We study multihop networks with flow-scheduling constraints, no constraints on simultaneous activation of different links, potentially multiple source-destination routes, and a mix of heavy-tailed and light-tailed traffic. In this setting, we analyze the delay performance of the widely studied class of Back-Pressure scheduling policies, known for their throughput optimality property, using as a performance criterion the notion of delay stability, i.e., whether the expected end-to-end delay in steady state is finite. Our analysis highlights the significance of “bottleneck links,” i.e., links that are allowed to serve the source queues of heavy-tailed flows. The main idea is that traffic that has to pass through bottleneck links experiences large delays under Back-Pressure. By means of simple examples, we provide insights into how the network topology, the routing constraints, and the link capacities may facilitate or hinder the ability of light-tailed flows to avoid bottlenecks. Our delay-stability analysis is greatly simplified by the use of fluid approximations, allowing us to derive analytical results that would have been hard to obtain through purely stochastic arguments. Finally, we show how to achieve the best performance with respect to the delay stability criterion, by using a parameterized version of the Back-Pressure policy.

We study the problem of scheduling over time varying links in a network that serves both heavy-tailed and light tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic (the heavy queue), and the other receives light-tailed traffic (the light queue). The queues are connected to the server through time-varying ON/OFF links, which model fading wireless channels. We first show that the policy that gives complete priority to the light-tailed traffic guarantees the best possible tail behavior of both queue backlog distributions, whenever the queues are stable. However, the priority policy is not throughput maximizing, and can cause undesirable instability effects in the heavy queue. Next, we study the class of throughput optimal max-weight-α scheduling policies. We discover a threshold phenomenon, and show that the steady state light queue backlog distribution is heavy-tailed for arrival rates above a threshold value, and light-tailed otherwise. We also obtain the exact tail coefficient of the light queue backlog distribution under max-weight-α scheduling. Finally, we study a log-max-weight scheduling policy, which is throughput optimal, and ensures that the light queue backlog distribution is light-tailed.

Motivated by challenging resource allocation issues arising in large-scale wireless and wireline communication networks, we study distributed network utility maximization problems with a mixture of concave (e.g., best-effort throughputs) and nonconcave (e.g., voice/video streaming rates) utilities. In the first part of the paper, we develop our methodological framework in the context of a locally coupled networked system, where nodes represent agents that control a discrete local state. Each node has a possibly nonconcave local objective function, which depends on the local state of the node and the local states of its neighbors. The goal is to maximize the sum of the local objective functions of all nodes. We devise an iterative randomized algorithm, whose convergence and optimality properties follow from the classical framework of Markov Random Fields and Gibbs Measures via a judiciously selected neighborhood structure. The proposed algorithm is distributed, asynchronous, requires limited computational effort per node/iteration, and yields provable convergence in the limit. In order to demonstrate the scope of the proposed methodological framework, in the second part of the paper we show how the method can be applied to two different problems for which no distributed algorithm with provable convergence and optimality properties is available. Specifically, we describe how the proposed methodology provides a distributed mechanism for solving nonconcave utility maximization problems: 1) arising in OFDMA cellular networks, through power allocation and user assignment; 2) arising in multihop wireline networks, through explicit rate allocation. Several numerical experiments are presented to illustrate the convergence speed and performance of the proposed method.

We consider the problem of scheduling in a single-hop switched network with a mix of heavy-tailed and light-tailed traffic and analyze the impact of heavy-tailed traffic on the performance of Max-Weight scheduling. As a performance metric, we use the delay stability of traffic flows: A traffic flow is delay-stable if its expected steady-state delay is finite, and delay-unstable otherwise. First, we show that a heavy-tailed traffic flow is delay-unstable under any scheduling policy. Then, we focus on the celebrated Max-Weight scheduling policy and show that a light-tailed flow that conflicts with a heavy-tailed flow is also delay-unstable. This is true irrespective of the rate or the tail distribution of the light-tailed flow or other scheduling constraints in the network. Surprisingly, we show that a light-tailed flow can become delay-unstable, even when it does not conflict with heavy-tailed traffic. Delay stability in this case may depend on the rate of the light-tailed flow. Finally, we turn our attention to the class of Max-Weight-α scheduling policies. We show that if the α-parameters are chosen suitably, then the sum of the α-moments of the steady-state queue lengths is finite. We provide an explicit upper bound for the latter quantity, from which we derive results related to the delay stability of traffic flows, and the scaling of moments of steady-state queue lengths with traffic intensity.

We consider the problem of packet scheduling in a single-hop network with a mix of heavy-tailed and light-tailed traffic, and analyze the impact of heavy-tailed traffic on the performance of Max-Weight scheduling. As a performance metric we use the delay stability of traffic flows: a traffic flow is delay stable if its expected steady-state delay is finite, and delay unstable otherwise. First, we show that a heavy-tailed traffic flow is delay unstable under any scheduling policy. Then, we focus on the celebrated Max-Weight scheduling policy, and show that a light-tailed flow that conflicts with a heavy-tailed flow is also delay unstable. This is true irrespective of the rate or the tail distribution of the light-tailed flow, or other scheduling constraints in the network. Surprisingly, we show that a light-tailed flow can be delay unstable, even when it does not conflict with heavy-tailed traffic. Furthermore, delay stability in this case may depend on the rate of the light-tailed flow. Finally, we turn our attention to the class of Max-Weight-α scheduling policies; we show that if the α-parameters are chosen suitably, then the sum of the α-moments of the steady-state queue lengths is finite. We provide an explicit upper bound for the latter quantity, from which we derive results related to the delay stability of traffic flows, and the scaling of moments of steady-state queue lengths with traffic intensity.

We investigate the asymptotic behavior of the steady-state queue-length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput-optimal max-weight-$\alpha $scheduling policies and derive an exact asymptotic characterization of the steady-state queue-length distributions. In particular, we show that the tail of the light queue distribution is at least as heavy as a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also shows that the celebrated max-weight scheduling policy leads to the worst possible tail coefficient of the light queue distribution, among all nonidling policies. Motivated by the above negative result regarding the max-weight-$\alpha $policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail while still being throughput-optimal.

We investigate the asymptotic behavior of the steady-state queue length distribution under generalized max-weight scheduling in the presence of heavy-tailed traffic. We consider a system consisting of two parallel queues, served by a single server. One of the queues receives heavy-tailed traffic, and the other receives light-tailed traffic. We study the class of throughput optimal max-weight-α scheduling policies, and derive an exact asymptotic characterization of the steady-state queue length distributions. In particular, we show that the tail of the light queue distribution is heavier than a power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic characterization also shows that the celebrated max-weight scheduling policy leads to the worst possible tail of the light queue distribution, among all non-idling policies. Motivated by the above `negative' result regarding the max-weight-α policy, we analyze a log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees an exponentially decaying light queue tail, while still being throughput optimal.