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Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Poisson process 1 Poisson process General Poisson process is one of the most important models used in queueing theory. • Often the arrival process of customers can be described by a Poisson process. • In teletraffic theory the “customers” may be calls or packets. Poisson process is a viable model when the calls or packets originate from a large population of independent users. In the following it is instructive to think that the Poisson process we consider represents discrete arrivals (of e.g. calls or packets). t í î ì N(t) t 1 t 2 Mathematically the process is described by the so called counter process N t or N ( t ). The counter tells the number of arrivals that have occurred in the interval (0 , t ) or, more generally, in the interval ( t 1 , t 2 ). N ( t ) = number of arrivals in the interval (0 , t ) (the stochastic process we consider) N ( t 1 , t 2 ) = number of arrival in the interval ( t 1 , t 2 ) (the increment process N ( t 2 ) N ( t 1 )) J. Virtamo 38.3143 Queueing Theory / Poisson process 2 General (continued) A Poisson process can be characterized in different ways: • Process of independent increments • Pure birth process – the arrival intensity λ (mean arrival rate; probability of arrival per time unit • The “most random” process with a given intensity λ J. Virtamo 38.3143 Queueing Theory / Poisson process 3 Definition The Poisson process can be defined in three different (but equivalent) ways: 1. Poisson process is a pure birth process: In an infinitesimal time interval dt there may occur only one arrival. This happens with the probability λdt independent of arrivals outside the interval. l í î ì dt l 2. The number of arrivals N ( t ) in a finite interval of length t obeys the Poisson( λt ) distribution, P { N ( t ) = n } = ( λt ) n n ! e λt Moreover, the number of arrivals N ( t 1 , t 2 ) and N ( t 3 , t 4 ) in nonoverlapping intervals ( t 1 ≤ t 2 ≤ t 3 ≤ t 4 ) are independent. í î ì í î ì l t 1 t 2 ~ Poisson( l t ) 1 ~ Poisson( l t ) 2 3. The interarrival times are independent and obey the Exp( λ ) distribution: P { interarrival time > t } = e λt í î ì l Exp( ~ l ) J. Virtamo 38.3143 Queueing Theory / Poisson process 4 The equivalence of the definitions The three definitions are equivalent: syntymäprosessi, l N(t) Poisson( l ~ t) väliajat Exp( l ~ ) 1 2 3 In the following we show the equivalence by showing the implications in the direction of the solid arrows. Then any of the three properties implies the other two ones. In fact, the implication 2 → 1 is not necessary for proving the equivalence (as it follows from the implications 2 → 3 and 3 → 1), but it can be shown very easily directly....
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This note was uploaded on 03/05/2010 for the course COMPUTER 343 taught by Professor Mac during the Spring '10 term at Punjab Engineering College.
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