Energy-Efficient Scheduling under Delay Constraints for Wireless Networks

Energy-Efficient Schedulingunder Delay Constraints forWireless Networks

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Energy-Efficient Scheduling under Delay Constraints for Wireless NetworksRandall Berry, Eytan Modiano, and Murtaza Zafer2012

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Energy-Efficient Scheduling under Delay Constraints for Wireless Networks

Randall Berry, Eytan Modiano, and Murtaza Zafer

www.morganclaypool.com

ISBN: 9781608458882 paperbackISBN: 9781608458899 ebook

DOI 10.2200/S00443ED1V01Y201208CNT011

A Publication in the Morgan & Claypool Publishers seriesSYNTHESIS LECTURES ON COMMUNICATION NETWORKS

Lecture #11Series Editor: Jean Walrand,University of California, BerkeleySeries ISSNSynthesis Lectures on Communication NetworksPrint 1935-4185 Electronic 1935-4193

Energy-Efficient Schedulingunder Delay Constraints forWireless Networks

Randall BerryNorthwestern University

Eytan ModianoMIT

Murtaza ZaferIBM Research

SYNTHESIS LECTURES ON COMMUNICATION NETWORKS #11

CM& cLaypoolMorgan publishers&

ABSTRACTPacket delay and energy consumption are important considerations in wireless and sensor networksas these metrics directly affect the quality of service of the application and the resource consumptionof the network; especially, for a rapidly growing class of real-time applications that impose strictrestrictions on packet delays.Dynamic rate control is a novel technique for adapting the transmissionrate of wireless devices, almost in real-time, to opportunistically exploit time-varying channel con-ditions as well as changing traffic patterns. Since power consumption is not a linear function of therate and varies significantly with the channel conditions, adapting the rate has significant benefits inminimizing energy consumption. These benefits have prompted significant research in developingalgorithms for achieving optimal rate adaptation while satisfying quality of service requirements. Inthis book, we provide a comprehensive study of dynamic rate control for energy minimization underpacket delay constraints. We present several formulations and approaches adopted in the literatureranging from discrete-time formulations and dynamic programming based solutions to continuous-time approaches utilizing ideas from network calculus and stochastic optimal control theory. Thegoal of this book is to expose the reader to the important problem of wireless data transmission withdelay constraints and to the rich set of tools developed in recent years to address it.

KEYWORDSenergy efficiency, transmission scheduling, deadline constraints, delay constraints

vii

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.1 Single-hop Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Service Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 Overview of Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Transmission Rate Adaptation under Deadline Constraints . . . . . . . . . . . . . . . . . . 112.1 Rate Control over Point-to-point Wireless Link . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Deterministic Setting without Channel Fading . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Stochastic BT -problem: Discrete-time Approach . . . . . . . . . . . . . . . . . . . . 232.1.4 Stochastic BT -problem: Continuous-time Approach . . . . . . . . . . . . . . . . . 292.1.5 Variable Packet Deadlines: Continuous-time Approach . . . . . . . . . . . . . . . 352.1.6 Online Policy with Arrivals and Deadlines . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Multi-user Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.2.1 Multi-user Time-sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2.2 Multi-access Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Average Delay Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Dynamic Programming Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.1 Constrained Markov Decision Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.2 Computational Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Structural Properties of Optimal Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3 Large Delay Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

viii

3.3.1 Large Delay Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3.2 Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Small Delay Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.1 Small Delay Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.2 Convergence Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Conclusions and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Authors Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

ix

PrefaceOver the past decade, much research has gone into the subject of energy-efficient networking,especially in the context of wireless and sensor networks. Energy savings in a wireless network canbe attained in many ways, and researchers have explored a plethora of avenues for achieving suchsavings, including intelligent battery management, energy-efficient routing, sleep-awake scheduling,and energy-efficient hardware designs. Moreover, in recent years, energy efficiency has become apressing consideration in network design in general, as the amount of electricity consumed bynetwork infrastructure is becoming a significant fraction of the overall electric power consumption.

