On a Bicriterion Server Allocation Problem for a Multidimensional Erlang Loss System

In this work an optimization problem on a classical elementary stochastic system system, modeled as an Erlang-B (M/M/x) loss system, is formulated by using a bicriteria approach. The problem is focused on the allocation of a given total of k servers to a number of groups of servers capable of carrying certain offered traffic processes assumed as Poissonian in nature. Two main objectives are present in this formulation. Firstly a criterion of equity in the grade of service, measured by the call blocking probabilities, entails that the absolute difference between the blocking probabilities experienced by the calls in the different service groups must be as small as possible. Secondly a criterion of system economic performance optimization requires the total traffic carried by the system, to be maximized. Relevant mathematical results characterizing the two objective functions and the set N of the non-dominated solutions, are presented. An algorithm for traveling on N based on the resolution of single criterion convex problems, using a Newton-Raphson method, is also proposed. In each iteration the two first derivatives of the Erlang-B function in the number of circuits (a difficult numerical problem) are calculated using a method earlier proposed. Some computational results are also presented

On Localized Control in QoS Routing

In this note, we study several issues in the design of localized quality-of-service (QoS) routing schemes that make routing decisions based on locally collected QoS state information (i.e., there is no network-wide information exchange among routers). In particular, we investigate the granularity of local QoS state information and its impact on the design of localized QoS routing schemes from a theoretical perspective. We develop two theoretical models for studying localized proportional routing: one using the link-level information and the other using path-level information. We compare the performance of these localized proportional routing models with that of a global optimal proportional model that has knowledge of the global network QoS state. We demonstrate that using only coarser-grain path-level information it is possible to obtain near-optimal proportions. We then discuss the issues involved in implementation of localized proportional routing and present some practical schemes that are simple and easy to implement

Equidade e eficiência na optimização de sistemas multidimensionais de Erlang-B

Doutoramento em MatemáticaFormula-se um problema de optimização relativo à afectação de servidores entre grupos funcionais de Erlang-B (M/M/n/0), que ocorre na modelação de sistemas de Teletráfego. Na prática, há normalmente dois objectivos a atingir no dimensionamento de tais sistemas: Maximizar o volume de tráfego transportado (minimizando o tráfego perdido pelo sistema global) e optimizar o grau de serviço (minimizando as probabilidades de bloqueio). Analisa-se o conflito potencial entre os dois objectivos e caracteriza-se o conjunto N das soluções não dominadas do problema. A abordagem do problema de optimização é antecedida de um estudo numérico (cálculo das derivadas da função de Erlang-B) e analítico (propriedades de convexidade do modelo). Estabelecem-se dois métodos numéricos de cálculo das derivadas parciais de primeira e segunda ordem na variável número de servidores (Método da Recursão Completa Generalizada e Método da Recursão Reduzida). Uma análise do erro propagado nessas recursões mostra a notável estabilidade numérica dos processos de cálculo. Extensos resultados computacionais mostram a grande eficiência dos métodos propostos, mesmo para valores elevados dos argumentos. No que toca às propriedades analíticas, estabelece-se o sinal das derivadas parciais de segunda ordem da função tráfego de transbordo num sistema de Erlang-B. A convexidade em cada uma das variáveis é demonstrada no sentido estrito e em todo o domínio, generalizando resultados já conhecidos. É proposto um algoritmo para percorrer N, que usa o método de Newton combinado com uma estratégia especial de determinação das restrições que estão activas em cada óptimo de Pareto.An optimization problem related to the server allocation between Erlang-B functional groups is formulated. This problem occurs in Teletraffic Systems modeling. In applied situations, the design of such a system usually involves two objectives: to minimize the overflow traffic of the global system and to minimize the blocking probabilities. The potential conflict between these objectives is shown and the set N of the non inferior solutions of the problem is characterized. An algorithm for traveling on N is given. This algorithm uses the Newtow- Raphson method together with a special strategy for determining the set of active restrictions for each Pareto optimal solution. Previously to the explanation of this optimization problem, a numerical and analytic study of the model, including the determination of the Erlang-B function derivatives and convexity properties is presented. Two methods are proposed for the numerical evaluation of the first and second order derivatives of the Erlang-B function: Complete Generalized Recursion Method and Reduced Recursion Method. A good numerical stability of both methods is ensured by including a detailed error analysis. Furthermore, extensive computations show the great efficiency of those methods even for high argument values. Concerning the analytic properties, the sign of the second order partial derivatives of the overflow traffic function is established and the strict convexity in each variables is shown