Iterative Methods for Queuing Systems with Batch Arrivals and Negative Customers

In this paper, we are interested in solving the stationary probability distributions of Markovian queuing systems having batch arrivals and negative customers by using the Preconditioned Conjugate Gradient Squared (PCGS) method. The preconditioner is constructed by exploiting the near-Toeplitz structure of the generator matrix of the system. We proved that under some mild conditions the preconditioned linear systems have singular values clustered around one when the size of the queue tends to infinity. Numerical results indicated that the convergence rate of the proposed method is very fast.

[1]  Raymond H. Chan,et al.  Circulant preconditioners for elliptic problems , 1992 .

[2]  Raymond H. Chan,et al.  Toeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch Arrivals , 1996, SIAM J. Sci. Comput..

[3]  R. Chan,et al.  Circulant preconditioners for complex Toeplitz matrices , 1993 .

[4]  P. Sonneveld.  CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems , 1989 .

[5]  T. Chan.  An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[6]  Erol Gelenbe,et al.  Queues with negative arrivals , 1991, Journal of Applied Probability.

[7]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[8]  X. Zhou,et al.  Circulant Preconditioners for Markov-Modulated Poisson Processes and Their Applications to Manufacturing Systems , 1997, SIAM J. Matrix Anal. Appl..

[9]  Wk Ching,et al.  Iterative Methods for Re-manufacturing Systems , 2002 .

[10]  W. Ching.  CIRCULANT PRECONDITIONERS FOR FAILURE PRONE MANUFACTURING SYSTEMS , 1997 .

[11]  W. K. Ching.  Circulant Preconditioning for Unreliable Manufacturing Systems with Batch Arrivals , 2000 .

[12]  O. Axelsson,et al.  Finite element solution of boundary value problemes - theory and computation , 2001, Classics in applied mathematics.

[13]  G. Strang.  A proposal for toeplitz matrix calculations , 1986 .

[14]  E. Gelenbe.  Product-form queueing networks with negative and positive customers , 1991 .

[15]  Raymond H. Chan,et al.  Circulant preconditioners for stochastic automata networks , 2000, Numerische Mathematik.

[16]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[17]  Eugene E. Tyrtyshnikov,et al.  Optimal and Superoptimal Circulant Preconditioners , 1992, SIAM J. Matrix Anal. Appl..

[18]  Wai-Ki Ching.  Iterative methods for manufacturing systems of two stations in tandem , 1998 .

[19]  J. Conway,et al.  Functions of a Complex Variable , 1964 .