Energy-aware capacity scaling in virtualized environments with performance guarantees

We investigate the trade-off between performance and power consumption in servers hosting virtual machines running IT services. The performance behavior of such servers is modeled through Generalized Processor Sharing (GPS) queues enhanced with a green speed-scaling mechanism that controls the processing capacity to use depending on the number of active virtual machines. When the number of virtual machines grows large, we show that the stochastic evolution of our model converges to a system of ordinary differential equations for which we derive a closed-form formula for its unique stationary point. This point is a function of the capacity and the shares that characterize the GPS mechanism. It allows us to show that speed-scaling mechanisms can provide large reduction in power consumption having only small performance degradation in terms of the delays experienced in the virtual machines. In addition, we derive the optimal choice for the shares of the GPS discipline, which turns out to be non-trivial. Finally, we show how our asymptotic analysis can be applied to the dimensioning and service partitioning in data-centers. Experimental results show that our asymptotic formulas are accurate even when the number of virtual machines is small

Asymptotically optimal parallel resource assignment with interference

Motivated by scheduling in cellular wireless networks and resource allocation in computer systems, we study a service facility with two classes of users having heterogeneous service requirement distributions. The aggregate service capacity is assumed to be largest when both classes are served in parallel, but giving preferential treatment to one of the classes may be advantageous when aiming at minimization of the number of users, or when classes have different economic values, for example. We set out to determine the allocation policies that minimize the total number of users in the system. For some particular cases we can determine the optimal policy exactly, but in general this is not analytically feasible. We then study the optimal policies in the fluid regime, which prove to be close to optimal in the original stochastic model. These policies can be characterized by either linear or exponential switching curves. We numerically compare our results with existing approximations based on optimization in the heavy-traffic regime. By simulations we show that, in general, our simple computable switching-curve strategies based on the fluid analysis perform well

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

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 [Rege, K. M., B. Sengupta. 1996. Queue length distribution for the discriminatory processor-sharing queue. Oper. Res. 44(4) 653-657], 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 independent and distributed 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 dk/E(B fwd k) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(B fwd k) 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

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

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

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