Transcription of 1 IEOR 6711: Notes on the Poisson Process
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1 IEOR 6711: Notes on the Poisson Process
Copyright c 2009 by Karl Sigman 1 IEOR 6711 : Notes on the Poisson Process We present here the essentials of the Poisson point Process with its many interesting properties. As preliminaries, we first define what a point Process is, define the renewal point Process and state and prove the Elementary Renewal Theorem. Point Processes Definition A simple point Process = {tn : n 1} is a sequence of strictly increas- ing points 0 < t1 < t2 < , (1). def with tn as n . With N (0) = 0 we let N (t) denote the number of points that fall in the interval (0, t]; N (t) = max{n : tn t}. {N (t) : t 0} is called the counting Process for . If the tn are random variables then is called a random point Process . def We sometimes allow a point t0 at the origin and define t0 = 0. Xn = tn tn 1 , n 1, is called the nth interarrival time. We view t as time and view tn as the nth arrival time (although there are other kinds of applications in which the points tn denote locations in space as opposed to time).)
We view tas time and view t n as the nth arrival time (although there are other kinds of applications in which the points t n denote locations in space as opposed to time). The word simple refers to the fact that we are not allowing more than one arrival to ocurr at the same time (as is stated precisely in (1)). In many applications there is a ...
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