Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T15:52:52.824Z Has data issue: false hasContentIssue false
  • English
  • Français

On entrance times of a homogeneous N-dimensional random walk: an identity

Published online by Cambridge University Press:  14 July 2016

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

Abstract

Present developments in computer performance evaluation require detailed analysis of N-dimensional random walks on the set of lattice points in the first 2N-ant of Recent research has shown that for the two-dimensional case the inherent mathematical problem can often be formulated as a boundary value problem of the Riemann–Hilbert type. The paper is concerned with a derivation and analysis of an identity for the first entrance times distributions into the boundary of such random walks. The identity formulates a relation between these distributions and the zero-tuples of the kernel of the random walk; the kernel contains all the information concerning the structure of the random walk in the interior of its stage space. For the two-dimensional case the identity is resolved and explicit expressions for the entrance times distributions are obtained.

Keywords

KERNELZERO-TUPLESBOUNDARY VALUE PROBLEMIDENTITY FOR ENTRANCE TIMESTABOO PROBABILITIESRIEMANN BOUNDARY VALUE PROBLEM
Type
Part 8 - Random Walks, Graphs and Networks
Information
Journal of Applied Probability , Volume 25 , Issue A , 1988 , pp. 321 - 333
Copyright
Copyright © Applied Probability Trust 1988 

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] Baccelli, F. (1986) Exponential martingales and Wald's formulas for two-queue networks. J. Appl. Prob. 23, 812–819.CrossRefGoogle Scholar
[2] Cohen, J. W. (1982) The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
[3] Cohen, J. W. (1987) On entrance time distributions. Proc. Workshop Appl. Math. Performance/Reliability Models of Computer Communication Systems, ed. Iazeolla, G., North-Holland, Amsterdam.Google Scholar
[4] Cohen, J. W. (1986) Entrance times of a homogeneous, semi-bounded N-dimensional random walk; an identity. Preprint, University of Utrecht.Google Scholar
[5] Cohen, J. W. and Boxma, O. J. (1983) Boundary Value Problems in Queueing System Analysis., Math. Studies 79, North-Holland, Amsterdam.Google Scholar
[6] Greenwood, P. E. and Shared, M. (1977) Fluctuations of random walks in Rd and storage systems. Adv. Appl. Prob. 9, 566–587.Google Scholar
[7] Mogulskii, A. A. and Pecherskii, E. A. (1977) On the first exit time out of a semigroup in Rm for a random walk. Theory Prob. Appl. 22, 818–825.Google Scholar