Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T15:17:04.815Z Has data issue: false hasContentIssue false
  • English
  • Français

The synchronization of Poisson processes and queueing networks with service and synchronization nodes

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Balaji Prabhakar ,
Nicholas Bambos  and
T. S. Mountford
Balaji Prabhakar*
Affiliation:
Stanford University
Nicholas Bambos*
Affiliation:
Stanford University
T. S. Mountford*
Affiliation:
University of California
*
∗ Postal address: Departments of Electrical Engineering and Computer Science, Stanford University, Stanford, CA 94305, USA. Email address: balaji@isl.stanford.edu
∗∗ Postal address: Department of Engineering-Economic Systems and Operations Research and (by courtesy) Department of Electrical Engineering, Stanford University, Stanford CA 94305, USA.
∗∗∗ Postal address: Department of Mathematics, UCLA, Los Angeles, CA 90024, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the dynamics of a synchronization node in isolation, and of networks of service and synchronization nodes. A synchronization node consists of M infinite capacity buffers, where tokens arriving on M distinct random input flows are stored (there is one buffer for each flow). Tokens are held in the buffers until one is available from each flow. When this occurs, a token is drawn from each buffer to form a group-token, which is instantaneously released as a synchronized departure. Under independent Poisson inputs, the output of a synchronization node is shown to converge weakly (and in certain cases strongly) to a Poisson process with rate equal to the minimum rate of the input flows. Hence synchronization preserves the Poisson property, as do superposition, Bernoulli sampling and M/M/1 queueing operations. We then consider networks of synchronization and exponential server nodes with Bernoulli routeing and exogenous Poisson arrivals, extending the standard Jackson network model to include synchronization nodes. It is shown that if the synchronization skeleton of the network is acyclic (i.e. no token visits any synchronization node twice although it may visit a service node repeatedly), then the distribution of the joint queue-length process of only the service nodes is product form (under standard stability conditions) and easily computable. Moreover, the network output flows converge weakly to Poisson processes. Finally, certain results for networks with finite capacity buffers are presented, and the limiting behavior of such networks as the buffer capacities become large is studied.

Keywords

SynchronizationPoisson processesqueueing networks

MSC classification

Primary: 60G55: Point processes
Secondary: 90B15: Network models, stochastic 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 824 - 843
Copyright
Copyright © Applied Probability Trust 2000 

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] Anantharam, V. (1988). Modelling the flow of coalescing data streams through a processor. J. Appl. Prob. 25, 184–193.Google Scholar
[2] Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
[3] Baccelli, F., Cohen, G., Oldser, G. J. and Quadrat, J.-P. (1992). Synchronization and Linearity—an Algebra for Discrete Event Systems. John Wiley, New York.Google Scholar
[4] Baccelli, F. and Makowski, A. M. (1989). Queueing models for systems with synchronization constraints. Proc. IEEE 77, 138–161.Google Scholar
[5] Baccelli, F., Massey, W. A. and Towsley, D. (1989). Acyclic fork-join queueing networks. J. Assoc. Comput. Mach. 36, 615–642.Google Scholar
[6] Baskett, F., Chandy, K., Muntz, R. R. and Palacios, F. G. (1975). Open, closed and mixed networks of queues with different classes of customers. J. Assoc. Comput. Mach. 22, 248–260.Google Scholar
[7] Brémaud, P (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.Google Scholar
[8] Chung, K. L. (1974). A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
[9] Commoner, F. (1972). Deadlocks in Petri Nets. Res. Rept CA-7606-2311, Applied Data Research Inc.Google Scholar
[10] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[11] Florin, G. and Natkin, S. (1985). On open synchronized queueing networks. In Proc. Int. Workshop on Timed Petri Nets, Torino, Italy.Google Scholar
[12] Harrison, J. M. (1973). Assembly-like queues. J. Appl. Prob. 10, 354–367.Google Scholar
[13] Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 15, 254–265.Google Scholar
[14] Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Heidelberg.Google Scholar
[15] Kelly, F. (1979). Reversibility and Stochastic Networks. John Wiley, London.Google Scholar
[16] Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Philos. Soc. 58, 497–520.Google Scholar
[17] Matthes, K., Kerstan, J. and Mecke, J. (1978). Infinitely Divisible Point Processes. Akademie Verlag, Berlin.Google Scholar
[18] Walrand, J. (1988). An Introduction to Queueing Networks. Prentice Hall, Englewood Cliffs.Google Scholar
[19](1989) Proc. 3rd Int. Workshop on Petri nets and performance models, Kyoto, Japan. IEEE Computer Society Press, Los Alamitos, California.Google Scholar
[20](1991) Proc. 4th Int. Workshop on Petri nets and performance models, Melbourne, Australia. IEEE Computer Society Press, Los Alamitos, California.Google Scholar