By L. Breuer, Dieter Baum

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

**Example text**

Thus we can from now on restrict the analysis to the transition probability matrices P (t) := P (0, t) with t ≥ 0. With this notation the Markov property yields the Chapman– Kolmogorov equations P (s + t) = P (s)P (t) for all time durations s, t ≥ 0. Thus the family {P (t) : t ≥ 0} of transition probability matrices forms a semi–group under the composition of matrix multiplication. In particular, we obtain for the neutral element of this semi–group P (0) = IE := (δij )i,j∈E with δij = 1 for i = j and zero otherwise.

Hence πP = π implies first π0 · (1 − p) = π0 ⇒ π0 = 0 since 0 < p < 1. Assume that πn = 0 for any n ∈ N0 . This and the condition πP = π further imply for πn+1 πn · p + πn+1 · (1 − p) = πn+1 ⇒ πn+1 = 0 22 AN INTRODUCTION TO QUEUEING THEORY which completes an induction argument proving πn = 0 for all n ∈ N0 . Hence the Bernoulli process does not have a stationary distribution. 21 The solution of πP = π and P = j∈E πj = 1 is unique for 1−p p p 1−p with 0 < p < 1. Thus there are transition matrices which have exactly one stationary distribution.

G = 0 for all t ≥ 0, which yields πG = 0 because of the uniqueness of the zero power series. 4) i=j for all j ∈ E. This system can be intepreted as follows. We call the value πi gij stochastic flow from state i to state j in equilibrium. Then the above equations mean that the accrued stochastic flow into any state j equals the flow out of this state. 4) are called the (global) balance equations. 1) is given by ⎛ ⎞ −λ λ 0 0 ... ⎟ 0 ⎜ 0 −λ λ ⎜ G=⎜ .. ⎟ 0 −λ λ ⎝ 0 ⎠ .. .. .. . . 48 AN INTRODUCTION TO QUEUEING THEORY This process has no stationary distribution, which can be seen as follows.