Consider the closed queuing network of single server queues with exponentially distributed service

times, as shown in the figure below.

The average service rates of Q$1,$ Q$2$ and Q$3$ are respectively ${}_1=1,{}_2=0\mathrm{.}5,$ and ${}_3=0\mathrm{.}5.$ The

system has a total user population of $4.$ Using the convolution $($Buzen$\text{'}$s Algorithm$)$ approach, obtain the following $-$

$($a$)$ The Normalization Constant

$($b$)$ The state probability distribution for the queuing network

$($c$)$ The actual throughput of each queue

$($d$)$ The mean number in each queue

Hint: This is a closed queing network related problem