2)?($h$)$ What is average size of the queue at machine $2?$ What is the probability of having less than $2$units in the queue at machine $2?$ Explain the differences from the results obtained in the case of unlimited waiting room. $($i$)$ Machine $2$ is replaced by two slower parallel machines that can process $25$ units per hour with exponentially distributed processing time. Assuming again, there is ample waiting room at machine $2,$ what is the throughput of the system? What is the expected time in queue? What is the expected queue size? What is expected total time in system? Explain differences in your answer with those for the single machine case.