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.)
1.1 Point Processes De nition 1.1 A simple point process = ft n: n 1gis a sequence of strictly increas-ing points 0 <t 1 <t 2 < ; (1) with t n!1 as n!1 . With N(0) def= 0 we let N(t) denote the number of points that fall in the interval (0;t]; N(t) = maxfn: t n tg. fN(t) : t 0gis called the counting process for . If the t
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