Consider the following queueing network that represents a $4-$station manufacturing system. The arrows show part flow through the system $($Stations A$,$ C & D have a capacity of $1,$ i$.$e$.c=1,$ while Station B has a capacity of $2,$ i$.$e$.c=2).$ The parameters are given as follows:

Part arrival rate $({}_A=100$ parts$/$hour$).$

Mean station service times $($all service times are exponentially distributed$)$:

A: $30$ seconds

B: $40$ seconds

C: $90$ seconds

D: $90$ seconds

Solve the queueing network to find the following steady$-$state values

a$)$ The expected utilizations for stations A$,$ B$,$ C$,$ and D

b$)$ The expected number of parts in the system $($L$)$

c$)$ The expected time that a part spends in the system $($W$)$

d$)$ Perform two $(2)$ sensitivity analysis for this system using Excel Data Tables.

i$.$ The first sensitivity analysis is to consider how the number of jobs in system $($L$)$ is impacted for the following values of service times $($in seconds$)$ for Station B: $25,30,35,40,45,50,55,60,65,70,$&$75,$ and the following values of mean station service times $($in seconds$)$ for station C: $75,80,85,90,95,100,105,$&$110$