Chap2._Renewalnotes

# Chap2._Renewalnotes - Renewal Theory Class Notes for ISEN...

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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 non-negative 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 non-negative 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 (non-terminating) 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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Chap2._Renewalnotes - Renewal Theory Class Notes for ISEN...

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