Qualitative Properties of alpha-Weighted Scheduling Policies

We consider a switched network, a fairly general constrained queueing network model that has been used successfully to model the detailed packet-level dynamics in communication networks, such as input-queued switches and wireless networks. The main operational issue in this model is that of deciding which queues to serve, subject to certain constraints. In this paper, we study qualitative performance properties of the well known $\alpha$-weighted scheduling policies. The stability, in the sense of positive recurrence, of these policies has been well understood. We establish exponential upper bounds on the tail of the steady-state distribution of the backlog. Along the way, we prove finiteness of the expected steady-state backlog when $\alpha<1$, a property that was known only for $\alpha\geq 1$. Finally, we analyze the excursions of the maximum backlog over a finite time horizon for $\alpha \geq 1$. As a consequence, for $\alpha \geq 1$, we establish the full state space collapse property.Comment: 13 page

Proportional fairness and its relationship with multi-class queueing networks

We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description.Comment: Published in at http://dx.doi.org/10.1214/09-AAP612 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

We consider a connection-level model of Internet congestion control, introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000) 185--201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted $\alpha$-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] [$\alpha\in (0,\infty)$]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.Comment: Published in at http://dx.doi.org/10.1214/08-AAP591 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Redundancy Scheduling with Locally Stable Compatibility Graphs

Redundancy scheduling is a popular concept to improve performance in parallel-server systems. In the baseline scenario any job can be handled equally well by any server, and is replicated to a fixed number of servers selected uniformly at random. Quite often however, there may be heterogeneity in job characteristics or server capabilities, and jobs can only be replicated to specific servers because of affinity relations or compatibility constraints. In order to capture such situations, we consider a scenario where jobs of various types are replicated to different subsets of servers as prescribed by a general compatibility graph. We exploit a product-form stationary distribution and weak local stability conditions to establish a state space collapse in heavy traffic. In this limiting regime, the parallel-server system with graph-based redundancy scheduling operates as a multi-class single-server system, achieving full resource pooling and exhibiting strong insensitivity to the underlying compatibility constraints.Comment: 28 pages, 4 figure

Concave Switching in Single and Multihop Networks

Switched queueing networks model wireless networks, input queued switches and numerous other networked communications systems. For single-hop networks, we consider a {($\alpha,g$)-switch policy} which combines the MaxWeight policies with bandwidth sharing networks -- a further well studied model of Internet congestion. We prove the maximum stability property for this class of randomized policies. Thus these policies have the same first order behavior as the MaxWeight policies. However, for multihop networks some of these generalized polices address a number of critical weakness of the MaxWeight/BackPressure policies. For multihop networks with fixed routing, we consider the Proportional Scheduler (or (1,log)-policy). In this setting, the BackPressure policy is maximum stable, but must maintain a queue for every route-destination, which typically grows rapidly with a network's size. However, this proportionally fair policy only needs to maintain a queue for each outgoing link, which is typically bounded in number. As is common with Internet routing, by maintaining per-link queueing each node only needs to know the next hop for each packet and not its entire route. Further, in contrast to BackPressure, the Proportional Scheduler does not compare downstream queue lengths to determine weights, only local link information is required. This leads to greater potential for decomposed implementations of the policy. Through a reduction argument and an entropy argument, we demonstrate that, whilst maintaining substantially less queueing overhead, the Proportional Scheduler achieves maximum throughput stability.Comment: 28 page