Iterative Methods for Queueing Models with Batch Arrivals

Raymond H. Chan and Wai-ki Ching
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Raymond H. Chan: Chinese University of Hong Kong, Department of Mathematics and Department of Systems Engineering
Wai-ki Ching: Chinese University of Hong Kong, Department of Mathematics and Department of Systems Engineering

Chapter 6 in Computations with Markov Chains, 1995, pp 81-93 from Springer

Abstract: Abstract We consider finding the stationary probability distribution vectors of Markovian queueing models having batch arrivals by using the preconditioned conjugate gradient (PCG) method. The preconditioners are constructed by exploiting the near-Toeplitz structure of the generator matrix of the model and are products of circulant matrices and band-Toeplitz matrices. We prove that if the number of servers s is fixed independent of the queue size n, then for sufficiently large n, the preconditioners are invertible and the preconditioned systems have singular values clustered around 1. Hence if the systems are solved by the preconditioned conjugate gradient method, we expect superlinear convergence. Our numerical results show that the PCG method with our preconditioner converges in finite number of steps independent of n and s whereas the numbers required by the Jacobi method or the PCG method without any preconditioner increase like O(n).

Keywords: Conjugate Gradient Method; Queue Size; Toeplitz Matrix; Toeplitz Matrice; Circulant Matrix (search for similar items in EconPapers)
Date: 1995
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DOI: 10.1007/978-1-4615-2241-6_6

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