Problem $1(30$ points$)$: 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 $r_{a_1}=6h{r}^{-1}$ 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 distributions 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.