Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T14:02:16.113Z Has data issue: false hasContentIssue false
  • English
  • Français

On a class of two-dimensional nearest-neighbour random walks

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

J. W. Cohen
Get access Rights & Permissions [Opens in a new window]

Abstract

For positive recurrent nearest-neighbour, semi-homogeneous random walks on the lattice {0, 1, 2, …} X {0, 1, 2, …} the bivariate generating function of the stationary distribution is analysed for the case where one-step transitions to the north, north-east and east at interior points of the state space all have zero probability. It is shown that this generating function can be represented by meromorphic functions. The construction of this representation is exposed for a variety of one-step transition vectors at the boundary points of the state space.

Keywords

STATIONARY DISTRIBUTIONSEMI-HOMOGENEOUSGENERATING FUNCTIONMEROMORPHIC FUNCTIONS

MSC classification

Secondary: 60K25: Queueing theory
Type
Part 4 Random Walks
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 207 - 237
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adan, I. J. B. F. (1991) A Compensation Approach for Queueing Problems. Doctoral thesis, Department of Mathematics, University of Eindhoven, The Netherlands.Google Scholar
[2] Adan, I. J. B. F., Wessels, J. and Zijm, W. H. M. (1993) Analysing multiprogramming queues by generating functions. SIAM J. Appl. Math. 53, 1123–1131.CrossRefGoogle Scholar
[3] Caratheodory, C. (1950) Funktionentheorie. Birkhauser, Basel.Google Scholar
[4] Cohen, J. W. (1988) Boundary value problems in queueing theory. QUESTA 3, 97–128.Google Scholar
[5] Cohen, J. W. (1992) Analysis of Random Walks. I.O.S. Press, Amsterdam.Google Scholar
[6] Cohen, J. W. and Boxma, O. J. (1983) Boundary Value Problems in Queueing Systems Analysis . North-Holland, Amsterdam.Google Scholar
[7] Fayolle, G. and Iasnogorodsky, R. (1979) Two coupled processors; reduction to a Riemann-Hilbert boundary value problem. Z. Wahrscheinlichkeitsth. 47, 325–351.CrossRefGoogle Scholar
[8] Fayolle, G., Iasnogorodsky, R. and Malyshev, V. A. (1990) Algebraic generating functions for two-dimensional random walks. Report, INRIA, Rocquencourt, France.Google Scholar
[9] Flatto, L. and Hahn, S. (1984) Two parallel queues created by arrivals with two demands. SIAM J. Appl. Math. 44, 1041–1054.Google Scholar
[10] Flatto, L. and Mckean, H. P. (1977) Two queues in parallel. Commun. Pure Appl. Math. 30, 255–263.Google Scholar
[11] Hofri, M. (1978) A generating-function analysis of multiprogramming queues. Internat. J. Comp. Inform. Sci. 7, 121–155.Google Scholar
[12] Jaffe, S. (1992) The equilibrium distribution for a clocked buffered switch. Prob. Eng. Inf. Sci. 6, 425–438.Google Scholar
[13] Kingman, J. F. C. (1961) Two similar queues in parallel. Ann. Math. Statist. 32, 1314–1323.Google Scholar
[14] Malyshev, V. A. (1972) An analytical method in the theory of two-dimensional positive random walks. Sibirskii Math. Zh. 13, 1314–1329.Google Scholar
[15] Nehari, Z. (1975) Conformal Mapping. Dover, New York.Google Scholar
[16] Saks, S. and Zygmund, A. (1952) Analytic Functions. Matematycznego, Warsaw.Google Scholar
[17] Takagi, H. (1991) Queueing Analysis , Vol. 1. North-Holland, Amsterdam.Google Scholar
[18] Wright, P. E. (1992) Two parallel processors with coupled input. Adv. Appl. Prob. 24, 986–1007.Google Scholar