Analysis Of The Poisson Process

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A traditional traffic model for circuit switched data as well as packet data is the Poisson process in which the number of incoming packets per time is according to Poisson distribution. It is one of the most widely used and oldest traffic models. In this model the length of each packet is based on exponential distribution and the number of simultaneously ongoing packets is according to Erlang distribution [16].
The Poisson process is characterized as a renewal process. In a Poisson process the inter-arrival times are exponentially distributed with a rate parameter λ: P{An ≤ t} = 1 – exp(-λt).
The Poisson distribution is appropriate if the arrivals are from a large number of independent sources, referred to as Poisson sources. The distribution has a mean and variance equal to the parameter λ. The Poisson distribution can be visualized as a limiting form of the binomial distribution, and is also used widely in queueing models. There is a number of interesting mathematical properties exhibited by Poisson processes [16].Primarily, superposition of independent Poisson processes results in a new Poisson process whose rate is the sum of the rates of the independent Poisson processes. Further, the independent increment property renders a Poisson process memoryless. Poisson processes are common in traffic applications scenarios that comprise of a large number of independent traffic streams. The reason behind the usage stems from Palm's Theorem which states that under suitable conditions, such large number of independent multiplexed streams approach a Poisson process as the number of processes grows, but the individual rates decrease in order to keep the aggregate rate constant. Nevertheless, it is to be noted that traffic aggregation need not always result in a Poisson process. The two primary assumptions that the Poisson model makes are: 1. the number of sources is infinite 2. the traffic arrival pattern is random. The probability distribution function and density function of the model are given as: F(t) = 1 – e -λt f(t) = λ e -λt There are also other variations of the Poisson distributed process that are widely used. There are for example, the Homogeneous Poisson process and Non-Homogeneous Poisson process that are used to represent traffic characteristics. An interesting observation in case of Poisson models is that as the mean increases, the properties of
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The traditional assumption of Poisson arrivals has been often justified by arguing that the aggregation of many independent and identically distributed renewal processes tends to a Poisson process when the number increases
Poisson arrivals with mean rate λ are separated by inter arrival times Equivalently, the number of arrivals up to time t is a Markov birth process with all birth rates of λ.
The Poisson arrival process has several properties that make it appealing for analysis.
• It is memoryless in the sense that, given the previous arrival occurred T time ago, the time to the next arrival will be exponentially distributed with mean 1/λ regardless of T.
In other words, the waiting time for the next arrival is independent of the time of the previous arrival. This memoryless property simplifies analysis because future arrivals do not need to take into account the past history of the arrival process.
• The number of arrivals in any interval of length t will have a Poisson probability distribution with mean λt .
• The sum of two independent Poisson arrival processes with rates λ1 and λ2 , is a Poisson process with rate λ1+ λ2. This is convenient for analysis because traffic flows are multiplexed in a

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