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Unformatted text preview: Copyright c 2010 by Karl Sigman 1 Renewal processes, Poisson processes, and com pound (batch) Poisson processes 1.1 Point Processes Definition 1.1 A simple point process ψ = { 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 ψ . If the t n are random variables then ψ 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. 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 made to a given phone or office, the times at which jobs are sent to a printer in a computer network, 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 ). 1.2 Renewal process A random point process ψ = { 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 ) denotes the common interarrival time distribution. The rate of the renewal process is defined as λ def = 1 /E ( X ). Such processes are very easy to simulate, assuming that one can generate samples from F which we shall assume is so here (for example by inversion, X = F 1 ( U )); the key point being that we can use the recursion t n +1 = t n + X n +1 , n ≥ 0, and thus simply generate the interarrival times sequentially: 1 Simulating a renewal process with interarrival time distribution F up to time T : 1. Set t = 0, N = 0 2. Generate an X distributed as F . 3. Set t = t + X . If t > T , then stop. 4. Set N = N + 1 and set t N = t ....
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 Fall '07
 sigman
 Probability theory, Poisson process, Tn

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