Sufficient conditions for stability of longest-queue-first scheduling: second-order properties using fluid limits

Abstract

We consider the stability of the longest-queue-first scheduling policy (LQF), a natural and low-complexity scheduling policy, for a generalized switch model. Unlike that of common scheduling policies, the stability of LQF depends on the variance of the arrival processes in addition to their average intensities. We identify new sufficient conditions for LQF to be throughput optimal for independent, identically distributed arrival processes. Deterministic fluid analogs, proved to be powerful in the analysis of stability in queueing networks, do not adequately characterize the stability of LQF. We combine properties of diffusion-scaled sample path functionals and local fluid limits into a sharper characterization of stability.

Delay Guarantees for Throughput-Optimal Wireless Link Scheduling

“…Subsequently, several policies have been shown to attain (in some cases, for specific topology classses and interference constraints), either the maximum throughput region [1], [5], [18], [21], [23], [24] or a guaranteed fraction thereof [2], [4], [13], [15], [26], while requiring lower computation time. However, whether these algorithms are able to provide low delay guarantees remains largely unknown.…”

Stability of linear EDF networks with resource sharing

“…Indeed, as was discussed in detail by Gurvich and Van Mieghem [19,20], such systems typically exhibit unavoidable bottleneck idleness. (Compare also Definition 2.1 of the stability region in [6] and the notion of a feasible arrival rate vector in [13].) We hope that a suitable counterpart of the inequality (51) holds for some more general networks, for example those satisfying the local pooling condition, and that it will turn out to be a key ingredient of the proof of the corresponding extension of Theorem 1.…”

“…In this context, let us consider an example, provided by Dimakis and Walrand [13], of a strictly subcritical six-cycle packet level model with a deterministic arrival of a single packet to each queue in every time slot, which is unstable under LQF with the "unbiased" random tie-breaking rule. If the initial queue lengths are all equal, LQF for this particular system coincides with FISFO, so the corresponding FISFO (and hence EDF) system is also unstable.…”

“…Nevertheless, it is plausible that an example of an EDF resource sharing network which is unstable under any tie-breaking rule may be constructed along this line (perhaps by replacing the six-cycle by a larger cyclic graph). Interestingly, Dimakis and Walrand [13] have observed that if there is any randomness in the packet arrivals, then the six-cycle system is actually stable, even with the "unbiased" random tie-breaking rule, as long as it is strictly subcritical. However, in the presence of any randomness, either in the arrivals (implied, for example, by each of our conditions (2)- (3)), or in the transmission times, LQF no longer coincides with FISFO (EDF), so the relation between their stability regions is not clear.…”

“…A natural service discipline for a generalized switch is the Longest Queue First (LQF) algorithm, selecting the set of served links iteratively, in a way somewhat similar to EDF and SRPT, but according to the queue lengths rather than lead times or residual service times. In a seminal paper, Dimakis and Walrand [13] showed stability for a class of LQF systems satisfying the local pooling condition (which is true for linear networks) and, under mild assumptions on stochastic arrivals, also for some more general network topologies, for which a suitable rank condition holds. The main ideas of their analysis were to use the maximal queue length as a Lyapunov function for the underlying Markov chain and to combine diffusion-scale properties of the sample paths with the fluid limit framework.…”