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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 !...
View Full Document

{[ snackBarMessage ]}

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online