Discrete-time stochastic processes

# Denition 21 a renewal process is an arrival process

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Unformatted text preview: the Bernoulli process is deﬁned by a sequence of IID binary rv’s Y1 , Y2 . . . , with PMF pY (1) = q specifying the probability of an arrival in each time slot i > 0. There is an associated counting process {N (t); t ≥ 0} giving the number of arrivals up to ¢ including ° t and time slot t. The PMF for N (t), for integer t > 0, is the binomial pN (t) (n) = n q n (1 − q )t−n . There is also a sequence S1 , S2 , . . . of integer arrival times (epochs), where the rv Si is the epoch of the ith arrival. Finally there is an associated sequence of interarrival times, X1 , X2 , . . . , which are IID with the geometric PMF, pXi (x) = q (1 − q )x−1 for positive integer x. It is intuitively clear that the Bernoulli process is fully speciﬁed by specifying that the interarrival intervals are IID with the geometric PMF. For the Poisson process, arrivals may occur at any time, and the probability of an arrival at any particular instant is 0. This means that there is no very clean way of describing a Poisson process in terms of the probability of an arrival at any given instant. It is more convenient to deﬁne a Poisson process in terms of the sequence of interarrival times, X1 , X2 , . . . , which are deﬁned to be IID. Before doing this, we describe arrival processes in a little more detail. 2.1.1 Arrival processes An arrival process is a sequence of increasing rv’s , 0 < S1 < S2 < · · · , where Si < Si+1 means that Si+1 − Si is a positive rv, i.e., a rv X such that Pr {X ≤ 0} = 0. These random variables are called arrival epochs (the word time is somewhat overused in this sub ject) and represent the times at which some repeating phenomenon occurs. Note that the process starts at time 0 and that multiple arrivals can’t occur simultaneously (the phenomenon of bulk arrivals can be easily handled by the simple extension of associating a positive integer rv to each arrival). We will often specify arrival processes in a way that allows an arrival at 58 2.1. INTRODUCTION 59 time 0 or simultaneous arrivals as events of zero probability, but such zero probability events can usually be ignored. In order to fully specify the process by the sequence S1 , S2 , . . . of rv’s, it is necessary to specify the joint distribution of the subsequences S1 , . . . , Sn for all n > 1. Although we refer to these processes as arrival processes, they could equally well model departures from a system, or any other sequence of incidents. Although it is quite common, especially in the simulation ﬁeld, to refer to incidents or arrivals as events, we shall avoid that here. The nth arrival epoch Sn is a rv and {Sn ≤ t}, for example, is an event. This would make it confusing to also refer to the nth arrival itself as an event. ✛ X1 0 ✛ r ✛ X2 ✲ r ✲ S1 X3 r ✲ ✻ (t) N t S2 S3 Figure 2.1: An arrival process and its arrival epochs {S1 , S2 , . . . }, its interarrival intervals {X1 , X2 , . . . }, and its counting process {N (t); t ≥ 0} As illustrated in Figure 2.1, any arrival process can also be speciﬁed by two other stochastic processes. The ﬁrst is the sequence of interarrival times, X1 , X2 , . . . ,. These are positive rv’s deﬁned in terms of the arrival epochs by X1 = S1 and Xi = Si − Si−1 for i > 1. Similarly, given the Xi , the arrival epochs Si are speciﬁed as Xn Sn = Xi . (2.1) i=1 Thus the joint distribution of X1 , . . . , Xn for all n > 1 is suﬃcient (in principle) to specify the arrival process. Since the interarrival times are IID, it is usually much easier to specify the joint distribution of the Xi than of the Si . The second alternative to specify an arrival process is the counting process N (t), where for each t > 0, the rv N (t) is the number of arrivals up to and including time t. The counting process {N (t); t > 0}, illustrated in Figure 2.1, is an uncountably inﬁnite family of rv’s {N (t); t ≥ 0} where N (t), for each t > 0, is the number of arrivals in the interval (0, t]. Whether the end points are included in these intervals is sometimes important, and we use parentheses to represent intervals without end points and square brackets to represent inclusion of the end point. Thus (a, b) denotes the interval {t : a < t < b}, and (a, b] denotes {t : a < t ≤ b}. The counting rv’s N (t) for each t > 0 are then deﬁned as the number of arrivals in the interval (0, t]. N (0) is deﬁned to be 0 with probability 1, which means, as before, that we are considering only arrivals at strictly positive times. The counting process {N (t), t ≥ 0} for any arrival process has the properties that N (τ ) ≥ N (t) for all τ ≥ t > 0 (i.e., N (τ ) − N (t) is a non-negative random variable). 60 CHAPTER 2. POISSON PROCESSES For any given integer n ≥ 1 and time t ≥ 0, the nth arrival epoch, Sn , and the counting random variable, N (t), are related by {Sn ≤ t} = {N (t) ≥ n}. (2.2) To see this, note that {Sn ≤ t} is the event that the nth arrival occurs by time t. This...
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## This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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