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A Stochastic Model with an Adaptive Proportional Controller for the Evolution of User-Router Bandwidth Demand for Quality of Service (QoS) Aspects
Iordanis K. Giannopoulos, Apostolos P. Leros, Assimakis K. Leros and Georgios Tsaramirsis

The router through traffic load tools which monitor network links, such as Multi Router Traffic Grapher (MRTG), gives information for the aggregate or cumulative sum (CUMSUM) values of the user’s bandwidth demand at the end of regular intervals, starting with a low handshake value at the beginning of every interval. However, there is no information on how this CUMSUM value evolves to its final value within each time interval. This paper presents a coarse methodology to model and control with an adaptive proportional controller the evolution of the user-router bandwidth demand within the interval given the CUMSUM value at the end of the interval. This modelling methodology is based on the concept of the mean reverting stochastic process (MRSP) with some modifications (M-MRSP) to accommodate the user-router behaviour within each time interval. Simulation results are presented to validate the M-MRSP evolution of the CUMSUM value along with the key characteristics and properties of the model. The CUMSUM bandwidth demand data generated by the M-MRSP model controlled adaptively by a proportional controller within the interval is statistically analyzed to determine whether it possesses the characteristics of non-stationarity, fat/short tails, and long range dependency (LRD), which as reported in the literature are present in real user-router data. This statistical analysis is based on the graphical techniques of histogram, QQ-plot, and autocorrelation function plots. The M-MRSP evolution of the CUMSUM value can be exploited by the network administrators to develop their own quality of service (QoS) policies and apply possible corrective actions within each time interval instead of waiting for the router to provide this information at the end of each interval. A suggestion for such a QoS exploitation is provided.

Keywords: Router bandwidth, cumulative sum, mean reverting stochastic process, proportional control, binomial-uniform distributions, quality of service (QoS), histogram, QQ-plot, autocorrelation function, non-stationarity, fat/short tails, long range dependency (LRD).

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