Ch3_queue_theory-2008-6

Ch3_queue_theory-2008-6 - Queueing Theory(Delay Models...

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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material Networks of Queues – Jackson’s theorem • Each queue is assumed to be M/M/1 • K queues • P ij : transition prob. from queues i to j • r j : external input rate at queue j • total arrival rate at queue j : j λ 1 = 1, , K j j i ij i r P j K λ λ = + = & • vector denoting the number of customers • state transition rate (Continuous Time MC) Arrival Departure Transition 1 2 ( , , , ) K n n n n = ( ) ( ) ( , ) (1 ) nn j j nn j ji j i nn j ji i j q r q P q P μ μ +- + - = = - = • Theorem: Assume and let the stationary distribution of the chain 1 < j ρ ) 1 ( ) ( w here ) ( ) ( ) ( ) ( Then ) , , ( 2 2 1 1 1 ≥- = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ j j n j j j k k n n P n P n P n P n P n n P j ρ ρ • Proof: using (P2) of reverse chain of CTMC, we show that with P(n) defined above, * * ' ' ( ') satisfies ( ) n n nn nm nm m m P n q q q q P n = = , ( ) 1 , ( ) , ( , ) 0, otherwise j j ji nm i j ji r m n j P m n j q P m n i j μ μ +- + - = &- = = & = *...
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This note was uploaded on 06/16/2008 for the course COMPUTER S 853 taught by Professor Choi during the Spring '08 term at Seoul National.

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Ch3_queue_theory-2008-6 - Queueing Theory(Delay Models...

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