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Unformatted text preview: Renewal Theory Class Notes for ISEN 618 October 2, 2007 1 Introduction and Preliminaries The physical basis on which renewal theory rests is very simple. Suppose we observe a particular (randomly occuring) phenomenon, e.g., the times at which a device or piece of equipment fails. Suppose that we begin observing this sys tem at the instant that the first device is installed, and suppose that when this device fails (i.e., at the end of its random lifetime), it is immediately replaced by a statistically identical (new) device and the process continues. Consider a sequence of random variables { X 1 ,X 2 ,... } describing the successive lifetimes of devices. Each lifetime is a nonnegative random variable, and because the devices are statistically identical, the sequence may be assumed to be indepen dent and identically distributed. Notice that at any point in time t , we can define a random variable N ( t ) to be the number of devices used up to time t . Then { N ( t ) ,t ≥ } is a counting process on < + , and such a counting processes (i.e. one whose interevent times are i.i.d. random variables, is called a renewal counting process . Formally, we define a renewal process as follows. Definition 1 A renewal process is a sequence of nonnegative random variables { X n ,n ≥ 1 } such that for all n = 1 , 2 ,... , and all x 1 ,x 2 ,...,x n P ( X 1 ≤ x 1 ,X 2 ≤ x 2 ,...,X n ≤ x n ) = P ( X 1 ≤ x 1 ) P ( X 2 ≤ x 2 ) ··· P ( X n ≤ x n ) . 1 Lemma 1 If P ( X 1 < ∞ ) = 1 , then { X n ,n ≥ 1 } is a sequence of i.i.d. random variables. Proof: left for the student. We will denote the common distribution function of the X i by F . Definition 2 A renewal process { X n ,n ≥ 1 } with P ( X 1 < ∞ ) = 1 is called a persistent (nonterminating) renewal process. If P ( X 1 < ∞ ) < 1 then we have a transient (terminating) renewal process. For our purposes, unless otherwise stated, we will consider persistent renewal processes. To avoid trivialities, we will also assume that P ( X 1 > 0) > 0; this condition ensures that X 1 has a mean μ > 0 (keep in mind that it may be + ∞ ). Now let us interpret X i in a point process context as the time between the i 1 st and the i th event. For n = 0 , 1 , 2 ,... , if we let T = 0 , T n = X 1 + X 2 + ... + X n , then clearly T n is the time, measured from the origin, at which the n th event (renewal) occurs. Note that by the strong law of large numbers, lim n →∞ T n n = μ a.s. and since we assume μ > 0, T n must approach infinity as n approaches infinity. Thus T n must be less than or equal to t for at most a finite number of values of n , and hence an infinite number of renewals cannot occur in a finite time....
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 Fall '07
 Klutke
 Probability theory, Tn, renewal process, renewal

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