Ching, Wai Ki, Chan, Raymond H. and Xun Yu, Zhou
(1997)
Circulant preconditioners for Markov-modulated Possion processes and their applications to manufacturing systems.
*SIAM Journal on Matrix Analysis and Applications*, 18 (2), 464-481.
(doi:10.1137/S0895479895293442).

## Abstract

The Markov-modulated Poisson process (MMPP) is a generalization of the Poisson process and is commonly used in modeling the input process of communication systems such as data traffic systems and ATM networks. In this paper, we give fast algorithms for solving queueing systems and manufacturing systems with MMPP inputs. We consider queueing systems where the input of the queues is a superposition of the MMPP which is still an MMPP. The generator matrices of these processes are tridiagonal block matrices with each diagonal block being a sum of tensor products of matrices. We are interested in finding the steady state probability distributions of these processes which are the normalized null vectors of their generator matrices. Classical iterative methods, such as the block Gauss--Seidel method, are usually employed to solve for the steady state probability distributions. They are easy to implement, but their convergence rates are slow in general. The number of iterations required for convergence increases like O(m), where m is the size of the waiting spaces in the queues. Here, we propose to use the preconditioned conjugate gradient method. We construct our preconditioners by taking circulant approximations of the tensor blocks of the generator matrices. We show that the number of iterations required for convergence increases at most like O(\log_2 m) for large m. Numerical results are given to illustrate the fast convergence.As an application, we apply the MMPP to model unreliable manufacturing systems. The production process consists of multiple parallel machines which produce one type of product. Each machine has exponentially distributed up time, down time, and processing time for one unit of product. The interarrival of a demand is exponentially distributed and finite backlog is allowed. We consider hedging point policy as the production control. The average running cost of the system can be written in terms of the steady state probability distribution. Our numerical algorithm developed for the queueing systems can be applied to obtain the steady state distribution for the system and hence the optimal hedging point. Furthermore, our method can be generalized to handle the case when the machines have a more general type of repairing process distribution such as the Erlangian distribution.

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