Heavy-traffic approximations for linear networks operating under α-fair bandwidth-sharing policies

Abstract

We consider the flow-level performance of a linear network supporting elastic traffic, where the service capacity is shared among the various classes of users according to a weighted alpha-fair policy. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the dynamics of the user population may be described by a Markov process. While valuable stability results have been established for the family of alpha-fair policies, the distribution of the number of active users has remained intractable in all but a few special cases. In order to gain further insight in the flow-level performance in more general scenarios, we develop approximations for the mean number of users based on the assumption that one or two of the nodes experience heavy-traffic conditions.In case of just a single 'bottleneck' node, we exploit the fact that this node approximately behaves as a two-class Discriminatory Processor-Sharing model. In the case that there are two nodes critically loaded, we rely on the observation that the joint workload process at these nodes is asymptotically independent of the fairness coefficient alpha, provided all classes have equal weights. In particular, the distribution of the joint workload process is roughly equal to that for an unweighted Proportional Fair policy, which is exactly known. In both cases, the numbers of users at nonbottleneck nodes can be approximated by that in an M/M/1 queue with reduced service capacity. Extensive numerical experiments indicate that the resulting approximations tend to be reasonably accurate across a wide range of parameters, even at relatively moderate load values. The approximations for the mean number of users also provide useful estimates for the mean transfer delays and user throughputs.

Monotonicity Properties for Multi-Class Queueing Systems

“…Proof: Equation (23) follows from (1) and equation (21). The first relation in equation (22) follows from equation (23) with m = 1, since the k-th class-1 user under both policies has the same (original) service requirement and the intra-class policy is FCFS.…”

“…For an arbitrary number of classes, the sufficient conditions in order for Conjecture 6.2 to hold for GPS can be obtained as well, however, the derivations become very cumbersome. In this section we used sample-path inequalities as given in (21) in order to compare the weighted mean number of users under two different policies. Property 6.4 is a sufficient (but not necessary) condition for these sample-path inequalities to hold.…”

“…For DPS and GPS, this property is not (always) satisfied, however, one might expect the sample-path inequalities still to hold. In the counterexample below we illustrate for the case of three classes that the sample-path inequalities (21) do not need to hold for either two DPS policies or two GPS policies that satisfy the conditions of Conjecture 6.2. This indicates that for more than two classes Conjecture 6.2 may not be proved using sample-path arguments and requires a different kind of approach.…”

“…Example 6.9 (Counterexamples for DPS and GPS) We give a counterexample for inequality (21) that is valid for both DPS and GPS, since the sample-path will exactly be the same under both policies. Consider a system with three classes, and consider the two policies with weight vectors φ = (2, 1, 1) andφ = (∞, 1, 1), respectively.…”

Comparison of bandwidth-sharing policies in a linear network

“…For α = 1, the joint equilibrium distribution of the number of users in a linear network is known [17]. In [12] approximations are given for the mean number of users under general weighted α-fair policies when one or more of the nodes are in heavy traffic.…”