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Unformatted text preview: 1 20. Extinction Probability for Queues and Martingales ( Refer to section 15.6 in text ( Branching processes) for discussion on the extinction probability). 20.1 Extinction Probability for Queues: • A customer arrives at an empty server and immediately goes for service initiating a busy period. During that service period, other customers may arrive and if so they wait for service. The server continues to be busy till the last waiting customer completes service which indicates the end of a busy period. An interesting question is whether the busy periods are bound to terminate at some point ? Are they ? PILLAI 2 Do busy periods continue forever? Or do such queues come to an end sooner or later? If so, how ? • Slow Traffic ( ) Steady state solutions exist and the probability of extinction equals 1. (Busy periods are bound to terminate with probability 1. Follows from sec 15.6, theorem 159.) • Heavy Traffic ( ) Steady state solutions do not exist, and such queues can be characterized by their probability of extinction. •Steady state solutions exist if the traffic rate Thus •What if too many customers rush in, and/or the service rate is slow ( ) ? How to characterize such queues ? . 1 < ρ { } lim ( ) 1. exists if k n p P X nT k ρ →∞ = = < 1 ≥ ρ 1 ≤ ρ 1 > ρ PILLAI 3 Extinction Probability for Population Models ) ( π 3 26 = 1 3 X = 2 9 X = ( ) 2 1 Y # ( ) 2 2 Y ( ) 2 3 Y ( ) 3 1 Y ( ) 3 2 Y ( ) 3 3 Y ( ) 3 5 Y ( ) 3 4 Y ( ) 3 6 Y ( ) 3 8 Y ( ) 3 7 Y ( ) 3 9 Y 1 X = 1 X = Fig 20.1 PILLAI 4 • Offspring moment generating function: ∑ ∞ = = ) ( k k k z a z P Queues and Population Models • Population models : Size of the n th generation : Number of offspring for the i th member of the n th generation. From Eq.(15287), Text n X ( ) n i Y ( 1 1 n X n n k k X Y + = = ∑ ) Let z ) ( z P a 1 1 PILLAI (201) Fig 20.2 ( ) = { } n k i a P Y k = ∆ 5 { } 1 1 1 1 1 ( ) { } { } ( { }) {[ ( )] } { ( )} { } ( ( )) n j i n i X k n n k Y X n n j j n n j P z P X k z E z E E z X j E E z X j E P z P z P X j P P z + + = ∞ + + = = = = ∑ = = = = = = = = ∑ ∑ )) ( ( )) ( ( ) ( 1 z P P z P P z P n n n = = + Extinction probability satisfies the equation which can be solved iteratively as follows: z z P = ) ( π " 2, 1, , ) ( 1 = = − k z P z k k { } ? n P X k = = lim { 0} ? Extinction probability n o n P X π →∞ = = = = and PILLAI (202) (203) (204) (205) (0) z P a = = ∆ 6 Let ( ) (1) { } i i k k k E Y P k P Y k ka ρ ∞ ∞ = = ′ = = = = = > ∑ ∑ 1 1 1 ( ) , 1 is the unique solution of P z z a ρ π ρ π π ≤ ⇒ = > ⇒ = < < • Left to themselves, in the long run, populations either die out completely with probability , or explode with probability 1 ( Both unpleasant conclusions )....
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 Fall '09
 Jacobs
 Probability theory, Stochastic process, Markov chain, martingales, PILLAI

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