Dynamic fluid-based scheduling in a multi-class abandonment queue

International audienceWe investigate how to share a common resource among multiple classes of customers in the presence of abandonments. We consider two different models: (1) customers can abandon both while waiting in the queue and while being served, (2) only customers that are in the queue can abandon. Given the complexity of the stochastic optimization problem we propose a fluid model as a deterministic approximation. For the overload case we directly obtain that the c˜µ/θ rule is optimal. For the underload case we use Pontryagin’s Maximum Principle to obtain the optimal solution for two classes of customers; there exists a switching curve that splits the two-dimensional state-space into two regions such that when the number of customers in both classes is sufficiently small the optimal policy follows the c˜µ-rule and when the number of customers is sufficiently large the optimal policy follows the c˜µ/θ-rule. The same structure is observed in the optimal policy of the stochastic model for an arbitrary number of classes. Based on this we develop a heuristic and by numerical experiments we evaluate its performance and compare it to several index policies. We observe that the suboptimality gap of our solution is small

Resource-sharing in a single server with time-varying capacity

We investigate the problem of sharing the resources of a single server with time-varying capacity with the objective of minimizing the mean delay. We formulate the resource allocation problem as a Markov Decision Process. The problem is not solvable analytically in full generality, and we thus set out to obtain an approximate solution. In our main contribution, we extend the framework of multi-armed bandits to develop a heuristic solution of index type. At every given time, the heuristic assigns an index to every user that depends solely on its current state, and serves the user with highest current index value. We show that in the case of constant capacity, the heuristic policy is equivalent to the so-called Gittins index rule, which is known to be optimal under the assumption of constant capacity

Convergence and Optimal Buffer Sizing for Window Based AIMD Congestion Control

We study the interaction between the AIMD (Additive Increase Multiplicative Decrease) 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 the hybrid model of the interaction between the AIMD congestion control and bottleneck router always converges to a cyclic behavior. We characterize the cycles. Necessary and sufficient conditions for the absence of multiple jumps of congestion window in the same cycle are obtained. Then, we propose an analytical framework for the optimal choice of the router buffer size. We formulate the problem of the optimal router buffer size 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. The solution to the optimization problem provides further evidence that the buffer size should be reduced in the presence of traffic aggregation. Our analytical results are confirmed by simulations performed with Simulink and the NS simulator

Interpolation approximations for the steady-state distribution in multi-class resource-sharing systems

International audienceWe consider a single-server multi-class queue that implements relative priorities among customers of the various classes. The discipline might serve one customer at a time in a non-preemptive way, or serve all customers simultaneously. The analysis of the steady-state distribution of the queue-length and the waiting time in such systems is complex and closed-form results are available only in particular cases. We therefore set out to develop approximations for the steady-state distribution of these performance metrics. We first analyze the performance in light traffic. Using known results in the heavy-traffic regime, we then show how to develop an interpolation-based approximation that is valid for any load in the system. An advantage of the approach taken is that it is not model dependent and hence could potentially be applied to other complex queueing models. We numerically assess the accuracy of the interpolation approximation through the first and second moments

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

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

Heavy-traffic analysis of a non-preemptive multi-class queue with relative priorities

International audienceWe study the steady-state queue-length vector in a multi-class single-server queue with relative priorities. Upon service completion, the probability that the next customer to be served is from class k is controlled by class- dependent weights. Once a customer has started service, it is served without interruption until completion. This is a generalization of the random-order-of-service discipline. We investigate the system in a heavy-traffic regime. We first establish a state-space collapse for the scaled queue length vector, that is, in the limit the scaled queue length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. As a direct consequence, we obtain that the scaled number of customers in the system reduces as classes with smaller mean service requirement obtain relatively larger weights. We then show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables. This allows us to determine the value of the weights that minimize the m-th moment of the scaled waiting time for a customer of arbitrary class. We simulate a system with two different classes of customers in order to numerically verify some of the analytical results

Asymptotically optimal index policies for an abandonment queue with convex holding cost.

International audienceWe investigate a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. In general, the optimal policy has a complex dependency on all the input parameters and state information. Our main contribution is to derive index policies that can serve as heuristics and are shown to give good performance. Our index policy attributes to each class an index, which depends on the number of customers currently present in that class. The index values are obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory. They can be expressed as a function of the steady-state distribution probabilities of a one-dimensional birth-and-death process. For linear holding cost, the index can be calculated in closed-form and turns out to be independent of the arrival rates and the number of customers present. In the case of no abandonments and linear holding cost, our index coincides with the $c\mu$-rule, which is known to be optimal in this simple setting. For general convex holding cost we derive properties of the index value in limiting regimes: we consider the behavior of the index (i) as the number of customers in a class grows large, which allows us to derive the asymptotic structure of the index policies, (ii) as the abandonment rate vanishes, which allows us to retrieve an index policy proposed for the multi-class M/M/1 queue with convex holding cost and no abandonments, and (iii) as the arrival rate goes to either 0 or $\infty$, representing light-traffic and heavy-traffic regimes, respectively. We show that Whittle's index policy is asymptotically optimal in both light-traffic and heavy-traffic regimes. To obtain further insights into the index policy, we consider the fluid version of the relaxed problem and derive a closed-form expression for the fluid index. The latter is shown to coincide with the index values for the stochastic model in asymptotic regimes. For arbitrary convex holding cost the fluid index can be seen as the $Gc\mu/\theta$-rule, that is, including abandonments into the generalized $c\mu$-rule ($Gc\mu$-rule). Numerical experiments for a wide range of parameters have shown that the Whittle index policy and the fluid index policy perform very well for a broad range of parameters

Sojourn time approximations for a discriminatory-processor-sharing queue

International audienceWe study a multi-class time-sharing discipline with relative priorities known as Discriminatory Processor Sharing (DPS), which provides a natural framework to model service differentiation in systems. The analysis of DPS is extremely challenging and analytical results are scarce. We develop closed-form approximations for the mean conditional (on the service requirement) and unconditional sojourn times. The main benefits of the approximations lie in its simplicity, the fact that it applies for general service requirements with finite second moments, and that it provides insights into the dependency of the performance on the system parameters. We show that the approximation for the mean conditional and unconditional sojourn time of a customer is decreasing as its relative priority increases. We also show that the approximation is exact in various scenarios, and that it is uniformly bounded in the second moments of the service requirements. Finally we numerically illustrate that the approximation for exponential, hyperexponential and Pareto service requirements is accurate across a broad range of parameters