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poisson

# poisson - 1 Poisson Processes An arrival is simply an...

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1 Poisson Processes An arrival is simply an occurance of some event — like a phone call, a job o ff er, or whatever — that happens at a particular point in time. We want to talk about a class of continuous time stochastic processes called arrival processes that describe when and how these events occur. Let be a sample space and P a probability. For any ω , for all t we de fi ne N t ( ω ) as the number of arrivals in the time interval [0 , t ] given the realization ω . We call N = { N t , t 0 } an arrivial process. Clearly, as time evolves N t ( ω ) jumps up by integer amounts with new arrivals. The type of arrivial process we are most interested in is called a Poisson process . A Poisson process is de fi ned as an arrival process that satis fi es the following three axioms: 1. for almost all ω , each jump is of size 1; 2. for all t, s > 0 , N t + s N t is independent of the history up to t , { N u , u t } ; 3. for all t, s > 0 , N t + s N t is independent of t . The fi rst axiom says that there is a zero probability of two arrivals at the exact same instant in time; the second says that the number of arrivals in the future is independant of what happened in the past; and the third says the number of arrivals in the future is indentically distributed over time, or that the process is stationary. What is interesting is that these simple qualitative feeatures of the process imply the following: Lemma 1 If N is a Poisson process then for all t 0 , P ( N t = 0) = e λt for some λ 0 . Proof. By the independence axiom, P ( N t + s = 0) = P ( N t = 0) P ( N t + s N t = 0) . By the stationarity axiom, P ( N t + s N t = 0) = P ( N s = 0) . Hence, if we let f ( t ) = P ( N t = 0) , we have just established f ( t + s ) = f ( t ) f ( s ) .

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poisson - 1 Poisson Processes An arrival is simply an...

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