Fractional Poisson process

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Abstract

A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process.

As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.

Introduction

The main experimentally observed features of anomalous kinetic phenomena in complex systems are non-exponential time and non-Gaussian space patterns [1], [2]. To describe the patterns fractional generalizations of the diffusion, diffusion–advection and Fokker–Planck type equations have been developed and studied recently [1], [2], [3]. From physical point of view the non-exponential evolution is caused by long-run memory effects in complex systems. From mathematical point of view fractional generalization of the kinetic equations results from substitution instead space and time derivatives the derivatives of fractional order. The current status and history of the fractional kinetic problem are presented in recent reviews [1], [2].

One of important feature of statistical analysis of a counting random process is analysis of statistics of interarrival times. It is well known that the Poisson model predicts exponential probability distribution of interarrival times (see, for example [4]). Recently the empirical observations of failure of the Poisson model were found. The sizes (in number of bytes) or durations (measured in seconds) of a set of the Web network sessions or connections exhibit the long-tailed property, see, for example Fig. 5 in [5] and references there. In other words, the probability of duration of network sessions decreases by power law at large session-times instead of exponential decay predicted by the standard Poisson model and well confirmed empirically for phone communication connections.

To understand the origin of the observed power law asymptotic behavior of probability distribution function of interarrival times we propose non-Markov fractional Poisson model based on fractional generalization of the Kolmogorov–Feller equation [3], [6], [9].

To explain where the fractional Kolmogorov–Feller equation comes from let us remind the standard Kolmogorov–Feller equation for probability distribution function P(x,t) [3], [4]P(x,t)t=∫−∞dyw(y)[P(x−y,t)−P(x,t)],P(x,t=0)=δ(x),here w(y) is probability density “to make a step” of the length y.

From point of view of the Montroll–Weiss continues time random walk (CTRW) model [1], [6], [7], [8], [9] the Kolmogorov–Feller equation (1) belongs to the type of master equations and describes situation when each consequent step of random length y is made after random waiting time t. Moreover randomness of step length is distributed in accordance with w(y) while waiting time t has exponential distribution ψ(t)ψ(t)=e−t,t⩾0.Therefore, the exponential waiting time distribution ψ(t) is the origin for the first order time derivative in the left side of Eq. (1). Fractional generalization of Eq. (1) is based on the fractional generalization of the waiting time distribution ψ(t). In [3] the fractional waiting time distribution function ψμ(t) was proposed as the following integral (see Eq. (7.6) Ref. [3])ψμ(t)=sinπμπ0dρe−ρtρμ−μ+2cosπμ,0<μ⩽1,and called a fractional Poissonian distribution.

When the parameter μ≠1 the fractional waiting time distribution becomes broader and possesses of non-exponential power-law behavior at large t. The spreading of the waiting time distribution leads to non-Markovian non-exponential evolution, the latter being a typical manifestation of temporal phenomena inherent in complex physical systems.

In this paper we have developed and elaborated the fractional Poisson distribution based on the fractional generalization of the Kolmogorov–Feller equation. We take the special form of w(y), w(y)=νδ(y−1), where δ is delta function and the parameter ν has physical dimension [ν]=secμ (see Eq. (19)). It results to probability distribution function of a counting process when the total number of “items” that have arrived up to time t is governed by the fractional stream. In comparison with the standard Poisson process the developed model includes additional parameter μ, 0<μ⩽1. Thus, the model provides fractional generalization of the standard Poisson process, to which it reduces at μ=1.

The paper is organized as follows.

In Section 1 we explain how and where the fractional Poisson comes from. The basic definitions of the standard Poisson random process are reminded in Section 2. In Section 3 we define the fractional Poisson model, obtain new probability distribution function Pμ(n,t), evaluate mean and variance, find probability distribution function of interarrival times. Fractional compound Poisson process is defined and elaborated in Section 4 as an application of the developed model. In Section 5 we discuss the relationships between the developed fractional model and the standard Poisson random process.

Section snippets

Generation function

The standard Poisson process is concerned with the distribution of arrivals under applicable assumptions. The probability of a single arrival during a small time interval Δt is nΔt, with rate n and more than a single arrival during Δt is negligible. Let P(n,t) be the probability of n items having arrived by time t. The probability P(n,t) satisfies the normalizing condition ∑n=0P(n,t)=1 because either nothing arrived or something must have arrived by time t. To see what happens during the

Generation function

We introduce the fractional Poisson process as the counting process with probability Pμ(n,t) of arriving n items (n=0,1,2,…) by time t. The probability Pμ(n,t) is governed by the following special form of the fractional Kolmogorov–Feller equation:0DtμPμ(n,t)=νPμ(n−1,t)−Pμ(n,t)+t−μΓ(1−μ)δn,0,0<μ⩽1,with normalization conditionn=0Pμ(n,t)=1,where the operator of fractional derivation 0Dtμ is defined as the Riemann–Liouville fractional integral,

Fractional compound Poisson process

We call stochastic process {X(t), t⩾0} a fractional compound Poisson process if it is represented byX(t)=∑i=1N(t)Yi,where {N(t), t⩾0} is a fractional Poisson process, and {Yi, i=1,2,…} is a family of independent and identically distributed random variables with probability distribution function p(Y) for each Yi. The process {N(t), t⩾0} and the sequence {Yi, i=1,2,…} are assumed to be independent.

We now calculate the moment generation function Jμ(s,t) of fractional compound Poisson processJμ

Conclusions

To explain empirically observed power law asymptotic behavior of probability distribution function of interarrival times we propose non-Markov fractional Poisson model based on fractional generalization of the Kolmogorov–Feller equation. We obtain analytical expression for the probability Pμ(n,t) (see Eq. (25)) that in the time interval [0,t] we observe n events governed by fractional Poisson stream with fractality parameter μ, 0<μ⩽1. The probability Pμ(n,t) is fractional generalization of the

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