Question $5($H&L$,17\mathrm{.}5-11).$

Suppose that a single$-$server queueing system fits all the assumptions of the birth$-$and$-$death process except that customers always arrive in pairs $($but are still served individually$).$ The mean arrival rate is $2$ pairs per hour $(4$ customers per hour$)$ and the mean service rate $($when

the server is busy$)$ is $5$ customers per hour.

a$)$ Construct the rate diagram for this queueing system.

b$)$ Develop the balance equations. $($You only need to write equations for nodes $0$ through $3.)$

c$)$ For comparison purposes, display the rate diagram for the corresponding queueing system that completely fits the birth$-$and$-$death process, i$.$e$.,$ where customers arrive individually at a mean rate of $4$ per hour.