Possibility of fractional calculus application for telecommunication traffic modelling

Abstract

Fractional calculus is a field of mathematical analysis concerned with research and application of derivatives and integrals of an arbitrary order. Many famous mathematicians studied the theory of fractional calculus such as Euler, Riemann, Liouville, Abel, Fourier and others. There are many proposed definitions for calculating derivatives and integrals of non-integer order. In this paper, several proposed definitions along with basic statements of fractional calculus are presented with an emphasis on a possibility of fractional calculus application in telecommunication traffic modelling.

The fact is that fractional calculus is widely used in various scientific disciplines in recent decades. Models based on fractional calculus have proved to be very useful in physics, mechanics, electrical engineering, biochemistry, medicine, economy, and probability theory. This paper analyses a possibility of application of fractional calculus for modelling telecommunication traffic. Many research studies have shown that traffic characteristics at a local and global level, such as self-similarity and long range dependence, can efficiently be described by fractional calculus instead of using conventional stochastic processes. Some proposed models based on fractional calculus that describe phenomena present in modern telecommunication networks are presented in this paper.

Introduction

Measurements and statistic analyses of telecommunication traffic have discovered that traffic in packet switched networks shows significant irregularities – burstiness, both in terms of traffic intensity variability and shape of autocorrelation function. It is noticed that traffic has fractional characteristics – self-similarity and long range dependence. As a result, a large bandwidth is required, and very often, this is one of the causes of network inefficiency.

In comparison with conventional models based on Poisson distribution widely used in circuit switched networks, in models based on self-similarity property there are problems that are difficult to predict, measure and control in telecommunication traffic. Thus, measurement, analysis and modelling of self-similar network traffic are a sort of challenge. Different research studies have shown that traffic in modern telecommunication networks can efficiently be described with statistical models based on fractional calculus.

Basic statements of fractional calculus

Although the term „fractional calculus”, is actually a misnomer, and a term „integration and differentiation of an arbitrary order” is more suitable, „fractional calculus” has been a common term since l'Hopital’s era. Fourier also studied derivatives and integrals of an arbitrary order. Abel applied fractional calculus for an integral equation which appears in a tautochrone problem formulation (find a shape of a curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point). The first serious attempt to obtain a logical definition of a fractional derivative belongs to Liouville. Let f be a locally integrable on (a, ∞). Usually, an n-fold iterated integration is marked as , and refers to as Riemann-Liouville fractional integral of order α of f. Definitions of integrals and derivatives of an arbitrary order can be consolidated into a differintegral. The process of obtaining a differintegral is referred to as fractional integro-differentiation.

Some proposed models of fractional calculus application for telecommunication traffic modelling

Recent research of telecommunication traffic has shown that it can efficiently be described by derivatives of an arbitrary order instead of conventional stochastic processes. A new interpretation of fractional calculus creates new fields of application of these mathematical tools in order to achieve understanding of local and global characteristics of network traffic. A fractional dimension and long range dependence can be seen in private, LAN and WAN networks.

Temporal-spatial model for packet transmission in TCP/IP networks

Defining Quality of Service parameters for Internet services requires appropriate virtual connections, depending on traffic flows. Generally, such a connection consists of several transit nodes and links. In modern TCP/IP networks, delay in a given virtual connection can be considered as a constant value. A buffering delay variation causes a propagation delay. The transmission process can be described by equations that can be solved by fractional calculus.

Fractal analysis modelling of telecommunication traffic-potentials and limitations

The fractal properties of telecommunication traffic in packet switched networks indicate the existence of periods of low or high activity concentrations in long time intervals. As a result, the correlation structure of traffic streams is in contradiction with common models. Therefore, the empirically confirmed fractal properties of aggregate traffic cannot be described by Poisson models. All streams are statistically multiplexed with local irregularities in nodes. Thus, each packet is stochastically delayed and non-trivial processes occur. In order to achieve a complete description of these processes, it is necessary to involve the properties existing in short and long intervals in such an analysis. The fractional analysis of packet flows in different time intervals ensures a qualitative and quantitative description of the telecommunication traffic fractional nature.

Conclusion

Fractional calculus has many applications. This paper is primarily concerned with fractional calculus application for telecommunication traffic modelling. It was shown that an indefinite packet delay in the transit node in packet switching networks adequately models the dropping of the packet in the node. The dynamics of packet transmission in virtual connections in TCP/IP networks can be described by the equations in fractional derivatives and it corresponds to processes with long-range-dependence. Also, fractional calculus can be a very useful tool in the research of local and global phenomena in telecommunication networks.