Fluid Limit Techniques Research Articles

Queue-proportional rate allocation with per-link information in multihop wireless networks

The backpressure scheduling algorithm for multihop wireless networks is known to be throughput optimal, but it requires each node to maintain per-destination queues. Recently, a clever generalization of processor sharing has been proposed which is also throughput optimal, but which only uses per-link queues. Here, we propose another algorithm, called Queue-Proportional Rate Allocation (QPRA), which also only uses per-link queues and allocates service rates to links in proportion to their queue lengths, and employs the Serve-In-Random-Order queueing discipline within each link. Through fluid limit techniques and using a novel Lyapunov function, we show that the QPRA algorithm achieves the maximum throughput. We demonstrate an advantage of QPRA by showing that, for the so-called primary interference model, it is able to develop a low-complexity scheduling scheme which approximates QPRA and achieves a constant fraction of the maximum throughput region, independent of network size.

Queue-Proportional Rate Allocation with Per-Link Information in Multihop Networks

The backpressure scheduling algorithm for multihop wireless networks is known to be throughput optimal, but it requires each node to maintain per-destination queues. Recently, a clever generalization of processor sharing has been proposed which is also throughput optimal, but which only uses per-link queues. Here we propose another algorithm called Queue Proportional Rate Allocation (QPRA) which also only uses per-link queues, and allocates service rates to links in proportion to their queue-lengths and employs the Serve-In-Random-Order (SIRO) discipline within each link. Through fluid limit techniques and using a novel Lyapunov function, we show that the QPRA achieves the maximum throughput. We demonstrate an advantage of QPRA by showing that, for the so-called primary interference model, it is able to develop a low-complexity scheduling scheme which approximates QPRA and achieves a constant fraction of the maximum throughput region, independent of network size.

Stability of longest-queue-first scheduling in linear wireless networks with multihop traffic and one-hop interference

We consider the stability of the longest-queue-first (LQF) scheduling policy in wireless networks with multihop traffic under the one-hop interference model. Although it is well known that the back-pressure algorithm achieves the maximal stability, its computational complexity is prohibitively high. In this paper, we consider LQF, a low-complexity scheduling algorithm, which has been shown to have near-optimal throughput performance in many networks with single-hop traffic flows. We are interested in the performance of LQF for multihop traffic flows. In this scenario, the coupling between queues due to multihop traffic flows makes the local-pooling-factor analysis difficult to perform. Using the fluid-limit techniques, we show that LQF achieves the maximal stability for linear networks with multihop traffic and a single destination on the boundary of the network under the one-hop interference model.

Throughput-Optimal Scheduling in Multihop Wireless Networks Without Per-Flow Information

In this paper, we consider the problem of link scheduling in multihop wireless networks under general interference constraints. Our goal is to design scheduling schemes that do not use per-flow or per-destination information, maintain a single data queue for each link, and exploit only local information, while guaranteeing throughput optimality. Although the celebrated back-pressure algorithm maximizes throughput, it requires per-flow or per-destination information. It is usually difficult to obtain and maintain this type of information, especially in large networks, where there are numerous flows. Also, the back-pressure algorithm maintains a complex data structure at each node, keeps exchanging queue-length information among neighboring nodes, and commonly results in poor delay performance. In this paper, we propose scheduling schemes that can circumvent these drawbacks and guarantee throughput optimality. These schemes use either the readily available hop-count information or only the local information for each link. We rigorously analyze the performance of the proposed schemes using fluid limit techniques via an inductive argument and show that they are throughput-optimal. We also conduct simulations to validate our theoretical results in various settings and show that the proposed schemes can substantially improve the delay performance in most scenarios.

Stability of a retrial queueing network with different classes of customers and restricted resource pooling

We consider a retrial queueing network with different classes of customers and several servers. Each customer class is associated with a set of servers who can serve the class of customers. Customers of each class exogenously arrive according to a Poisson process. If an exogenously arriving customer finds upon his arrival any idle server who can serve the customer class, then he begins to receive a service by one of the available servers. Otherwise he joins the retrial group, and then tries his luck again after exponential time, the mean of which is determined by his customer class. Service times of each server are assumed to have general distribution. The retrial queueing network can be represented by a Markov process, with the number of customers of each class, and the customer class and the remaining service time of each busy server. Using the fluid limit technique, we find a necessary and sufficient condition for the positive Harris recurrence of the representing Markov process. This work is the first that applies the fluid limit technique to a model with retrial phenomena.

The ODE method for stability of skip-free Markov chains with applications to MCMC

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization. In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in ``most'' of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.

Cell switching versus packet switching in input-queued switches

Input Queued (IQ) switches have been well studied in the past two decades by researchers. The main problem concerning IQ switches is scheduling the switching fabric in order to transfer packets from input ports to output ports. Scheduling is relatively easier when all packets are of the same size. However, in practice, packets are of variable length. In the current implementation of switches, variable length packets are segmented into fixed length packets-also knowns as cells-for the purpose of scheduling. However, such cell-based switching comes with some significant disadvantages: (a) loss of bandwidth due to the existence of incomplete cells; and (b) additional overhead of segmentation of packets and re-assembly of cells. This is a strong motivation to study packet-based scheduling, i.e., scheduling the transfer of packets without segmenting them. The problem of packet scheduling was first considered by Marsan et al. They showed that under any admissible Bernoulli IID (independent and identically distributed) arrival traffic, a simple modification of the Maximum Weight Matching (MWM) algorithm achieves 100% throughput. In this paper, we first show that no work-conserving (i.e., maximal) packet-based algorithm is stable for arbitrary admissible arrival processes. Thus, the results of Marsan et al. are strongly dependent on the arrival distribution. Next, we propose a new class of waiting algorithms. We show that the waiting-MWM algorithm is stable for any admissible traffic using the fluid limit technique. We would like to note that the algorithms presented in this paper are distribution independent or universal. The algorithms and proof methods of this paper may be useful in the context of other scheduling problems.