Problem $1(30$ points$)$: Consider a single$-$server workstation that pro$-$

cesses two types of jobs, type $1$ and type $2.$ Each job type arrives according

to a Poisson process with corresponding rates $r_{a_1}=6h{r}^{-1}$ and $r_{a_2}=9h{r}^{-1}.$

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 distri$-$

butions being $N_1(5(min),4mi{n}^2)$ and $N_2(3(min),1mi{n}^2).$

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 $r_a$ of this process.

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

can be modeled as an $\frac{M}{G}\frac{?}{1}$ queue. Let r$.$v$.T_e$ denote the effective

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

the variance ${}_e^2$ and the squared coefficient of variation $c_e^2$ 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 $C{T}_1$ and $C{T}_2$ for the two job types, and the expected number

of jobs of each type, $WI{P}_{q_1}$ and $WI{P}_{q_2},$ in the waiting queue.

Hint: The analysis of this system was briefly outlined in class as another

case that can be modeled by an M$/$G$/1$ queue where the effective processing

times follow a mixing distribution.