Matrix geometric approach for random walks: Stability condition and equilibrium distribution

ABSTRACT In this paper, we analyze a sub-class of two-dimensional homogeneous nearest neighbor (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions[30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions.[13] Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix R appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix R.

[1]  Ivo J. B. F. Adan,et al.  The shorter queue polling model , 2013, Annals of Operations Research.

[2]  G. Fayolle,et al.  Random Walks in the Quarter Plane: Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics , 2018 .

[3]  Peter G. Taylor,et al.  Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators , 2006, Advances in Applied Probability.

[4]  Naoki Makimoto,et al.  GEOMETRIC DECAY OF THE STEADY-STATE PROBABILITIES IN A QUASI-BIRTH-AND-DEATH PROCESS WITH A COUNTABLE NUMBER OF PHASES , 2001 .

[5]  P. Shivakumar,et al.  A review of infinite matrices and their applications , 2009 .

[6]  Ben Atkinson,et al.  A Compensation Approach for Queueing Problems. , 1996 .

[7]  J. Cohen.  Analysis of the asymmetrical shortest two-server queueing model , 1995 .

[8]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[9]  Johan van Leeuwaarden,et al.  Steady-state analysis of shortest expected delay routing , 2015, Queueing Systems.

[10]  Michael N. Katehakis,et al.  A SUCCESSIVE LUMPING PROCEDURE FOR A CLASS OF MARKOV CHAINS , 2012, Probability in the Engineering and Informational Sciences.

[11]  D. P. Kroese,et al.  Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process , 2003 .

[12]  Onno Boxma,et al.  Boundary value problems in queueing system analysis , 1983 .

[13]  Peter G. Taylor,et al.  Queues with boundary assistance: the effects of truncation , 2011, Queueing Syst. Theory Appl..

[14]  Richard J. Boucherie,et al.  A linear programming approach to error bounds for random walks in the quarter-plane , 2014, Kybernetika.

[15]  Masakiyo Miyazawa,et al.  Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks , 2009, Math. Oper. Res..

[16]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[17]  Johan van Leeuwaarden,et al.  Erlang arrivals joining the shorter queue , 2013, Queueing Syst. Theory Appl..

[18]  Richard J. Boucherie,et al.  Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric terms , 2015, Queueing Syst. Theory Appl..

[19]  Jacob Cohen.  On a class of two-dimensional nearest neighbouring random walks , 1992 .

[20]  Whm Henk Zijm,et al.  A compensation approach for two-dimensional Markov processes , 1993, Advances in Applied Probability.

[21]  G. Fayolle,et al.  Two coupled processors: The reduction to a Riemann-Hilbert problem , 1979 .

[22]  Yanting Chen,et al.  Random walks in the quarter-plane: invariant measures and performance bounds , 2015 .

[23]  Liming Liu,et al.  SUFFICIENT CONDITIONS FOR A GEOMETRIC TAIL IN A QBD PROCESS WITH MANY COUNTABLE LEVELS AND PHASES , 2005 .

[24]  Hui Li,et al.  Geometric Decay in a QBD Process with Countable Background States with Applications to a Join-the-Shortest-Queue Model , 2007 .

[25]  Ilya Gertsbakh,et al.  The shorter queue problem: A numerical study using the matrix-geometric solution☆ , 1984 .

[26]  A. Krall.  Applied Analysis , 1986 .

[27]  Michael N. Katehakis,et al.  A comparative analysis of the successive lumping and the lattice path counting algorithms , 2015, Journal of Applied Probability.

[28]  Sheldon M. Ross,et al.  Introduction to Probability Models, Eighth Edition , 1972 .

[29]  M. Katehakis,et al.  Level product form QSF processes and an analysis of queues with Coxian interarrival distribution , 2015, 1507.05298.

[30]  Ivo J. B. F. Adan,et al.  Queueing Models with Multiple Waiting Lines , 2001, Queueing Syst. Theory Appl..

[31]  Larisa Beilina,et al.  Numerical Linear Algebra: Theory and Applications , 2017 .

[32]  J. Kingman.  Two Similar Queues in Parallel , 1961 .

[33]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[34]  Ijbf Ivo Adan,et al.  A compensation approach for queueing problems , 1991 .

[35]  S. G. Mohanty,et al.  Lattice Path Counting and Applications. , 1980 .

[36]  Anders C. Hansen,et al.  Infinite-dimensional numerical linear algebra: theory and applications , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[37]  Anders C. Hansen,et al.  On the approximation of spectra of linear operators on Hilbert spaces , 2008 .