This book broadly explores one key aspect of energy-efficient networking: the tradeoff betweenenergy consumption and packet delay over a wireless link and,more specifically, explores the problemof energy-efficient transmission rate control at the link layer. Energy efficiency at the wireless linklayer can be achieved through transmission rate control by exploiting two fundamental properties.First, it is known that transmitting an amount of data at a low rate, but over a longer durationconsumes less energy as compared to a fast rate of transmission over a shorter duration.Thus, givena certain amount of data to be transmitted, transmission at the lowest possible rate will be the mostenergy efficient.Second, the amount of transmission energy required to achieve a certain transmissionrate is a function of the underlying channel state; thus, it is possible to exploit the time-varying natureof a wireless channel through opportunistic scheduling and achieve significant energy savings byjudiciously scheduling transmissions during times that the channel is in a good state.

Lowering the transmission rate and opportunistically scheduling transmissions provide toolsfor a wireless transmitter to improve energy efficiency; however, these tools can also lead to unac-ceptably large delay. In this book, we survey rate-adaptation algorithms that exploit these tools forenergy savings in the presence of various delay constraints.The delay constraints considered includeboth hard-deadline and average-delay constraints. Several problem formulations are studied in thebook with different assumptions regarding the packet arrival process, channel fading and delay con-straints. A complementary aim of the book is to highlight the rich set of mathematical tools thatcan be used to address these problems.

We begin inChapter 1 with amotivation and preliminary description of the wireless setup thatis considered throughout the book. We also outline the preliminaries on transmission rate controland the various facets of quality-of-service constraints considered in the literature. In Chapter 2,we address the problem of transmission rate control subject to strict deadline constraints on thedata. In particular, considering first a deterministic setup, where the arrival times are known inadvance and the channel is not time varying, we discuss a simple and intuitive mechanism formeeting the deadline constraints using the minimum amount of energy. We then consider thecase of a stochastic time-varying channel, and study the application of a wide range of mathematical

x PREFACE

techniques, fromdiscrete-time dynamic programming to continuous-time stochastic optimal controland convex optimization, for developing optimal algorithms to minimize the total expected energycost subject to the deadline constraints. In Chapter 3, we turn our attention to the stochastic arrivalsetting where data arrives to the link at random times and must be queued for transmission. In thiscontext,we consider the problemofminimizing average energy consumption subject to average-delayconstraints. In particular, we develop a dynamic programming formulation for optimally solving theproblem, and consider the asymptotic behavior of the optimal average energy needed both in thehigh-delay and low-delay regimes.

Randall Berry, Eytan Modiano, and Murtaza ZaferAugust 2012

xi

AcknowledgmentsThis book is based on the work of the authors, their co-authors, and other researchers in the field.As such, its writing would not have been possible without their significant efforts. In particular,we are grateful to all of our co-authors, whose works served as the basis for this book, including:Alvin Fu, Robert Gallager, Jun Sun, Alessandro Tarello, and John Tsitsiklis.We are also grateful toElif Uysal-Biyikoglu and Abbas El Gamal, whose work was the basis for the FlowRight algorithmdescribed in Section 2.2.2, and to Juyul Lee and Nihar Jindal whose work was the basis for theoptimal policies described in Section 2.1.3. We would also like to acknowledge the work of B. E.Collins and Rene Cruz and the work of Munish Goyal, Anurag Kumar, and Vinod Sharma, bothof which influenced our treatment of dynamic programing approaches in Section 3.2. Finally, wethank the reviewers and editors for helping improve the quality and presentation of this book.

