Stability of Parallel Queueing Systems with Coupled Service Rates

Abstract

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.

Asymptotic Stability Region of Slotted Aloha

“…These probabilities are unknown in general, and so is the stability region. The results of [26] have been recently generalized to more general systems of interacting queues [8]. The only previous explicit stability condition for arbitrary N is given in [2]; unfortunately, to obtain this condition, the author has to assume that the arrival processes of the different users are correlated, which is unrealistic in practice.…”

Stability, Fairness, and Performance: A Flow-Level Study on Nonconvex and Time-Varying Rate Regions

“…These papers study the stability of networks where the rate region reduces to a single point depending on the set of classes with active flows. Some other papers aim at providing exact stability conditions: in [9], [20], [29], a recursive (with respect to the number of flow classes) stability condition is given for a particular class of networks, including those studied in [3], [25]. Unfortunately, this kind of recursive formula often proves difficult to exploit: the stability condition of networks with classes of flows depends on that of the network with classes and also on more detailed characterizations such as the probability that a given class has no active flows.…”

“…We consider the same function defined in (9), but given the assumption on , the sign of cannot be guaranteed. We write as defined in (11) and (12) (6), the trajectory of also converges as .…”

Modeling the impact of interference on wireless ad hoc network performance

“…The model discussed in this chapter however does not fall under the set of allocation functions, as the overall capacity of the network is not constant due to interference. For a network of parallel servers with coupled service rates, necessary and sufficient conditions for stability are derived in [BJL08]. Stability and performance of networks where the service rate depends on the network state is also analyzed in [VLK01], where transmissions over links with a fixed capacity are considered.…”

“…Our work is related to [BJL08] where parallel systems with coupled service rates are studied (a special case of our model in which there is no forwarding) in which the service rate is a function of the number of packets in the other queue.…”