System-theoretical algorithmic solution to waiting times in semi-Markov queues

Cataloged from PDF version of article.Markov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples. © 2009 Elsevier B.V. All rights reserve

Traffic matrix estimation in IP networks

An Origin-Destination (OD) traffic matrix provides a major input to the design, planning and management of a telecommunications network. Since the Internet is being proposed as the principal delivery mechanism for telecommunications traffic at the present time, and this network is not owned or managed by a single entity, there are significant challenges for network planners and managers needing to determine equipment and topology configurations for the various sections of the Internet that are currently the responsibility of ISPs and traditional telcos. Planning of these sub-networks typically requires a traffic matrix of demands that is then used to infer the flows on the administrator's network. Unfortunately, computation of the traffic matrix from measurements of individual flows is extremely difficult due to the fact that the problem formulation generally leads to the need to solve an under-determined system of equations. Thus, there has been a major effort from among researchers to obtain the traffic matrix using various inference techniques. The major contribution of this thesis is the development of inference techniques for traffic matrix estimation problem according to three different approaches, viz: (1) deterministic, (2) statistical, and (3) dynamic approaches. Firstly, for the deterministic approach, the traffic matrix estimation problem is formulated as a nonlinear optimization problem based on the generalized Kruithof approach which uses the Kullback distance to measure the probabilistic distance between two traffic matrices. In addition, an algorithm using the Affine scaling method is developed to solve the constrained optimization problem. Secondly, for the statistical approach, a series of traffic matrices are obtained by applying a standard deterministic approach. The components of these matrices represent estimates of the volumes of flows being exchanged between all pairs of nodes at the respective measurement points and they form a stochastic counting process. Then, a Markovian Arrival Process of order two (MAP-2) is applied to model the counting processes formed from this series of estimated traffic matrices. Thirdly, for the dynamic approach, the dual problem of the multi-commodity flow problem is formulated to obtain a set of link weights. The new weight set enables flows to be rerouted along new paths, which create new constraints to overcome the under-determined nature of traffic matrix estimation. Since a weight change disturbs a network, the impact of weight changes on the network is investigated by using simulation based on the well-known ns2 simulator package. Finally, we introduce two network applications that make use of the deterministic and the statistical approaches to obtain a traffic matrix respectively and also describe a scenario for the use of the dynamic approach