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Perturbation analysis for denumerable Markov chains with application to queueing models

We study the parametric perturbation of Markov chains with denumerable state spaces. We consider both regular and singular perturbations. By the latter we mean that transition probabilities of a Markov chain, with several ergodic classes, are perturbed such that (rare) transitions among the different ergodic classes of the unperturbed chain are allowed. Singularly perturbed Markov chains have been studied in the literature under more restrictive assumptions such as strong recurrence ergodicity or Doeblin conditions. We relax these conditions so that our results can be applied to queueing models (where the conditions mentioned above typically fail to hold). Assuming ν-geometric ergodicity, we are able to explicitly express the steady-state distribution of the perturbed Markov chain as a Taylor series in the perturbation parameter. We apply our results to quasi-birth-and-death processes and queueing models.

The impact of the service discipline on delay asymptotics

This paper surveys the M/G/1 queue with regularly varying service requirement distribution. It studies the effect of the service discipline on the tail behavior of the waiting-time and/or sojourn-time distribution, demonstrating that different disciplines lead to quite different tail behavior. The orientation of the paper is methodological: We outline four different methods for determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS.

Beyond processor sharing

While the (Egalitarian) Processor-Sharing (PS) discipline offers crucial insights in the performance of fair resource allocation mechanisms, it is inherently limited in analyzing and designing differentiated scheduling algorithms such as Weighted Fair Queueing and Weighted Round-Robin. The Discriminatory Processor-Sharing (DPS) and Generalized Processor-Sharing (GPS) disciplines have emerged as natural generalizations for modeling the performance of such service differentiation mechanisms. A further extension of the ordinary PS policy is the Multilevel Processor-Sharing (MLPS) discipline, which has captured a pivotal role in the analysis, design and implementation of size-based scheduling strategies. We review various key results for DPS, GPS and MLPS models, highlighting to what extent these disciplines inherit desirable properties from ordinary PS or are capable of delivering service differentiation.

Stability of size-based scheduling disciplines in resource-sharing networks

Size-based scheduling strategies such as Shortest Remaining Processing Time first (SRPT) and Least Attained Service first (LAS) provide popular mechanisms for improving the overall delay performance by favoring smaller service requests over larger ones. The performance gains from these disciplines have been thoroughly investigated for single-server systems, and have also been experimentally demonstrated in web servers for example. In the present paper we explore the fundamental stability properties of size-based scheduling strategies in multi-resource systems, such as bandwidth-sharing networks, where users require service from several shared resources simultaneously. In particular, we establish the exact stability conditions for the SRPT and LAS disciplines in various limiting regimes. The results indicate that size-based scheduling strategies may fail to use the available resources efficiently, and in fact cause instability effects, even at arbitrarily low traffic loads, and will therefore not yield optimal delay performance. The qualitative findings confirm the tendency for users with long routes and large service requirements to experience severe performance degradation.

Asymptotic regimes and approximations for discriminatory processor sharing

We study the joint queue length distribution of the Discriminatory Processor Sharing model, assuming all classes have phase-type service requirement distributions. We show that the moments of the joint queue length distribution can be obtained by solving linear equations. We use this to study the system in two asymptotic regimes. In the first regime, the different user classes operate on strictly separated time scales. Then we study the system in heavy traffic.

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 a is simply fixed, provided it is not too 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.

User-level performance of elastic traffic in a differentiated-services environment

We consider a system with two service classes, one of which supports elastic traffic. The traffic characteristics of the other class can be completely general, allowing streaming applications as an important special case. The link capacity is shared between the two traffic classes in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queuing, provide a flexible mechanism for service differentiation and prioritization.We examine the user-level performance of the elastic traffic. The elastic traffic users randomly initiate file transfers with a heavy-tailed distribution. Within the elastic traffic class, the active flows share the available bandwidth in an ordinary processor-sharing (PS) fashion. The PS discipline has emerged as a natural paradigm for evaluating the user-perceived performance of bandwidth sharing algorithms like TCP. For a certain parameter range, we establish that the transfer delay incurred by the elastic traffic flows is asymptotically equivalent to that in an isolated PS system with constant service rate. This service rate is only affected by the streaming traffic through its average rate. Specifically, the elastic traffic is largely immune from possible adverse traffic characteristics or performance degradation due to prioritization of the streaming traffic. This confirms that GPS-based multiplexing mechanisms achieve significantly better performance for both traffic classes than a static bandwidth partitioning approach.

Dynamic Profit Optimization of Composite Web Services with SLAs

In this paper we investigate sequential decision mechanisms for composite web services. After executing each sub-service within a sequential workflow, decisions are made whether to terminate or continue the execution of the workflow. These decisions are based on observed response times, expected rewards, and typical Service Level Agreement parameters such as costs, penalties, and agreed response-time objectives. We propose a model for the sequential decision-making process within which we explore a couple of decision algorithms. We benchmarked these algorithms against the profit made when executing the workflow without decision-making. We show that algorithm based on backward recursion principle of dynamic programming is optimal with respect to profit. Next, we analyse the structure of erroneous decisions for both algorithms and show that significant profit gains can be obtained by sequential decision making.

Performance of TCP-friendly streaming sessions in the presence of heavy-tailed elastic flows

We consider a fixed number of streaming sessions which share a bottleneck link with a dynamic population of elastic flows. Motivated by extensive measurement studies, we assume that the sizes of the elastic flows exhibit heavy-tailed characteristics. The elastic flows are TCP-controlled, while the transmission rates of the streaming applications are governed by a so-called TCP-friendly rate control protocol. TCP-friendly rate control protocols provide a promising mechanism for avoiding severe fluctuations in the transmission rate, while ensuring fairness with competing TCP-controlled flows. Adopting the processor-sharing (PS) discipline to model the bandwidth sharing, we investigate the tail distribution of the deficit in service received by the streaming sessions compared to a nominal service target. The latter metric provides an indication for the quality experienced by the streaming applications. The results yield valuable qualitative insight into the occurrence of persistent quality disruption for the streaming users. We also examine the delay performance of the elastic flows by exploiting a useful relationship with a processor-sharing queue with permanent customers.