Consider a single$-$server workstation that processes two types of jobs, type $1$ and type $2.$ Each job type arrives according

to a Poisson process with corresponding rates ra$1=6$ hr$1$ and ra$2=9$ hr$1$

$.$

These two arrival processes are independent from each other, and jobs are

processed according to a first$-$come$-$first$-$serve policy. The processing times

of the two job types are normally distributed with the corresponding distributions being N$1(5$min, $4$min$2$

$)$ and N$2(3$min, $1$min$2$

$).$

i$.(5$ pts$)$ Argue that the counting process that counts the arriving jobs

to the station, irrespective of their type, is Poisson, and determine the

rate ra of this process.

ii$.(10$ pts$)$ Use the result from part $($i$)$ to argue that this workstation

can be modeled as an M$/$G$/1$ queue. Let r$.$v$.$ Te denote the effective

processing time for this queueing station, and determine the mean te$,$

the variance $\backslash$sigma

$2$

e and the squared coefficient of variation c

$2$

e

for this r$.$v$.$

iii. $(5$ pts$)$ Use your results from part $($ii$)$ to show that the operation of

this workstation is stable.

iv$.(10$ pts$)$ Use your results from the previous parts to perform a Mean

Value Analysis $($MVA$)$ of this workstation. More specifically, compute

the station throughputs T H$1$ and T H$2$ with respect to each job type,

the corresponding server utilizations u$1$ and u$2,$ the expected cycle

times CT$1$ and CT$2$ for the two job types, and the expected number

of jobs of each type, W IPq$1$

and W IPq$2$

$,$ in the waiting queue