By Mark A. Pinsky, Samuel Karlin

ISBN-10: 0123814162

ISBN-13: 9780123814166

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**Additional resources for An introduction to stochastic modeling**

**Example text**

Zn ) means that xi ≥ zi , i = 1, 2, . . , n, and there exists at least one i such that xi > zi . Deﬁnition 7 (Minimal cut vector). A vector z is a cut vector to level c if Φ(z) < c. A cut vector to level c, z, is minimal if Φ(x) ≥ c for all x > z. Deﬁnition 8 (Minimal path vector). A vector y is a path vector to level c if Φ(y) ≥ c. A path vector to level c, y, is minimal if Φ(x) < c for all x < y. 6 shows a simple example of a ﬂow network model. The system comprises three components. Flow (gas/oil) is transmitted from a to b.

There are also other reliability operations, among them mixtures of distributions or forming the sum of random variables, and the question arises whether certain distribution classes are closed under these operations. For example, convolutions arise in connection with the addition of lifetimes and cold reserves. Before we come to the IFRA Closure Theorem we need a preparatory lemma to prove a property of the reliability function h(p) = P (Φ(X) = 1) of a monotone structure. Lemma 3 Let h be the reliability function of a monotone structure.

Let X and Y be two random variables. Then X is said to be smaller in the stochastic order, denoted X ≤st Y, if P (X > t) ≤ P (Y > t) for all t ∈ R+ . In reliability terms we say that X is stochastically smaller than Y , if the probability of surviving a given time t is smaller for X than for Y for all t. Note that the stochastic order compares two distributions, the random variables could even be deﬁned on diﬀerent probability spaces. One main point is now to compare a given lifetime distribution with the exponential one.

### An introduction to stochastic modeling by Mark A. Pinsky, Samuel Karlin

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