Advanced Renewal Theory

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80 INTRODUCTION

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A renewal process is a counting process that generalizes the Poisson process In the Poisson process the interoccurrence times between the events are independent random variables with an exponential distribution, whereas in a renewal process the interoccurrence times have a general distribution A rst introduction to renewal theory has been already given in Section 21 In that section several limit theorems were given without proof These limit theorems will be proved in Section 82 after having discussed the renewal function in more detail in Section 81 A key tool in proving the limit theorems is the so-called key renewal theorem Section 83 deals with the alternating renewal model and gives an application of this model to a reliability problem In queueing and insurance problems it is often important to have asymptotic estimates for the waiting-time probability and the ruin probability In Section 84 such estimates are derived by using renewal-theoretic methods This derivation illustrates the simplicity of analysis to be achieved by a general renewaltheoretic approach to hard individual problems

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THE RENEWAL FUNCTION

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Let us rst repeat some de nitions and results that were given earlier in Section 21 The starting point is a sequence X1 , X2 , of non-negative independent random variables having a common probability distribution function F (x) = P {Xk x}, x 0

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for k = 1, 2, Letting 1 = E(Xk ), it is assumed that 0 < 1 < The random variable Xk denotes the interoccurrence time between the (k 1)th and kth events in some speci c probability problem; see Section 21 for examples Letting

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S0 = 0 and Sn =

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n = 1, 2, ,

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A First Course in Stochastic Models HC Tijms c 2003 John Wiley & Sons, Ltd ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)

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ADVANCED RENEWAL THEORY

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we have that Sn is the epoch at which the nth event occurs For each t 0, let N (t) = the largest integer n 0 for which Sn t Then the random variable N (t) represents the number of events up to time t The counting process {N (t), t 0} is called the renewal process generated by the interoccurrence times X1 , X2 , It is said that a renewal occurs at time t if Sn = t for some n Since F (0) < 1 the number of renewals up to time t is nite with probability 1 for any t 0 The renewal function M(t) is de ned by M(t) = E[N (t)], t 0

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For n = 1, 2, , de ne the probability distribution function Fn (t) by Fn (t) = P {Sn t}, t 0

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The function Fn (t) is the n-fold convolution of F (t) with itself Using the important observation that N (t) n if and only if Sn t, it was shown in Section 21 that

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E[N (t)] =

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Fn (t),

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t 0

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(811)

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Moreover, it was established in Section 21 that M(t) < for all t 0 Another important quantity introduced in Section 21 is the excess or residual life at time t This random variable is de ned by t = SN (t)+1 t and denotes the waiting time from time t onwards until the rst occurrence of an event after time t Using Wald s equation, it was shown in Section 21 that E( t ) = 1 {1 + M(t)} t The following bounds apply to the renewal function: t 2 t 1 M(t) + 2, 1 1 1

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2 where 2 = E(X1 ) The left inequality is an immediate consequence of (812) and the fact that t 0 The proof of the other inequality is demanding and lengthy The interested reader is referred to Lorden (1970)

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(812)

811 The Renewal Equation A useful characterization of the renewal function is provided by the so-called renewal equation