Randall Berry, Eytan Modiano, and Murtaza ZaferAugust 2012

1C H A PT E R 1

IntroductionEnergy efficiency is of prime importance in many wireless systems such as sensor and ad-hoc net-works.Over the past decade,much research has gone into the subject of energy-efficient networking.Energy savings in a network can be attained in many ways, and researchers have explored a plethoraof avenues for achieving such savings. In sensor networks, where energy savings are paramount forextending the life of the network, much research has gone into addressing the problem of sleep-awake-scheduling for the purpose of reducing energy consumption [66]. Others have exploredenergy savings through intelligent battery management [20], energy-efficient routing [17; 57; 67],stochastic network control [4; 46; 51], and energy-efficient hardware designs [26]. Moreover, inrecent years, researchers have started to pay attention to energy efficiency in wired networks as well,as the amount of electricity consumed by network infrastructure is becoming a significant fraction ofelectric power consumption [37]. Indeed, energy-efficient networking is of paramount importance,and is the subject of hundreds of articles and a number of books [61].

A somewhat less explored avenue, and the subject of this book, is that of energy-efficienttransmission at the link level. In many settings, the wireless link forms the first hop that connectswireless users to wired infrastructure. In such settings, battery life can be significantly extendedthrough the efficient transmission over this wireless access link. In this context, transmission energyoften forms a significant proportion of the total energy expenditure and hence, one way to maximizebattery lifetime is by reducing transmission energy. It has been shown [12; 52] that transmittingan amount of data at a low rate but over a longer duration consumes less transmission energy ascompared to a fast rate of transmission over a shorter duration.That is, the power required to transmitat a certain rate is a convex function of the rate, and this relationship can be exploited to extractsignificant energy savings. Thus, given a certain amount of data to be transmitted, transmission atthe lowest possible rate will be the most energy efficient, although it may result in unacceptably largedelays. Additionally, in most modern wireless systems the underlying transmission rate can be variedover time, for example by changing the modulation and/or coding scheme used. Such techniquesenable one to dynamically exploit this trade-off based, for example, on the available traffic. Anotherimportant characteristic that can be exploited for energy savings is the time varying nature of thewireless link. Typically, channel quality varies with time due to fading. By judiciously schedulingtransmissions during times that the channel is in a good state, significant additional energy savingscan be attained. Here, again, it may be possible to obtain significant energy savings by delayingtransmissions to wait for such good states.

Generally, however, data to be transmitted must satisfy certain quality-of-service (QoS) re-quirements, such as constraints on the maximum or the average delay. For example, real-time voice

2 1. INTRODUCTION

may have somewhat strict constraints on delay, whereas Internet file transfer may impose less rigidaverage delay constraints. In principle, one can meet these QoS requirements by transmitting athigher rates, using higher transmission power and an increased energy expenditure. In contrast, alower rate transmission may not always be able to meet the QoS constraints. Thus, it is clear that atradeoff exists between the energy consumption and the service or delay requirements.Over the pastdecade, the authors of this book have explored this tradeoff and developed algorithms for minimumenergy transmission scheduling subject to service requirements on delay. This book explores thissubject, reviewing in detail the recent work of the authors and others.

The preceding discussion and the remainder of this book focuses on transmission energyas the dominant energy cost in a device. This is reasonable for example in cellular systems thatcommunicate over relatively long distances, but may not be for systems that operate with a shortrange of communication. This is because the needed transmission energy increases with distancewhile the energy needed for other functions, such as processing, does not. In short-range scenarioswhere processing becomes significant, the energy/delay trade-offs may change: one may prefer tosend data faster so that the device can enter into sleep mode and save on processing power. Suchscenarios are not the focus of this book, though some of the techniques used here might be usefulto address such settings as well. Additionally, we note that improvements in processing technologycontinually reduce these costs, while transmission energy is constrained by the Shannon limit.Thus,over time, one expects that transmission energy will dominate in an even greater number of scenarios.

