A note on the computational cost of the linearizer algorithm for queueing networks

Linearizer is one of the best known approximation algorithms for obtaining numeric solutions for product form queueing networks. In the original exposition of Linarizer, the computational cost was started to be O(MK³) for a model with M queues and K job classes. We show in this note that with som stright forward algebraic manipulation Linearizer can be modified to require only O (MK²) computational cost.Linearizer é um dos mais conhecidos algoritmos de aproximação para se obter soluções numéricas para redes de filas em solução em forma de produto. Na explanação original de Linearizer, o custo computacional foi mostrado ser de O(MK³) para um modelo com M filas e K classes de jobs. Nesta nota mostramos que, com algumas manipulações algébricas simples, Linearizer pode ser modificado de tal maneira que o custo computacional seja reduzido para O(MK²)

A Unifying Framework for the Approximate Solution of Closed Multiclass Queuing Networks

Queuing network models of modern computing systems must consider a large number of components (e.g., Web servers, DB servers, application servers, firewall, routers, networks) and hundreds of customers with very different resource requirements. The complexity of such models makes the application of exact solution techniques prohibitively expensive, motivating research on approximate methods. This paper proposes an interpolation-matching framework that allows a unified view of approximate solution techniques for closed product-form queuing networks. Depending upon the interpolating functional form and the matching populations selected, a large versatile family of new approximations can be generated. It is shown that all the known approximation strategies, including Linearizer, are instances of the interpolation-matching framework. Furthermore, a new approximation technique, based on a third-order polynomial, is obtained using the interpolation-matching framework. The new technique is shown to be more accurate than other known methods

The MVA Priority Approximation

A Mean Value Analysis (MVA) approximation is presented for computing the average performance measures of closed-, open-, and mixed-type multiclass queuing networks containing Preemptive Resume (PR) and nonpreemptive Head-Of-Line (HOL) priority service centers. The approximation has essentially the same storage and computational requirements as MVA, thus allowing computationally efficient solutions of large priority queuing networks. The accuracy of the MVA approximation is systematically investigated and presented. It is shown that the approximation can compute the average performance measures of priority networks to within an accuracy of 5 percent for a large range of network parameter values. Accuracy of the method is shown to be superior to that of Sevcik's shadow approximation

Queueing networks: solutions and applications

During the pasttwo decades queueing network models have proven to be a versatile tool for computer system and computer communication system performance evaluation. This chapter provides a survey of th field with a particular emphasis on applications. We start with a brief historical retrospective which also servesto introduce the majr issues and application areas. Formal results for product form queuenig networks are reviewed with particular emphasis on the implications for computer systems modeling. Computation algorithms, sensitivity analysis and optimization techniques are among the topics covered. Many of the important applicationsof queueing networks are not amenableto exact analysis and an (often confusing) array of approximation methods have been developed over the years. A taxonomy of approximation methods is given and used as the basis for for surveing the major approximation methods that have been studied. The application of queueing network to a number of areas is surveyed, including computer system cpacity planning, packet switching networks, parallel processing, database systems and availability modeling.Durante as últimas duas décadas modelos de redes de filas provaram ser uma ferramenta versátil para avaliação de desempenho de sistemas de computação e sistemas de comunicação. Este capítulo faz um apanhado geral da área, com ênfase em aplicações. Começamos com uma breve retrospectiva histórica que serve também para introduzir os pontos mais importantes e as áreas de aplicação. Resultados formais para redes de filas em forma de produto são revisados com ênfase na modelagem de sistemas de computação. Algoritmos de computação, análise de sensibilidade e técnicas de otimização estão entre os tópicos revistos. Muitas dentre importantes aplicações de redes de filas não são tratáveis por análise exata e uma série (frequentemente confusa) de métodos de aproximação tem sido desenvolvida. Uma taxonomia de métodos de aproximação é dada e usada como base para revisão dos mais importantes métodos de aproximação propostos. Uma revisão das aplicações de redes de filas em um número de áreas é feita, incluindo planejamento de capacidade de sistemas de computação, redes de comunicação por chaveamento de pacotes, processamento paralelo, sistemas de bancos de dados e modelagem de confiabilidade

Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence

We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid, or customers, on each route eventually concentrates on the bottleneck queues only, and that the long-term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network.Comment: 22 page

Heavy-traffic revenue maximization in parallel multiclass queues

Motivated by revenue maximization in server farms with admission control, we investigate the optimal scheduling in parallel processor-sharing queues. Incoming customers are distinguished in multiple classes and we define revenue as a weighted sum of class throughputs. Under these assumptions, we describe a heavy-traffic limit for the revenue maximization problem and study the asymptotic properties of the optimization model as the number of clients increases. Our main result is a simple heuristic that is able to provide tight guarantees on the optimality gap of its solutions. In the general case with M queues and R classes, we prove that our heuristic is (1+1M-1)-competitive in heavy-traffic. Experimental results indicate that the proposed heuristic is remarkably accurate, despite its negligible computational costs, both in random instances and using service rates of a web application measured on multiple cloud deployments

Hierarchical Analyses Applied to Computer System Performance: Review and Call for Further Studies

We review studies based on analytic and simulation methods for hierarchical performance analysis of Queueing Network - QN models, which result in an order of magnitude reduction in performance evaluation cost with respect to simulation. The computational cost at the lower level is reduced when the computer system is modeled as a product-form QN. A Continuous Time Markov Chain - CTMC or discrete-event simulation can then be used at the higher level. We first consider a multiprogrammed transaction - txn processing system with Poisson arrivals and predeclared locks requests. Txn throughputs obtained by the analysis of multiprogrammed computer systems serve as the transition rates in a higher level CTMC to determine txn response times. We next analyze a task system where task precedence relationships are specified by a directed acyclic graph to determine its makespan. Task service demands are specified on the devices of a computer system. The composition of tasks in execution determines txn throughputs, which serve as transition rates among the states of the higher level CTMC model. As a third example we consider the hierarchical simulation of a timesharing system with two user classes. Txn throughputs in processing various combinations of requests are obtained by analyzing a closed product-form QN model. A discrete event simulator is provided. More detailed QN modeling parameters, such as the distribution of the number of cycles in central server model - CSM affects the performance of a fork/join queueing system. This detail can be taken into account in Schwetman's hybrid simulation method, which counts remaining cycles in CSM. We propose an extension to hybrid simulation to adjust job service demands according to elapsed time, rather than counting cycles. An example where Equilibrium Point Analysis to reduce computaional cost is privided