Stability conditions for a decentralised medium access algorithm: single- and multi-hop networks

Abstract

We consider a decentralised multi-access algorithm, motivated primarily by the control of transmissions in a wireless network. For a finite single-hop network with arbitrary interference constraints we prove stochastic stability under the natural conditions. For infinite and finite single-hop networks, we obtain broad rate-stability conditions. We also consider symmetric finite multi-hop networks and show that the natural condition is sufficient for stochastic stability.

Stability and moment bounds under utility-maximising service allocations: Finite and infinite networks

“…Service-rate allocationsψ(X), under many natural service allocation algorithms, are such that ψ i (X) is decreasing in each X j for j = i, as for these algorithms a higher load in queue j usually leads to all other queues receiving less service. This property is in fact satisfied by the rates defined by algorithms introduced in [11] and [15] that we will study here as examples. We would like to emphasise, however, that for our general results we are not going to make this assumption.…”

“…In this paper we study service allocation algorithms (in both discrete and continuous time) such that the corresponding service-rate allocationψ(X) maximises some utility function within some set C. In some cases, the set C arises naturally as the set of all feasible instantaneous ratesμ given the model structure, but this is not necessarily true in all cases. For instance, in the networks considered in [15], as well as in Section 3 here, the set C is a subset of the set of all feasible rates. Our main goal is to obtain network stability conditions, in terms of the set C. For example, our main stability results (Theorems 1 and 7) for single-hop networks show that the network is stable when the exogenous arrival ratesλ = (λ i , i ∈ N ) are ('strictly') within the set C. In addition to stability, we are able to obtain some steady-state moment bounds (Theorems 2 and 8).…”

“…In the second part of the paper, we apply the moment bounds established in this paper to obtain stability results for some infinite networks in discrete and continuous time that were considered in the recent papers [15] and [11], respectively. The models considered in the two papers are motivated by different wireless networks but share similar service-rate allocations.…”

“…The so-called rate stability (guaranteeing that the queue lengths do not grow linearly in time) is demonstrated in both discrete and continuous settings in [15] for arrival ratesλ within some natural set C. The authors of [11] considered a continuous-time model where arrival rates into all nodes are the same and equal to λ, say. They considered the system dynamics on intervals of the form (−n, .…”

“…This was not established in [11], but was left as a conjecture (Conjecture 1.12). Our analysis is based on showing that the rates of [11] and [15] are in fact utility-maximising (or 2-fair in the α-fair terminology) in a certain natural set C-a fact already mentioned in [15]. This allows us to use our results on stability and moment bounds for finite systems.…”