$17\mathrm{.}5-13.$ Consider a queueing system that has two classes of customers,$$two clerks providing service, and no queue. Potential customers$$from each class arrive according to a Poisson process, with$$a mean arrival rate of $10$ customers per hour for class $1$ and $5$ customers$$per hour for class $2,$ but these arrivals are lost to the system$$if they cannot immediately enter service.$$Each customer of class $1$ that enters the system will receive$$service from either one of the clerks that is free, where the$$service times have an exponential distribution with a mean of$5$ minutes.$$Each customer of class $2$ that enters the system requires the$$simultaneous use of both clerks $($the two clerks work together as a$$single server$),$ where the service times have an exponential distribution$$with a mean of $5$ minutes. Thus, an arriving customer of$$this kind would be lost to the system unless both clerks are free to$$begin service immediately.$($a$)$ Formulate the queueing model in terms of transitions that only$$involve exponential distributions by defining the appropriate$$states and constructing the rate diagram.$($b$)$ Now describe how the formulation in part $($a$)$ can be fitted into$$the format of the birth$-$and$-$death process.$($c$)$ Use the results for the birth$-$and$-$death process to calculate the$$steady$-$state joint distribution of the number of customers of$$each class in the system.$($d$)$ For each of the two classes of customers, what is the expected$$fraction of arrivals who are unable to enter the system?