Urtzi Ayesta

Discriminatory processor sharing revisited

As a natural multi-class generalization of the well-known (egalitarian) processor sharing (PS) service discipline, discriminatory processor sharing (DPS) is of great interest in many application areas, including telecommunications. Under DPS, the mean response time conditional on the service requirement is only known in closed form when all classes have exponential service requirement distributions. For generally distributed service requirements, Fayolle et al. (1980) showed that the expected conditional response times satisfy a system of integro-differential equations. In this paper, we exploit that result to prove that, provided the system is stable, for each class the expected unconditional response time is finite and that the expected conditional response time has an asymptote. The asymptotic bias of each class is found in closed form, involving the mean service requirements of all classes and the second moments of all classes but the one under consideration. In the course of the development we prove two other results that are of independent interest: we establish a conservation law for the time average unfinished work of all classes and, using a stochastic coupling argument, we show that the response times of different classes are stochastically ordered according to the DPS weights. Finally, we study DPS as a tool to achieve size based scheduling and we provide guidelines as to how the weights of DPS must be chosen such that DPS outperforms PS.

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.

Two-level processor-sharing scheduling disciplines: mean delay analysis

Inspired by several recent papers that focus on scheduling disciplines for network flows, we present a mean delay analysis of Multilevel Processor Sharing (MLPS) scheduling disciplines in the context of M/G/1 queues. Such disciplines have been proposed to model the effect of the differentiation between short and long TCP flows in the Internet. Under MLPS, jobs are classified into classes depending on their attained service. We consider scheduling disciplines where jobs within the same class are served either with Processor Sharing (PS) or Foreground Background (FB) policy, and the class that contains jobs with the smallest attained service is served first. It is known that the FB policy minimizes (maximizes) the mean delay when the hazard rate of the job size distribution is decreasing (increasing). Our analysis, based on pathwise and meanwise arguments of the unfinished truncated work, shows that Two-Level Processor Sharing (TLPS) disciplines, e.g., FB+PS and PS+PS, are better than PS scheduling when the hazard rate of the job size distribution is decreasing. If the hazard rate is increasing and bounded, we show that PS outperforms PS+PS and FB+PS. We further extend our analysis to study local optimality within a level of an MLPS scheduling discipline.

A modeling framework for optimizing the flow-level scheduling with time-varying channels

We introduce a comprehensive modeling framework for the problem of scheduling a finite number of finite-length jobs where the available service rate is time-varying. The main motivation comes from wireless data networks where the service rate of each user varies randomly due to fading. We employ recent advances on the restless bandit problem that allow us to obtain an opportunistic scheduling rule for the system without arrivals. When the objective is to minimize the mean number of users in the system or to minimize the mean waiting time, we obtain a priority-based policy which we call the Potential Improvement (PI) rule, since the priority index equals the ratio between the current available service rate and the expected potential improvement of the service rate. We also show that for certain objective functions, the index rule takes the form of known opportunistic scheduling rules like Relatively Best (RB) or Proportionally Best (PB). Thus our model provides a formal justification for the deployment of opportunistic scheduling rules in order to improve the flow-level performance in the presence of time-varying capacities. We further analyze the performance of the PI rule in the presence of randomly arriving users. When the service rates are constant, PI is equivalent to the [emailxa0protected], which is known to be optimal with any distribution of arrivals. Using a recent characterization for the stability region of flow-level scheduling rules under random arrivals, we show that PI achieves the maximum stability region. We perform numerical experiments in a wide range of scenarios and compare the performance of PI with other popular disciplines like RB, PB, Score-Based (SB) and the [emailxa0protected] Our results show that RB, PB, SB or the [emailxa0protected] might outperform the others depending on the scenario, but regardless of this, the performance of PI is always superior or equivalent to the best of these opportunistic rules.

