Consider the following combined queueing system. The first queue system is M/M/1 with service rate µ 1 . The second queue system has IID exponentially distributed service times with rate µ 2 . Each...

Consider the following combined queueing system. The first queue system is M/M/1 with service rate µ₁. The second queue system has IID exponentially distributed service times with rate µ₂. Each departure from system 1 independently 1 - Q₁. System 2 has an additional Poisson input of rate 2, independent of inputs and outputs from the first system. Each departure from the second system independently leaves the combined system with probability Q₂
and re-enters system 2 with probability 1 - Q₂. For parts 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
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