1.1 SINGLE-HOPWIRELESSNETWORKS

This book focuses on studying energy delay trade-offs in single-hop wireless networks and in manycases on a single wireless link. There are two reasons for this. First, while much attention has beenpaid to multi-hop wireless networks, such as ad-hoc networks, or sensor networks, the predominantuse of wireless transmission is for single-hop transmission. Nearly all wireless local area networksuse a single-hop wireless link to a wired hub node, and wireless phones and PDA devices transmitover a single wireless link to a base station. Second, having a good understanding of how to manageper-link energy and delay trade-offs will likely be a key ingredient in addressing these issues inmulti-hop settings.

1.1.1 CHANNELMODELWe next discuss a basic model for a single wireless link that will be used in most of the followingsections. This is based on a common assumption in the communication literature that the power Prequired to reliably transmit at a certain rate r is a convex increasing function of the rate, i.e.,P = g(r)where g(.) is a convex function such that g(r) 0 and strictly increasing for r 0 [12; 20; 28; 52].Physically, this means that as the transmission rate increases the required power has a faster rate ofincrease, and hence lower rate transmissions are preferable for energy efficiency.One justification forthis claim is based on the Shannon capacity formula over an additive white Gaussian noise (AWGN)

1.1. SINGLE-HOPWIRELESSNETWORKS 3

channel. The capacity is a concave function of the transmission power and is given by,

C = log(

1 + PN

)bits/transmission, (1.1)

whereN denotes the noise power. Assuming that one can transmit at rate C and rewriting the aboveequation it easily follows that the expended power is a convex function of the transmission rate.Note that in practice the actual functional relationship between power and rate would depend onthe specific modulation and coding schemes employed, but we assume that convexity holds, namely,the convex relationship between power and transmission rate. This assumption is commonly usedin the literature and serves as a good model in many practical coding schemes.

Generally, the rate-power relationship is time-varying due to characteristics of wireless links,such as fading. The time-varying nature of the link can again be exploited to improve energyutilization by waiting for the channel to be in a good state and only transmitting opportunisticallyat times when the channel gains are high. In order to illustrate this point, consider a discrete-time,block fading wireless channel with additive white Gaussian noise and frequency-flat fading. Lett N denote discrete time slots, where each time slot corresponds to a transmission block of Nconsecutive channel symbols. Let H(t) denote the channel gain process and assume a block fadingmodel so thatH(t) is constant over a time slot, which is a reasonable approximation if the coherencetime of the channel exceeds a time slot. Let X = (X1, , XN) and Y = (Y1, , YN) denote,respectively, the input and output channel symbols during a time slot, Z = (Z1, , ZN) be theadditive white Gaussian noise, and H be the corresponding fading coefficient (Xi, Yi, Zi,H C).Then,

Y = HX + Z. (1.2)Assuming that the block length N is sufficiently large and that H is known at the transmitter andreceiver, the maximum reliable transmission rate r during a given time slot is given by [64],

r = log(

1 + P |H(t)|2

N0

)bits/transmission, (1.3)

where P = E[|X|2] denotes the transmission power used by the input symbols and N0 is the noisepower.1 By varying the average power used for transmission, one can control the transmission rate.Inverting the above relationship gives the minimum amount of power necessary to transmit reliablyat a certain rate, i.e.,

P = N0|H |2(2r 1) . (1.4)

1To be precise the expression in (1.3) upper bounds the maximum reliable rate per time slot. This bound can be approachedarbitrarily close as the number of channel uses per time slot becomes large. This gives a reasonable indication of the rate inmodern wireless systems where each channel use is on the order of 106 seconds and channel coherence times are on the orderof 102 seconds.

4 1. INTRODUCTION

Note that transmission power required per time slot is a convex function of the rate in that time slot.Furthermore, this power-rate relationship depends on the channel gain |H |2 in a particular timeslot.

More generally, in this book, we consider a power-rate function of the form,

P = g(r(t), c(t)), r(t) 0 (1.5)where g(., c(t)) is a convex, non-decreasing function of the transmission rate r . Dependence ontime t is through the time-varying channel state c(t) (e.g., |H |2 in the previous example).While theabove relationship was motivated using a simple AWGN channel model, it applies more generallyto most practical non-capacity achieving modulation techniques.

