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

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

Comparison of bandwidth-sharing policies in a linear network

In bandwidth-sharing networks, users of various classes require service from different subsets of shared resources simultaneously. These networks have been proposed to analyze the performance of wired and wireless networks. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies in a linear bandwidth-sharing network. This allows us to compare the performance of the system under various policies in terms of stability, the mean overall delay and the weighted mean number of users. For the important family of weighted α-fair policies, we derive stability results and establish monotonicity of the weighted mean number of users with respect to the fairness parameter α and the relative weights. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments

Heavy-traffic analysis of a multiple-phase network with discriminatory processor sharing

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 that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result 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 Rege and Sengupta (1996) who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically i.i.d. according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for d_k/E(B_k^fwd) obtain a larger share of the capacity, where d_k is the cost associated to class k, and E(B_k^fwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments

Heavy-traffic analysis of a multiple-phase network with discriminatory processor sharing

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 that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result 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 Rege and Sengupta (1996) who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically i.i.d. according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for d_k/E(B_k^fwd) obtain a larger share of the capacity, where d_k is the cost associated to class k, and E(B_k^fwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments

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