Draw a graph for the states of the process

Reference no: EM13908966

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

Define the 'state' of the system at time 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 and the other of type B; the type 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 arrives.

ABB if another type arrives.

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 to another state S∗, how is the number of transitions from to S∗ related to the number from S∗ to S?

(c) Characterize the process of type 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 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 ρλAand ρλBB.

(e) Let Qbe the probability of customers in the system, as a function of Q0:

(f) Pr{E}. Show that Q= (1 - ρ)ρn where ρ ρρB.

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

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