Matthieu Jonckheere

Stability of Parallel Queueing Systems with Coupled Service Rates

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.

Flow-level performance and capacity of wireless networks with user mobility

The performance evaluation of wireless networks is severely complicated by the specific features of radio communication, such as highly variable channel conditions, interference issues, and possible hand-offs among base stations. The latter elements have no natural counterparts in wireline scenarios, and create a need for novel performance models that account for the impact of these characteristics on the service rates of users.Motivated by the above issues, we review several models for characterizing the capacity and evaluating the flow-level performance of wireless networks carrying elastic data transfers. We first examine the flow-level performance and stability of a wide family of so-called α-fair channel-aware scheduling strategies. We establish that these disciplines provide maximum stability, and describe how the special case of the Proportional Fair policy gives rise to a Processor-Sharing model with a state-dependent service rate. Next we turn attention to a network of several base stations with inter-cell interference. We derive both necessary and sufficient stability conditions and construct lower and upper bounds for the flow-level performance measures. Lastly we investigate the impact of user mobility that occurs on a slow timescale and causes possible hand-offs of active sessions. We show that the mobility tends to increase the capacity region, both in the case of globally optimal scheduling and local α-fair scheduling. It is additionally demonstrated that the capacity and user throughput improve with lower values of the fairness indexxa0α.

Scheduling in a random environment: stability and asymptotic optimality

We investigate the scheduling of a common resource between several concurrent users when the feasible transmission rate of each user varies randomly over time. Time is slotted, and users arrive and depart upon service completion. This may model, for example, the flow-level behavior of end-users in a narrowband HDR wireless channel (CDMA 1xEV-DO). As performance criteria, we consider the stability of the system and the mean delay experienced by the users. Given the complexity of the problem, we investigate the fluid-scaled system, which allows to obtain important results and insights for the original system: 1) We characterize for a large class of scheduling policies the stability conditions and identify a set of maximum stable policies, giving in each time-slot preference to users being in their best possible channel condition. We find in particular that many opportunistic scheduling policies like Score-Based, Proportionally Best, or Potential Improvement are stable under the maximum stability conditions, whereas the opportunistic scheduler Relative-Best or the cμ-rule are not. 2) We show that choosing the right tie-breaking rule is crucial for the performance (e.g., average delay) as perceived by a user. We prove that a policy is asymptotically optimal if it is maximum stable and the tie-breaking rule gives priority to the user with the highest departure probability. We will refer to such tie-breaking rule as myopic. 3) We derive the growth rates of the number of users in the system in overload settings under various policies, which give additional insights on the performance. 4) We conclude that simple priority-index policies with the myopic tie-breaking rule are stable and asymptotically optimal. All our findings are validated with extensive numerical experiments.

Flow-Level Stability of Channel-Aware Scheduling Algorithms

Channel-aware scheduling strategies provide an effective mechanism for improving the throughput performance in wireless data networks by exploiting channel fluctuations. The performance of channel-aware scheduling algorithms has mainly been examined at the packet level for a static user population, often assuming infinite backlogs. Recently, some studies have also explored the flow-level performance in a scenario with user dynamics governed by the arrival and completion of random service demands over time. Although in certain cases the performance may be evaluated by means of a Processor-Sharing model, in general the flow-level behavior has remained largely intractable, even basic stability properties. In the present paper we derive simple necessary stability conditions, and show that these are also sufficient for a wide class of utility-based scheduling policies. This contrasts with the fact that the latter class of strategies generally fail to provide maximum-throughput guarantees at the packet level.

Stability of two interfering processors with load balancing

We examine the stability of two interfering processors with service rates depending on the number of users present of each of the classes and subject to static or dynamic load balancing. Such models arise in several contexts, especially in wireless networks, or multiprocessing. In case of static load balancing, we extend existing stability results by deriving Lyapunov functions that are connected to the solutions of one dimensional Poisson equation. We then characterize the optimal static load balancing. The Lyapunov function found for the static load balancing is used to derive the exact stability condition of an interesting class of dynamic load balancing policies. We show that for certain properties of the state-dependent service rates, simple dynamic load balancing schemes improve the stability condition.