Assessing the efficiency of resource allocations in bandwidth-sharing networks

Resource allocation in bandwidth-sharing networks is inherently complex: The distributed nature of resource allocation management prohibits global coordination for efficiency, i.e., aiming at full resource usage at all times. In addition, it is well recognized that resource efficiency may be conflicting with other critical performance measures such as flow delay. Without a notion of optimal (or “near-optimal”) behavior, the performance of resource allocation schemes can not be assessed properly. In previous work, we showed that optimal workload-based (or queue-length based) strategies have certain structural properties (they are characterized by so-called switching curves), but are too complex in general to be determined exactly. In addition, numerically determining the optimal strategy often requires excessive computational effort. This raises the need for simpler strategies with “near-optimal” behavior that can serve as a sensible bench-mark to test resource allocation strategies. We focus on flows traversing the network, sharing the resources on their common path with (independently generated) cross-traffic. Assuming exponentially distributed flow sizes, we show that in many scenarios optimizing the "drain time" under a fluid scaling gives a simple linear switching strategy that accurately approximates the optimal strategy. When two nodes on the flow path are equally congested, however, the fluid scaling is not appropriate, and the corresponding strategy may not even ensure stability. In such cases we show that the appropriate scaling for efficient workload-based allocations follows a square-root law. Armed with these, we then assess the potential gain that any sophisticated strategy can achieve over standard alpha-fair strategies, which are representations of common distributed allocation schemes, and confirm that alpha-fair strategies perform excellently among non-anticipating policies. In particular, we can approximate the optimal policy with a weighted alpha-fair strategy

Asymptotically optimal parallel resource assignment with interference

Motivated by scheduling in multi-cell wireless networks and resource allocation in computer systems, we study a service facility with two types of users (or jobs) having heterogen

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

Delay optimization in bandwidth-sharing networks

Bandwidth-sharing networks as considered by Massoulie & Roberts provide a natural modeling framework for describing the dynamic flow-level interaction among elastic data transfers. Although valuable stability results have been obtained, crucial performance metrics such as flow-level delays and throughputs in these models have remained intractable in all but a few special cases. In particular, it is not well understood to what extent flow-level delays and throughputs achieved by standard bandwidth-sharing mechanisms such as alpha-fair strategies leave potential room for improvement. In order to gain a better understanding of the latter issue, we set out to determine the scheduling policies that minimize the mean delay in some simple linear bandwidth-sharing networks. We compare the performance of the optimal policy with that of various alpha-fair strategies so as to assess the efficacy of the latter and gauge the potential room for improvement. The results indicate that the optimal policy achieves only modest improvements, even when the value of alpha is simply fixed, provided it is not too smal

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