Download PDF by L. Breuer, Dieter Baum: An introduction to queueing theory and matrix-analytic

By L. Breuer, Dieter Baum

ISBN-10: 1402036302

ISBN-13: 9781402036309

ISBN-10: 1402036310

ISBN-13: 9781402036316

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Additional resources for An introduction to queueing theory and matrix-analytic methods

Example text

Since p, q > 0, the chain Q is irreducible. If p < q, then h(n) := n defines a function which satisfies the conditions for Foster’s criterion, as ∞ k=0 pik h(k) = q(1 − p) · (i − 1) + (qp + (1 − q)(1 − p)) · i + p(1 − q) · (i + 1) = i − q(1 − p) + p(1 − q) = i − q + p ≤ i − ε 33 Markov Chains and Queues in Discrete Time for all i ∈ N, with ε = q − p > 0, and ∞ k=0 p0k · h(k) = p < ∞ show. The ratio p/q is called the load of the queue. Thus the system process Q is positive recurrent if the queue load is less than one.

Clearly, the structure of the matrix G shows that the process Q is irreducible and hence there is at most one stationary distribution π for Q. 3) n=0 of equations, where the latter is simply the normalization of the distribution π. The first two equations are the global balance equations and can be illustrated by the following scheme: λ λ 0 λ 1 µ λ ... 2. λ µ m µ ... µ Transition rates for the M/M/1 queue This gives the rates of jumps between the states of the system. If we encircle any one state, then the sum of the rates belonging to the arcs reaching into this state must equal the sum of the rates which belong to the arcs that go out of this state.

Note that r < 1 if and only if p < q. Further, the mean inter–arrival time is E(T T1 ) = 1/p and the mean service time is E(S1 ) = 1/q. Thus the geometric approach is successful if the so–called stability condition ρ= p E(S1 ) = <1 q E(T T1 ) holds. This condition simply postulates that the mean service time be shorter than the mean inter–arrival time. In this case, the stationary distribution π of Q has the form π0 = 1 − ρ and πn = (1 − ρ) ρ n−1 r 1−p for all n ≥ 1. It thus is a modified geometric distribution with parameter r = p′ /q ′ < 1.

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An introduction to queueing theory and matrix-analytic methods by L. Breuer, Dieter Baum


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