Stability of join-the-shortest-queue networks

Authors:
J. G. Dai;John J. Hasenbein;Bara Kim
Affiliations:
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA 30332;Graduate Program in Operations Research and Industrial Engineering, Department of Mechanical Engineering, University of Texas at Austin, Austin, USA 78712;Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, Seoul, Republic of Korea 136-701
Venue:
Queueing Systems: Theory and Applications
Year:
2007

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Abstract

This paper investigates stability behavior in a variant of a generalized Jackson queueing network. In our network, some customers use a join-the-shortest-queue policy when entering the network or moving to the next station. Furthermore, we allow interarrival and service times to have general distributions. For networks with two stations we derive necessary and sufficient conditions for positive Harris recurrence of the network process. These conditions involve only the mean values of the network primitives. We also provide counterexamples showing that more information on distributions and tie-breaking probabilities is needed for networks with more than two stations, in order to characterize the stability of such systems. However, if the routing probabilities in the network satisfy a certain homogeneity condition, then we show that the stability behavior can be explicitly determined, again using the mean value parameters of the network. A byproduct of our analysis is a new method for using the fluid model of a queueing network to show non-positive recurrence of a process. In previous work, the fluid model was only used to show either positive Harris recurrence or transience of a network process.