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Ch3_queue_theory-2008-7

Ch3_queue_theory-2008-7 - Queueing Theory(Delay Models...

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Unformatted text preview: Queueing Theory (Delay Models) Sunghyun Choi Adopted from Prof. Saewoong Bahk’s material Closed Queueing Networks – No customers are allowed to arrive or depart – M : the number of customers in the system – : the state dependent service rate at queue j when there are m customers : the total arrival rates (need to be found) : the constant related with M : a particular solution ) m ( j μ 1 1 1 K ij j p i , , K = = = & ) m ( j λ 1 1, , ( ) ( ) K j i ij i j j p j K m m λ λ λ α λ = = = = ) m ( α where j λ ⋅ ⋅ ⋅ = = = if ) ( ) 2 ( ) 1 ( if 1 ) ( ˆ ) ( ) ( Let j j j j j j j j j j j n n n n P m m ρ ρ ρ μ λ ρ Closed queueing networks – states for the network of K queues: n =(n 1 ,n 2 , …,n K ) – total of M customers in the system: where G(M) is a normalization constant that ensures that P[ n ] is a probability distribution P n P n P n G M K K [ ] ( ) ( ) ( ) = 1 1 G M P n P n K K n n n n M K K ( ) ( ) ( ) {( , , ) } = + + = ∑ 1 1 1 1 • Example ( closed queueing networks) –...
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Ch3_queue_theory-2008-7 - Queueing Theory(Delay Models...

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