# lec23 - The M/M/c queue Again X t the number of individuals...

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Unformatted text preview: The M/M/c queue Again, X ( t ) the number of individuals in the queueing system can be modeled as a birth & death process. The transition state diagram for the X ( t ) is: 1 λ μ 2 λ λ 2μ 3μ K λ ( c-1 )μ c-1 1 λ c μ c λ c μ Clearly, the critical thing here in terms of whether or not a steady state exists is whether or not λ/ ( cμ ) < 1 . Let a = λ/μ and ρ = a/c = λ/ ( cμ ) . The balance equations for steady state are: 1 The M/M/c queue: balance equations p 1 = ap p 2 = a 2 2 · 1 p p 3 = a 3 3! p ... p c = a c c ! p p c +1 = ρ · a c c ! p ... p n = ρ n − c · a c c ! p for n ≥ c. In order to get an expression for p , we use the condition, that the overall sum of probabilities must be 1. This gives: 1 = ∞ summationdisplay k =0 p k = p parenleftBigg c − 1 summationdisplay k =0 a k k ! + a c c ! ∞ summationdisplay k = c ρ k − c parenrightBigg = p c − 1 summationdisplay k =0 a k k !...
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lec23 - The M/M/c queue Again X t the number of individuals...

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