Useful application all birthdeath processes with j pj

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Unformatted text preview: ible if and only if the embedded chain is. 7 The guessing theorem: Suppose a MP is irreducible and {pi} is a set of probabilities that satisfies piqij = ￿ pj qji for all i, j and satisfies i piνi < ∞. Then (1), pi > 0 for all i, (2), pi is the sample-path fraction of time in state i WP1, (3), the process is reversible, and (4), the embedded chain is p ositive recurrent. ￿ Useful application: All birth/death processes with j pj νj < ∞ are reversible. Similarly, if the Markov graph is a ￿ tree with j pj νj < ∞, the process is reversible. 8 ✉ ✉ a2 d1 a1 ✉ ✉ a3 d2 ✉ d3 ✉ ✉ a4 d4 ✉ Right moving (forward) M/M/1 pro cess ✉ d3 a4 d4 ✉ ✉ d2 ✉ ✉ a3 ✉ a2 d1 ✉ ✉ a1 Left moving (backward) M/M/1 process Burke’s thm: Given an M/M/1 queue in steady-state with (arrival rate) λ < µ (departure rate), (1) Departure process is Poisson with rate λ (2) State X (t) is independent of departures b efore t (3) For FCFS, a customer’s arrival time, given its de­ parture at t, is independent of departures b efore t. 9 ✉ ✉ a2 d1 a1 ✉ ✉ d3 a4 d4 ✉ ✉ ✉ a3 d2 ✉ d2 ✉ ✉ a3 t d3 ✉ t ✉ a2 ✉ a4 d1 ✉ d4 ✉ ✉ a1 A departure at t in (right-moving) sample path is an arrival in the M/M/1 (left-moving) sample path. For FCFS left-moving process, departure time of ar­ rival at t depends on arrivals (and their service req.) to the right of t; independent of arrivals to left. For corresponding (right-moving) process, the arrival time of that departure is independent of departures b efore t. 10 P...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.

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