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372 An Introduction to Stochastic Modeling and the renewal theorem states that t lim [M D (t) - M D (t - h)] = h , µ provided X 2 , X 3 , . . . are continuous random variables. 7.5.2 Stationary Renewal Processes x A delayed renewal process for which the first life has the distribution function G(x) = µ -1 0 {1 - F(y)}dy is called a stationary renewal process. We are attempting to model a renewal process that began indefinitely far in the past, so that the remaining life of the item in service at the origin has the limiting distribution of the excess life in an ordinary renewal process. We recognize G as this limiting distribution. It is anticipated that such a process exhibits a number of stationary, or time- invariant, properties. For a stationary renewal process, M D (t) = E[N(t)] = and Pr t D x = G(x), for all t. Thus, what is in general only an asymptotic renewal relation becomes an identity, holding for all t, in a stationary renewal process. t µ (7.26) 7.5.3 Cumulative and Related Processes Suppose associated with the ith unit, or lifetime interval, is a second random variable Y i ({Y i } identically distributed) in addition to the lifetime X i . We allow X i and Y i to be dependent but assume that the pairs (X 1 , Y 1 ), (X 2 , Y 2 ), . . . are independent. We use the notation F(x) = Pr{X i x}, G(y) = Pr{Y i y}, µ = E[X i ], and v = E[Y i ]. A number of problems of practical and theoretical interest have a natural formula- tion in those terms. Renewal Processes Involving Two Components to Each Renewal interval Suppose that Y i represents a portion of the duration X i . Figure 7.5 illustrates the model. There we have depicted the Y portion occurring at the beginning of the interval, but this assumption is not essential for the results that follow.