The application of non-linear dynamics to teletraffic modelling.

Telecoms Research Group. Queen Mary and Westfield College, University of London 3 Abstract Self-similar traffic has been observed in teletraffic networks over all time scales of engineering interest. This type of traffic has no characteristic time scale due to its burstiness and causes the network buffers to overflow affecting Quality of Service (QoS). Self-similar traffic has been modelled via stochastic methods. It has also been modelled using non-linear dynamics. In this thesis we use techniques from non-linear dynamics in the teletraffic modelling of modern packetised telecommunications networks. We develop a novel teletraffic framework for the modelling of self-similar traffic in a parsimonious, parameterisable and predictable manner based on the use of non-linear dynamics models in the form of a chaotic map family. This family consists of models, based on intermittency maps, for the accelerated simulation of self-similar behaviour of individual sources and aggregated traffic in such networks. We have significantly extended the characterisation of the individual source map models of Pruthi and Erramilli. The extension accounts for the impact of all five parameters (ε and m for both states, and d) on H, the parameterisation for load via the invariant density, and the parameterisation for heavy tailed sojourn times in the ON and OFF states via the transit-time. These new aggregate traffic models provide up to two orders of magnitude speed-up over FBM/FGN and Pruthi’s N one-step methods. We perform mathematical analysis of the map family with respect to H proving the conjecture put forward by Pruthi that asymptotically H is only dependent on the dominant value of m. Numerical results show that convergence is slow and that for the coupled map H differs substantially from the theory. However the deviation from the theoretical is predictable and this leads to an empirical fit for the asymptotic dependence of H on m. These results also show that the underlying dynamics of the map persist in all of the map interpretations. The numerical results also show limitations of parameter ranges on H, particularly for d (0.1 < d <0.9). However, this limit can be overcome practically to some degree by manipulating the time resolution of the iterates. Transit-time analysis of the map family highlights a further parameter limitation which stems from ε: any value of ε >0 effectively limits the range of time-scales over which LRD occurs. These numerical limitations apply to all map interpretations. We have developed a method of measuring H via the map’s variance that is promising for measuring H on-line. We have also found, by comparing for accuracy this and the AbryVeitch method for measuring H, that H by itself as a parameter for modelling self-similar traffic in a “parsimonious” manner may not be enough. This conclusion is drawn from observing different queueing behaviour with input traffic having the same H. This leads to the suggestion that reliance on a single method for determining H on-line may prove unwise. This comparison also shows the flexibility that these map models have in specifying key LRD behaviour that determines the impact on queueing. This flexibility is derived from the intuitive relationship that these map models have to their underlying physical ON-OFF process. The Application of Non-linear Dynamics to Teletraffic Modelling Telecoms Research Group. Queen Mary and Westfield College, University of London 4 Acknowledgments There are a number of people to who I am indebted. Undertaking a PhD is not easy especially for those connected indirectly to the task, in particular my wife Rosi: a constant source of encouragement and an outstanding display of patience well beyond the call of duty. For those who are directly connected to the task I especially would like to thank Dr Raul Mondragón. His mathematical supervision, encouragement, enthusiasm and friendship has been a constant source of inspiration throughout this PhD. Dr Jonathan Pitts, his engineering supervision has always ensured that at least one leg be fixed to the ground, and above all ensured timely completion of the project. Professor David Arrowsmith and Dr John Schormans, for their enlivening conversations in, on and around the subject. Professor Laurie Cuthbert, paymaster general to the Telecoms Group, for that essential ingredient without which this project would surely not have started: finding funding. EPSRC and NORTEL for financial support. Dr Dragan Boscovic of Motorola for encouraging me to rise up to the task. My colleagues in the Lab: Arif Al-Hammadi, Husam Awadalla, Eliane Bodanese, Babul Miah, Sutha Sivagnanasundaram, Steven Winstanley, and Tijana Timotijevic, for hours of endless fun. A special thank you goes to Rebecca Whiting and Arif Al-Hammadi, for without the use of their lap tops this project would certainly not have finished on time. Finally, I would like to thank the Royal Navy: In their infinite wisdom they decreed that I did not posses the intelligence to complete a degree, let alone a PhD. It is comforting to know that such an august body can still make errors in judgement. Stand fast Captain Mike Bowker (RN). The Application of Non-linear Dynamics to Teletraffic Modelling Telecoms Research Group. Queen Mary and Westfield College, University of London 5 Abbreviations and Acronyms ABR Available Bit Rate ATM Asynchronous Transfer mode CML Coupled Map Lattices FBM Fractional Brownian Motion FGN Fractional Gaussian Noise FPDI Floating Point Double Intermittency map