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Unformatted text preview: Copyright c 2009 by Karl Sigman 1 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. 1.1 Point Processes Definition 1.1 A simple point process = { t n : n 1 } is a sequence of strictly increas ing points < t 1 < t 2 < , (1) with t n as n . With N (0) def = 0 we let N ( t ) denote the number of points that fall in the interval (0 ,t ] ; N ( t ) = max { n : t n t } . { N ( t ) : t } is called the counting process for . If the t n are random variables then is called a random point process. We sometimes allow a point t at the origin and define t def = 0 . X n = t n t n 1 , n 1 , is called the n th interarrival time. We view t as time and view t n as the n th 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 system to which customers are arriving over time (classroom, bank, hospital, supermarket, airport, etc.), and { t n } denotes the arrival times of these customers to the system. But { t n } could also represent the times at which phone calls are received by a given phone, the times at which jobs are sent to a printer in a computer network, the times at which a claim is made against an insurance company, the times at which one receives or sends email, the times at which one sells or buys stock, the times at which a given web site receives hits, or the times at which subways arrive to a station. Note that t n = X 1 + + X n , n 1 , the n th arrival time is the sum of the first n interarrival times. Also note that the event { N ( t ) = 0 } can be equivalently represented by the event { t 1 > t } , and more generally { N ( t ) = n } = { t n t,t n +1 > t } , n 1 . In particular, for a random point process, P ( N ( t ) = 0) = P ( t 1 > t ). 1 1.2 Renewal process A random point process = { t n } for which the interarrival times { X n } form an i.i.d. sequence is called a renewal process . t n is then called the n th renewal epoch and F ( x ) = P ( X x ) , x , denotes the common interarrival time distribution. To avoid trivialities we always assume that F (0) < 1, hence ensuring that wp1, t n . The rate of the renewal process is defined as def = 1 /E ( X ) which is justified by Theorem 1.1 (Elementary Renewal Theorem (ERT)) For a renewal process, lim t N ( t ) t = w . p . 1 ....
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This note was uploaded on 10/16/2010 for the course IEOR 4701 taught by Professor Karlsigma during the Summer '10 term at Columbia.
 Summer '10
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