Consider the following combined queueing system. The first queue sys- tem is M/M/1 with service rate μ1. The second queue system has IID exponential service times with rate μ2. Each departure from system 1 leaves the entire system with prob- ability 1 - Q1: and enters system 2 with the remaining probability Q1: System 2 has an additional Poisson input of rate λ2, independent of inputs and outputs from the first system. Each departure from the second system independently leaves the combined sys- tem with probability Q2: and reenters system 2 with probability 1 - Q2: For (a), (b), (c) assume that Q2: = 1 (i.e., there is no feedback).
(a) Characterize the process of departures from system 1 that enter system 2 and characterize the overall process of arrivals to system 2.
(b) Assuming FCFS service in each system, find the steady-state distribution of time that a customer spends in each system.
(c) For a customer that goes through both systems, show why the time in each system is independent of that in the other; find the distribution of the combined system time for such a customer.
(d) Now assume that Q2:
(e) <>1. Is the departure process from the combined system Poisson? Which of the three input processes to system 2 are Poisson? Which of the input processes are independent? Explain your reasoning, but do not attempt formal proofs.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.