On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground-Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground-Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic. © Springer Science+Business Media, LLC 2009

Price of anarchy in non-cooperative load balancing games

We investigate the price of anarchy of a load balancing game with K dispatchers. The service rates and holding costs are assumed to depend on the server, and the service discipline is assumed to be processor-sharing at each server. The performance criterion is taken to be the weighted mean number of jobs in the system, or equivalently, the weighted mean sojourn time in the system. Independent of the state of the servers, each dispatcher seeks to determine the routing strategy that optimizes the performance for its own traffic. The interaction of the various dispatchers thus gives rise to a non-cooperative game. For this game, we first show that, for a fixed amount of total incoming traffic, the worst-case Nash equilibrium occurs when each player routes exactly the same amount of traffic, i.e., when the game is symmetric. For this symmetric game, we provide the expression for the loads on the servers at the Nash equilibrium. Using this result, we then show that, for a system with two or more servers, the price of anarchy, which is the worst-case ratio of the global cost of the Nash equilibrium to the global cost of the centralized setting, is lower bounded by K(2√K-1) and upper bounded by √K, independent of the number of servers

Convergence of trajectories and optimal buffer sizing for AIMD congestion control

We study the interaction between the AIMD (Additive Increase Multiplicative Decrease) multi-socket congestion control and a bottleneck router with Drop Tail buffer. We consider the problem in the framework of deterministic hybrid models. First, we show that trajectories always converge to limiting cycles. We characterize the cycles. Necessary and sufficient conditions for the absence of multiple jumps in the same cycle are obtained. Then, we propose an analytical framework for the optimal choice of the router buffer size. We formulate this problem as a multi-criteria optimization problem, in which the Lagrange function corresponds to a linear combination of the average goodput and the average delay in the queue. Our analytical results are confirmed by simulations performed with MATLAB Simulink

Bandwidth-sharing networks under a diffusion scaling

This paper considers networks operating under alpha-fair bandwidth sharing. When imposing a peak rate (i.e., an upper bound on the users' transmission rates, which could be thought of as access rates), the equilibrium point of the fluid limit is explicitly identified, for both the single-node network as well as the linear network. More specifically, a criterion is derived that indicates, for each specific class, whether or not it is essentially transmitting at peak rate. Knowing the equilibrium point of the fluid limit, the steady-state behavior under a diffusion scaling is determined. This allows an explicit characterization of the correlations between the number of flows of the various classes

Optimal policy for multi-class scheduling in a single server queue

In this paper we apply the Gittins optimality result to characterize the optimal scheduling discipline in a multi-class M/G/1 queue. We apply the general result to several cases of practical interest where the service time distributions belong to the set of decreasing hazard rate distributions, like Pareto or hyper-exponential. When there is only one class it is known that in this case the Least Attained Service policy is optimal. We show that in the multi-class case the optimal policy is a priority discipline, where jobs of the various classes depending on their attained service are classified into several priority levels. Using a tagged-job approach we obtain, for every class, the mean conditional sojourn time. This allows us to compare numerically the mean sojourn time in the system between the Gittins optimal and popular policies like Processor Sharing, First Come First Serve and Least Attained Service (LAS). We implement the Gittins' optimal algorithm in NS-2 and we perform numerical experiments to evaluate the achievable performance gain. We find that the Gittins policy can outperform by nearly 10% the LAS policy

Heavy-traffic analysis of the M/PH/1 Discriminatory Processor Sharing queue with phase-dependent weights

We analyze a generalization of the Discriminatory Processor Sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume phase-type distributed service requirements and allow that customers have different weights in various phases of their service. We establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by [12] who considered a DPS queue with exponentially distributed service requirements. We finally discuss some implications for residual service requirements and monotonicity properties in the ordinary DPS model

A unifying conservation law for single server queues.

In this paper we develop a conservation law for a work conserving multi-class $GI/GI/1$ queue operating under a general scheduling discipline. In the context of single-class queues, conservation laws have been obtained for both non-anticipating and anticipating disciplines with general service time distributions. In the context of multi-class queues, conservation laws have been previously obtained for (i) non-anticipating disciplines and exponential service time distribution and (ii) non-preemptive disciplines and general service time distribution. The conservation law we develop generalizes already existing conservation laws, and includes in particular popular multi-class time-sharing disciplines such as Discriminatory Processor Sharing (DPS) and Generalized Processor Sharing (GPS). In the literature, the conservation laws for single-class and multi-class queues are presented as if they were different in nature. The conservation law we develop includes existing conservation laws as special case