##### Reference no: EM13908966

Consider a preemptive resume LCFS queueing system with two classes of customers. Type *A *customer arrivals are Poisson with rate *λ**A *and Type *B *customer arrivals are Poisson with rate *λ**B*. The service time for type *A *customers is exponen- tial with rate *μ**A *and that for type *B *is exponential with rate *μ**B*. Each service time is independent of all other service times and of all arrival epochs.

Define the 'state' of the system at time *t *by the string of customer types in the system at *t*, in order of arrival. Thus state *AB *means that the system contains two customers, one of type *A *and the other of type *B*; the type *B *customer arrived later, so is in service. The set of possible states arising from transitions out of *AB *is as follows:

*ABA *if another type *A *arrives.

*ABB *if another type *B *arrives.

*A *if the customer in service (*B*) departs.

Note that whenever a customer completes service, the next most recently arrived resumes service, so the state changes by eliminating the final element in the string.

**(a) **Draw a graph for the states of the process, showing all states with two or fewer customers and a couple of states with three customers (label the empty state as *E*). Draw an arrow, labeled by the rate, for each state transition. Explain why these are states in a Markov process.

**(b) **Is this process reversible? Explain. Assume positive recurrence. Hint: If there is a transition from one state *S *to another state *S*∗, how is the number of transitions from *S *to *S*∗ related to the number from *S*∗ to *S*?

**(c) **Characterize the process of type *A *departures from the system (i.e., are they Poisson, do they form a renewal process, at what rate do they depart etc.?)

**(d) **Express the steady-state probability Pr{*A*} of state *A *in terms of the probability of the empty state Pr{*E*}. Find the probability Pr{*AB*} and the probability Pr{*ABBA*} in terms of Pr{*E*}. Use the notation *ρ**A *= *λ**A**/μ**A *and *ρ**B *= *λ**B**/μ**B*.

*(e) *Let *Q**n *be the probability of *n *customers in the system, as a function of *Q*0*:*

**(f) **Pr{*E*}. Show that *Q**n *= (1 - *ρ*)*ρ**n** *where *ρ *= *ρ**A *+ *ρ**B*.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.