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4703-10-Fall-h9

# 4703-10-Fall-h9 - IEOR 4703 Homework 9 1 Gibbs sampler for...

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Unformatted text preview: IEOR 4703: Homework 9 1. Gibbs sampler for a closed Jackson queueing network: (READ LECTURE NOTES 10 FOR REFERENCE HERE; SECTION 2.2) Consider a closed queueing network with c = 10 nodes (single-server FIFO queues), and M = 50 customers, in which the 10 × 10 transition matrix P for customer routing is known to be irreducible and doubly stochastic , so that the unique probability solution to λ = λP is the discrete uniform distribution over the set { 1 , 2 ,..., 10 } ; λ i = 1 / 10 , i ∈ { 1 , 2 ,..., 10 } . Further assume that the service times at node i are iid exponentially distributed with rate μ i = 1 /i, i ∈ { 1 , 2 ,..., 10 } . For then ρ i = λ i /μ i = i/ 10, and the largest one is ρ c = ρ 10 = 1. As a result, the stationary distribution π of ( X 1 ( t ) ,...,X 9 ( t )) is of the form π ( x ) = Kg ( x ) def = K 9 Y i =1 ρ i x i , x ∈ S , (1) where S = { x ∈ N 9 + : 9 X i =1 x i ≤ M } , and K- 1 = X x ∈S g ( x ) , (a constant for which we typically do not know, and will not need in what follows).(a constant for which we typically do not know, and will not need in what follows)....
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4703-10-Fall-h9 - IEOR 4703 Homework 9 1 Gibbs sampler for...

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