Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T14:30:44.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential martingales and Wald's formulas for two-queue networks

Published online by Cambridge University Press:  14 July 2016

François Baccelli
François Baccelli*
Affiliation:
INRIA
*
∗Postal address: INRIA, Rocquencourt — BP 105, 78153 Le Chesnay Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

An exponential martingale is defined for a class of random walks in the positive quarter lattice which are associated with a wide variety of Markovian two-queue networks. Balance formulas generalizing Wald's exponential identity are derived from the regularity of several types of hitting times with respect to this martingale. In a queuing context, these formulas can be interpreted as functional relations of practical interest between the number of customers at certain epochs and the utilization of the queues up to these epochs.

Keywords

TWO-DIMENSIONAL RANDOM WALKSERGODICITYHITTING TIMESBUSY PERIOD
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 3 , September 1986 , pp. 812 - 819
Copyright
Copyright © Applied Probability Trust 1986 

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.)

Footnotes

This work was done when the author was visiting the Applied Probability Group, Bell Communications Research, Morristown, NJ, USA.

References

[1] Baccelli, F. and Makowski, A. M. (1985) Direct martingale arguments for stability: the M/G/1 case. Syst. Control Letters 6, 181–186.CrossRefGoogle Scholar
[2] Baccelli, F. and Makowski, A. M. (1985) Exponential martingales for M/G/1 queues in random environment. Presented at the Atlanta ORSA/TIMS Joint National Meeting, Atlanta.Google Scholar
[3] Baras, J. S., Dorsey, A. J. and Makowski, A. M. (1984) Discrete time competing queues with geometric service requirements: stability, parameter estimation and adaptive control. Technical Research Report, University of Maryland, College Park.Google Scholar
[4] Cohen, J. W. and Boxma, O. (1983) Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.Google Scholar
[5] Fayolle, G. and Iasnogorodski, R. (1979) Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325–352.CrossRefGoogle Scholar
[6] Fuchs, B. A. and Shabat, B. V. (1964) Functions of A Complex Variable and Their Applications, Vol. 2. Pergamon Press, Oxford.Google Scholar
[7] Jackson, J. R. (1963) Jobshop-like queueing systems. Management Sci. 10, 131–142.CrossRefGoogle Scholar
[8] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[9] Malyshev, V. A. (1972) Classification of two-dimensional positive random walks and almost linear semimartingales. Soviet Math. Dokl. 13 (1).Google Scholar
[10] Mensikov, ?. V. (1974) Ergodicity and transience conditions for random walks of the positive octant of space. Dokl. Akad. Nauk SSSR 217 (4).Google Scholar
[11] Neveu, J. (1975) Discrete Parameter Martingales. North-Holland, Amsterdam.Google Scholar