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1. E. Kuumola, Modeling the joint dynamics of instantaneous and exponentially averaged queue lengths, Masters Thesis, Networking Laboratory, Helsinki University of Technology, 2002 (bib)

   Abstract: Differentiated Services architecture is proposed to provide a variety of quality of service levels over the packet switched Internet network. In the DiffServ network complex flow measuring and classification functions are implemented at the network boundary nodes. At the interior nodes the distinctive for- warding behavior between flow aggregates is obtained with simple Per-Hop Behavior (PHB) packet scheduling mechanisms. The effect of the suggested PHB mechanisms and their parameterization on the traffic quality issues has still remained an open question. The focus of the thesis is on constructing a modeling methodology for the Assured Forwarding scheme that is one of the suggested PHB mechanisms for the DiffServ architecture. The framework for the AF model is adapted from the previous approaches of modeling DiffServ PHB mechanisms. The model aims at capturing the packet level dynamics of the AF buffer in order to evaluate the effect of the AF parameterization on the traffic QoS measures. The joint dynamics of instantaneous and exponentially averaged queue length involved in the Random Early Detection (RED) congestion control mechanism of the AF buffer is modeled using Markovian fluid queue models. The time evolution of the joint distribution functions of the instantaneous and averaged queue length is governed by Kolmogrov equations that reduce to an ordinary differential equation system in the stationary case. An analytical solution for the equations is derived in a few special cases, though the general solution is not reached. Solving the ODE system numerically with traditional integration schemes turns out to be unstable. Therefore, two new numerical approaches, method of characteristics and embedded process ap- proach, are constructed. The approaches are based on following the time evolution of the distribution functions until the solution converges to the sta- tionary state. The numerical methods are briefly evaluated and compared. Based on the results both methods are found to give the stationary distribu- tions accurately. Two extensions for the basic model are briefly illustrated. The Markov Mod- ulated Poisson process is attached to the basic model to capture the behavior of bursty traffic sources. The second extension illustrates how the basic model can be extended to buffers with coupled queues, such as the AF buffer. The nu- merical methods can be used with slight modifications also with the extended models.