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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.
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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 

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