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Mathematical Problems in Engineering
Volume 2011, Article ID 702834, 18 pages
Research Article

Upper Bounds on Performance Measures of Heterogeneous 𝑀 / 𝑀 / 𝑐 Queues

1Centro de Estudos da Fala, Acústica, Linguagem e músicA, Departamento de Engenharia Eletrônica, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil
2Fundação Dom Cabral, 30140-083 Belo Horizonte, MG, Brazil
3Departamento de Estatística, Universidade Federal de Minas Gerais, 31270-901 Belo Horizonte, MG, Brazil
4Department of OMIT and the Research Center GREGHEC, HEC School of Management, 78351 Paris, France

Received 22 February 2011; Accepted 11 May 2011

Academic Editor: Ben T. Nohara

Copyright © 2011 F. S. Q. Alves et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In many real-life queueing systems, the servers are often heterogeneous, namely they work at different rates. This paper provides a simple method to compute tight upper bounds on two important performance measures of single-class heterogeneous multi-server Markovian queueing systems, namely the average number in queue and the average waiting time in queue. This method is based on an expansion of the state space that is followed by an approximate reduction of the state space, only considering the most probable states. In most cases tested, we were able to approximate the actual behavior of the system with smaller errors than those obtained from traditional homogeneous multiserver Markovian queues, as shown by GPSS simulations. In addition, we have correlated the quality of the approximation with the degree of heterogeneity of the system, which was evaluated using its Gini index. Finally, we have shown that the bounds are robust and still useful, even considering quite different allocation strategies. A large number of simulation results show the accuracy of the proposed method that is better than that of classical homogeneous multiserver Markovian formulae in many situations.