By Gunter Bolch; et al

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**Additional resources for Queueing networks and Markov chains : modeling and performance evaluation with computer science applications**

**Example text**

17) If the unique steady-state probability vector of a DTMC exists, it can be determined by the solution of the system of linear Eqs. 15), so that I? need MARKOV PROCESSES 43 not be determined explicitly. From Eq. 14), we have y(n) = v(n - l)P. If the limit exists, we can take it on both sides of the equation and get: lim v(n) = ZI = lim v(n - l)P = C/p. 18) n+cc In the steady-state case no ambiguity can arise, so that, for the sake of convenience, we may drop the annotation and refer to steady-state probability vector by using the notation u instead of t.

7 Vn E T. 6) For the sake of convenience, we usually drop the superscript, so that pij = ~13) refers to a one-step transition probability of a homogeneous DTMC. Starting with state i, the DTMC will go to some state j (including the possibility of j = i), so that it follows that Cj pij = 1, where 0 5 pij 5 1. The one-step transition probabilities pij are usually summarized in a nonnegative, stochastic’ transition matrix P: p = p(l) = bij] = PO0 PO1 po2 **’ ;;; ;;; g:; : 1: I . . . . . I Graphically, a finite-state DTMC is represented by a state transition diagram, a finite directed graph, where state i of the chain is depicted by a vertex, and a one-step transition from state i to state j by an edge marked with one-step transition probability pij.

Statistics Let X1, X2 . . ). Let Yr, Ys,. . , Y, be random variables obtained by permuting the set ,X, so as to be in increasing order. Thus, for instance: x1,x2,... Yr = min{Xr, X2,. . ,X,} and: Y, = max{Xr,X2,. . ,Xn}. The random variable Yk is generally called the kth -order statistic. , . . < Y, (as opposed to Yr 5 Y2 < . . < Yn) with a probability of one. BASICS OF PROBABILITY AND STATISTICS 31 As examples of the use of order statistics, let Xi be the time to failure of the ith component in a system of n independent components.