Ch3_queue_theory-2008-5-rev

Ch3_queue_theory-2008-5-rev - Queueing Theory(Delay Models...

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Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material

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Time Reversibility - Burke’s theorem Birth-death process: Markov chain with integer states in that transitions occur only between neighboring states • Consider a discrete-time Markov chain X n , X n +1 ,… p P X j all n j n = = [ ] for
P X j X i X i X i m m m m k k [ , , , ] = = = = + + + 1 2 2 = = = = = = = = + + + + + + P X j X i X i X i P X i X i X i m m m m k k m m m k k [ , , , , ] [ , , , ] 1 2 2 1 2 2 = = = = = = = = = = = = + + + + + + + + P X j P X i X j P X i X i X i X j P X i P X i X i X i m m m m m k k m m m m m k k m [ ] [ ] [ , , , ] [ ] [ , , ] 1 2 2 1 1 2 2 1 = p P p j ji i ,...) 3 , 2 at times states of nt (independe + + m m

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The backward transition probability If for all i,j the chain is time reversible!!! P P X j X i p P p i j ij m m j ji i * [ ] , = = = = ≥ + 1 0 P P ij ij * =
Properties of the reversed chain (discrete time) (P1) Irreducible, aperiodic, the same stationary

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Ch3_queue_theory-2008-5-rev - Queueing Theory(Delay Models...

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