Consider the following motor vehicle registration process with four stations $($reception$,$ clerk $1,$

clerk $2,$ cashier$)$:

$1)$ Reception: Customers arrive in a Poisson process with rate $12$ per hour. The receptionist takes

an amount of time that is exponentially distributed with $20$ seconds to direct each customer to

either one of two processing clerks with probabilities $0\mathrm{.}3($clerk $1)$ and $0\mathrm{.}7($clerk $2).$

$2)$ Clerk $1$ handles out$-$of$-$state and new licenses and takes on average $10$ minutes, the amount of

time having an exponential distribution.

$3)$ Clerk $2$ handles standard in$-$state renewal applications and takes on average $5$ minutes. An

exponential distribution assumption is appropriate for this time as well.

$4)20\%$ of applications processed by clerk $1$ go to clerk $2$ and $10\%$ of applications processed by

clerk $2$ go to clerk $1$ for further processing. When the processing is completed by the two clerks

$(80\%$ by clerk $1$ and $90\%$ by clerk $2),$ the customers move to the cashier for paying the fees.

$5)$ The amount of time spent by the cashier with a customer is exponential with mean $1$ minute.

a$)$ Model this system as an open network and obtain the limiting distribution of the number of

customers at each station.

b$)$ Determine the average total amount of time a customer spends in the system.