Optimal choice of the buffer size in the Internet routers

We study an optimal choice of the buffer size in the Internet routers. The objective is to determine the minimum value of the buffer size required in order to fully utilize the link capacity. The reare some empirical rules for the choice of the buffer size. The most known rule of thumb states that the buffer length should be set to the bandwidth delay product of the network, i.e., the product between the average round trip time in the network and the capacity of the bottleneck link. Several recent works have suggested that as a consequence of the traffic aggregation, the buffer size should be set to smaller values. In this paper we propose an analytical framework for the optimal choice of the router buffer size. We formulate this problem as a multi-criteria optimization problem, in which the Lagrange function corresponds to a linear combination of the average sending rate and average delay in the queue. The solution to this optimization problem provides further evidence that indeed the buffer size should be reduced in the presence of traffic aggregation. Furthermore, our result states that the minimum required buffer is smaller than what previous studies suggested. Our analytical results are confirmed by simulations performed with the NS simulator.

Scheduling in a random environment: stability and asymptotic optimality

We investigate the scheduling of a common resource between several concurrent users when the feasible transmission rate of each user varies randomly over time. Time is slotted, and users arrive and depart upon service completion. This may model, for example, the flow-level behavior of end-users in a narrowband HDR wireless channel (CDMA 1xEV-DO). As performance criteria, we consider the stability of the system and the mean delay experienced by the users. Given the complexity of the problem, we investigate the fluid-scaled system, which allows to obtain important results and insights for the original system: 1) We characterize for a large class of scheduling policies the stability conditions and identify a set of maximum stable policies, giving in each time-slot preference to users being in their best possible channel condition. We find in particular that many opportunistic scheduling policies like Score-Based, Proportionally Best, or Potential Improvement are stable under the maximum stability conditions, whereas the opportunistic scheduler Relative-Best or the cμ-rule are not. 2) We show that choosing the right tie-breaking rule is crucial for the performance (e.g., average delay) as perceived by a user. We prove that a policy is asymptotically optimal if it is maximum stable and the tie-breaking rule gives priority to the user with the highest departure probability. We will refer to such tie-breaking rule as myopic. 3) We derive the growth rates of the number of users in the system in overload settings under various policies, which give additional insights on the performance. 4) We conclude that simple priority-index policies with the myopic tie-breaking rule are stable and asymptotically optimal. All our findings are validated with extensive numerical experiments.

Batch Arrival Processor-Sharing with Application to Multi-Level Processor-Sharing Scheduling

We analyze a Processor-Sharing queue with Batch arrivals. Our analysis is based on the integral equation derived by Kleinrock, Muntz and Rodemich. Using the contraction mapping principle, we demonstrate the existence and uniqueness of a solution to the integral equation. Then we provide asymptotical analysis as well as tight bounds for the expected response time conditioned on the service time. In particular, the asymptotics for large service times depends only on the first moment of the service time distribution and on the first two moments of the batch size distribution. That is, similarly to the Processor-Sharing queue with single arrivals, in the Processor-Sharing queue with batch arrivals the expected conditional response time is finite even when the service time distribution has infinite second moment. Finally, we show how the present results can be applied to the Multi-Level Processor-Sharing scheduling.

Competition yields efficiency in load balancing games

We study a nonatomic congestion game with N parallel links, with each link under the control of a profit maximizing provider. Within this load balancing game, each provider has the freedom to set a price, or toll, for access to the link and seeks to maximize its own profit. Given prices, a Wardrop equilibrium among users is assumed, under which users all choose paths of minimal and identical effective cost. Within this model we have oligopolistic price competition which, in equilibrium, gives rise to situations where neither providers nor users have incentives to adjust their prices or routes, respectively. In this context, we provide new results about the existence and efficiency of oligopolistic equilibria. Our main theorem shows that, when the number of providers is small, oligopolistic equilibria can be extremely inefficient; however as the number of providers N grows, the oligopolistic equilibria become increasingly efficient (at a rate of 1/N) and, as N->~, the oligopolistic equilibrium matches the socially optimal allocation.

On the Gittins index in the M/G/1 queue

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground–Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground–Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.

Early Retransmit for TCP and Stream Control Transmission Protocol (SCTP)

This document proposes a new mechanism for TCP and Stream ControlnTransmission Protocol (SCTP) that can be used to recover lost segmentsnwhen a connections congestion window is small. The EarlynRetransmit mechanism allows the transport to reduce, in certainnspecial circumstances, the number of duplicate acknowledgmentsnrequired to trigger a fast retransmission. This allows the transportnto use fast retransmit to recover segment losses that would otherwisenrequire a lengthy retransmission timeout. [STANDARDS-TRACK]