$($Queueing Model III$)$ Consider the two tandem network as the one described in the

lecture $($this was motivated as a Starbucks drive through$).$ Customer arrive according

Poisson process with rate lambda$>0.$ The service requirement in the first station $($when people place their orders$)$ is exponential with ratemu$1>0$ and the service requirement in the second station $($when people pickup their orders and pay$)$ is exponential with rate mu$2>0.$ We assume that once the customers are served in the first station, they instantly transition to the second station. Each station has an infinite capacity waiting room. All service times and inter$-$arrival times are mutually independent. Customers are served in the order in which they arrive and servers are work conserving. Now, the difference between this system and the one discussed in class is that in this problem, we assume that when a customer is served by the second station, they return to the system $($to the back of the queue in the $1$st station$)$ to be served again with probability p in $(0,1)$ and leave the system with probability $1-$p: Any returning customer, once he joins the system again, is treated as any new arriving customer $($i$.$e$.$ independent exponential service times with rates mu$1$ andmu$2,$ for the first and second stations, respectively$).$ To be clear, in class we covered the case p $=0.$

a$.)$ Let Y $($t$)=($Q$1($t$),$Q$2($t$))$ where Qi $($t$)$ is the number of people in the system in station i at time t$.$ Is Y $($t$)$ a continuous time Markov chain? Explain your reasoning.

b$.)$ Provide the transition diagram for the chain Y $(.),$ use the lattice repres$-$

entation given used in class for the case p $=0.$