Fast solvers for queueing systems with negative customers

In this paper, we are interested in solving queueing systems having Poisson batch arrivals, exponential servers and negative customers. Preconditioned Conjugate Gradient (PCG) method is applied to solving the steady-state probability distribution of the queueing system. Preconditioners are constructed by exploiting near-Toeplitz structure of the generator matrix and the Gohberg-Semumcul formula. We proved that the preconditioned system has singular values clustered around one. Therefore Conjugate Gradient (CG) methods when applied to solving the preconditioned system, we expect fast convergence rate. Numerical examples are given to demonstrate our claim.

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

[2]  R. Chan,et al.  Preconditioners for non‐Hermitian Toeplitz systems , 2001 .

[3]  Wai-Ki Ching,et al.  Inverse Toeplitz preconditioners for Hermitian Toeplitz systems , 2005, Numer. Linear Algebra Appl..

[4]  Erol Gelenbe,et al.  Product form networks with negative and positive customers , 1991 .

[5]  Michael K. Ng,et al.  Markov Chains: Models, Algorithms and Applications (International Series in Operations Research & Management Science) , 2005 .

[6]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[7]  George Labahn,et al.  Inversion of Toeplitz Matrices with Only Two Standard Equations , 1992 .

[8]  Wai-Ki Ching,et al.  Iterative Methods for Queuing Systems with Batch Arrivals and Negative Customers , 2003 .

[9]  Thomas Strohmer.  Four short stories about Toeplitz matrix calculations , 2000 .

[10]  Raymond H. Chan,et al.  Preconditioners for non-Hermitian Toeplitz systems , 2001, Numer. Linear Algebra Appl..

[11]  Michael K. Ng,et al.  On inversion of Toeplitz matrices , 2002 .

[12]  Georg Heinig.  On the reconstruction of Toeplitz matrix inverses from columns , 2002 .

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

[14]  Tak Kuen Siu,et al.  Markov Chains: Models, Algorithms and Applications , 2006 .

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

[16]  Tamir Shalom,et al.  On inversion of Toeplitz and close to Toeplitz matrices , 1986 .

[17]  Hong Liu,et al.  A note on the stability of Toeplitz matrix inversion formulas , 2004, Appl. Math. Lett..