## New traffic model on the base of fractional calculusV.Zaborovski (vlad@stu.neva.ru)State Technical University of Saint-Petersburg, Russia Y.Podgurski (podg@stu.neva.ru)Institute of Robotics and Technical Cybernetics, Russia S.Yegorov (ye@stu.neva.ru)State Technical University of Saint-Petersburg, Russia
The model represents network traffic as a stochastic process with possible infinite mean delay is proposed. Such model may be used to explain the appearance of long-range dependency and fractal-like feature of a network data flow. The formal relationship between heavy-tailed delay distributions, hyperbolically decaying of the packet delay auto-covariance function and fractional differential equation is shown. The new interpretation of fractional calculus opens up the new area of using this well-developed mathematical tool in order to understand the local and global characteristics of the packet traffic behaviour.
Recent measurements of network traffic have shown that traffic characteristics include features that are more
efficiently described in terms of fractal than conventional stochastic processes. Fractal dimension and long-range
dependency in statistical moments have been found in corporate, local and wide-area networks [1-3].
As a result a correlation function of network process
R(k)~Ak^{-b},
where
A functional behaviour one can be find in different areas of natural science. In several publications its
authors try to explain dependence like (1) on the base of analysis of the diffusion processes and describes
the evolution of the some system with a information loss property. In such case, the fractional parameter
The QoS engineering for Internet services require the adequate models of a virtual connection, formed by packet flow. In general case such connection consists of several intermediate nodes and links. Without loss of generality we shall assume that the connection under investigation is created by the TCP protocol and there is end-to-end transmission control. Despite of it we shall consider processes in intermediate sites and communication lines (Fig.1), while constructing the packet flow model. These processes show themselves in different types of packet delays and the actions of packet drops. Let us turn our mind to the packet delivery time. At detailed elaboration up to a level of intermediate sites (Fig.2b), the sender-to-receiver packet transmission time T may be presented as sum of packet delays at each hop.
where i and i+1
(link propagation delay), t- processing/buffering delay of a packet in the node
_{i}^{B}i, M - total number of nodes in the connection.
In the modern TCP/IP networks one can consider the link propagation delay
t.
^{B}
The packet may be dropped due to buffer overflow or routing error in the intermediate node. From the
receiver point of view the intermediate site acts as a "trap" of packages, and detains them forever. In
this case one may assign to
The virtual channel is given by a finite sequence of intermediate nodes
For construction of mathematical model of such processes we shall enter function of probability
density
It appears, that exactly such structure of probability density function characterises the long range
statistical dependence, which are widely observed in computer networks [1-3, 7]. For the further
construction of model we shall enter function
where - time of a presence of a packet
in the
Let's evaluate the most probable number of packets in the site
where x before
the packets arrival from the x-1 site.
The value In these designations the equation of packet migration may be represented as
where the left part of the equation (6) is the fractional derivative from function
Taking into account the discrete character of change of coordinate n, _{0}(k)=0k=1,2,...,
Then we shall receive
The dependences
The dependency
For
Thus, the calculations can be continued for any
For the initial conditions
This expression shows, that the correlation function has hyperbolically decaying concerning change of
characteristic for processes with long range dependency and asymptotic second-order self-similarity.
The behaviours of D(t)/n
as functions of time _{0}^{2} t are shown at the Fig.5 and Fig.6.
Thus, the properties of the network traffic, widely discussed in the modern literature, can directly be connected with the packet switching methods, used in computer network. The processes of packet transfer can be described with operator equations, which decision turns out with use of the fractional calculus tool. Using the temporal asymptotic decisions, differential equations with fractional derivatives can be formulated, that allows to receive expressions for the second order statistical moments of arising processes. Distinctive feature of such traffic characteristics is the hyperbolically decaying of their correlation functions which should be taken into account at construction of control systems of the traffic.
We have shown, that the assumption about a possibility of the indefinitely large packet delay in the intermediate node in the packet switching networks adequate models the dropping of physical packets in the node. The "heavy-tailed" function of probability density of a packet buffering delay satisfies to this assumption. We have shown, that dynamics of the packet passing in the virtual connection in TCP/IP networks is described by the equations in fractional derivative and corresponds to processes with long-range-dependency. We hope the usage of the well-developed tool of fractional calculus will promote understanding how packet traffic behaves. We assume, that LRD property noticed at measurement of the traffic in computer networks, is general property, characteristic for processes of carry in the mediums and systems with losses.
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Figure 1. Packet delay/drop processes on a TCP connection. Figure 2. Packet transfer. Figure 3. Figure 4. Figure 5. Figure 6. |