Themodels considered in this book apply to settings inwhich the channel state c(t) is availableat the transmitter. This is appropriate for settings in which the fading process is slow enough thatit can be estimated and fed back to the transmitter. For example, modern wireless systems providepilot symbols for this estimation and provide feed-back on the order of 5 msec, enabling one to trackrelatively fast fading and adapt the transmission rate on a similar time scale.Of course, this estimationand feedback process is subject to errors, but we will typically make the simplifying assumption thatit is error-free so that we can focus on the scheduling aspects of these problems.

1.1.2 SERVICEREQUIREMENTSIn the context of a point-to-point link, the service requirements of interest are typically throughputand delay. Until recently, most researchers have focused on maximizing network throughput orlink capacity, through effective network control. However, delay considerations are critical for arapidly growing class of real-time applications that impose strict restrictions on packet delays.Manyapplications such as real-time audio and video, gaming, and time-critical financial transactions havehard delay constraints, i.e., the data must be delivered by the deadline or else it is of no value.Other applications, including traditional Internet services, such as web browsing, file transfers, andemail exchanges, can tolerate some delays, but increasingly have come to expect good (average) delayperformance. Below we briefly discuss models for throughput and delay over a wireless link, and therelationship between energy and throughput and delay performance.

1.1.3 THROUGHPUTThroughput refers to the (average) achievable data rate over the link.Over an additivewhiteGaussiannoise channel (AWGN) the maximum achievable throughput is given by the Shannon Capacityformula (1.1). When the channel is non-time-varying, the relationship between throughput andenergy is rather straightforward, as energy is simply the product of power over time.However, whenthe channel is time-varying, its capacity will be given by an average of instantaneous capacityexpressions as in (1.3), a quantity sometimes referred to as the ergodic capacity of the channel. Thus,it is clear from (1.3) that throughput can be increased by expending more energy when the channelis in a good state, and less when the channel is in a poor state. In fact, it was shown in [34] that the


ergodic capacity of such a time-varying channel, for a given average power constraint, is achievedby the well known water-filling formula; where the amount of power used at each point in time isallocated as a function of the channel gain |H(t)|2, so that the marginal throughput with respect topower is equal at all times, subject to the average power constraint.

While the water-filling solution maximizes ergodic capacity, it requires full knowledge of thechannel statistics, an infinite time horizon, and an average power constraint. Over a finite timehorizon, it may be possible to achieve higher throughputs by opportunistically waiting for goodchannel instances. Suppose for example, that a transmitter has a battery with a small amount ofenergy that must be expended over a certain amount of time. In that case, the transmitter may wantto wait for the channel to be in a good state before expending any energy. In fact, it was shownin [28] that given a finite amount of energy, and a deadline by which the energy must be spent,throughput can be maximized using a dynamic programming solution that takes into account boththe channel state, and the proximity to the pending deadline; thus, introducing a trade-off betweenthroughput, energy, and delay. More recent work exploring such a trade-off with limited channelstate information can be found in [1].

1.1.4 DELAYThroughput is considered to be a first-order performance metric in a communications systems.Delay is a finermetric, that in general ismuchmore difficult to characterize and optimize.Typically,delay refers to the amount of time that a message or packet takes to be transmitted from the sourceto its destination. Over a single-hop communication link, delay refers to the amount of time thatelapses from the moment that the message arrives at the link, until it is delivered in its entirety acrossthe link. In systems with random message arrivals, delay has two components queueing delay: theamount of time that the message has to wait for other messages that are ahead of it in the queue,and transmission delay: the amount of time that the message requires for transmission.

Both queueing delay and transmission delay are directly impacted by the data rate used overthe channel, and that quantity is related to the transmission power, as given by (1.5). In this bookwe explore this trade-off between energy and delay in both systems with random packet arrivals aswell as in systems where the packet arrival process is deterministic and known in advance. Whenthe arrival process is deterministic, delay is a deterministic function of the transmission rate. Forexample, consider a simple problem of a packet that needs to be transmitted over a single link.The delay experienced by that packet is a function of the data rate, which, in turn, is a functionof the transmission power. Thus, there exists a trade-off between the transmission delay and thetransmission power or energy. In this context, the problem of interest is: given a (deterministic)packet arrival stream with deadline constraints on the packets, by which they must be delivered, howshould transmission energy be allocated so as to minimize the amount of energy needed to meet thedeadline constraints?

In contrast, when the packets arrive at random, delay is not only a function of the transmissionrate (or power) but also of the random arrival process.Hence, in the context of randompacket arrivals,

6 1. INTRODUCTION

the performance metric of interest is average delay. Again, a trade-off arises between average delayand transmission energy. In this context, the problem of interest is: given an arrival process, and anaverage delay constraint, how should transmission energy be allocated so as to minimize the averageamount of energy needed to meet the average delay constraint?

1.1.5 OVERVIEWOFBOOKWhile there has been much work on energy-efficient networking over the past decade, the focusof this book is on saving energy via effective transmission rate control. This problem is particularlymeaningful when data is subjected to certain quality-of-service requirements such as constraintson transmission and queueing delays. In Chapter 2 we address the problem of transmission ratecontrol subject to a strict deadline constraint on the data; related work on the general topic canbe found in [5; 18; 19; 28; 32; 41; 43; 49; 53; 60; 65; 68; 70; 72; 73; 74]. In particular, we assumethat the arrival times, and deadline constraints on the data are known in advance and that thechannel is not time-varying.With such assumptions, we develop a simple and intuitive mechanismfor meeting the deadline constraints using the minimum amount of energy. We also consider thecase of a time-varying channel, and using techniques from stochastic optimal control we developoptimal algorithms for minimizing expected energy subject to the deadline constraints. In Chapter3 we turn our attention to the stochastic setting where data arrives to the link at random and mustbe queued for transmission. In this context, we consider the problem of minimizing expected energyconsumption subject to average delay constraints. In particular, we develop a dynamic programmingformulation for optimally solving the problem, and consider the asymptotic behavior of the optimalaverage energy needed both in the high-delay and low-delay regimes.

Transmission Rate Adaptation under Deadline ConstraintsIn many applications, data must be transmitted by a certain time in order to be of any value.This is particularly the case for real-time applications such as voice, interactive games, and militaryapplications that require timely delivery of command, control, situational awareness, or positioninginformation. InChapter 2 we focus our attention on the case of hard deadline constraints.We assumethat data arrives and must be transmitted within a certain time interval.

As mentioned earlier, in principle, such a deadline constraint can be met by transmitting withhigher power. As implied by equation (1.5), using higher power will result in higher data rates, thusdeadlines can be met by simply increasing the transmission power.However, since power is a convexfunction of rate, higher data rates require more energy per bit. Thus, there is a trade-off betweenthe ability to meet the deadline, and the amount of energy required. It was first observed in [52]that, in fact, due to this concave relationship between rate and power, the optimal way to meet thedeadline constraint with minimum energy is to transmit at the lowest possible data rate, while stillmeeting the deadline.This notion, which we explore in Chapter 2 in great detail, was called in [52]lazy packet scheduling, a term that is now commonly used in the literature.


The idea of lazy packet schedulingwas generalized in [73] to a general setting of (deterministic)packet arrivals, with various deadlines, where a very physically intuitive transmission rate adaptationscheme was developed. In particular, [73] developed an interesting geometric visualization for theoptimal departure curve, as shown in Figure 1.1, where data arrivals are represented by the curveA(t) and deadlines by the curve Dmin(t).

A(t)

Dmin(t)

string

A(t)

Dmin(t)Dopt(t)

0 T 0 TFig. (a) Fig. (b)

Figure 1.1: Optimal departure curve depicted as a taut string: (a) string lying between the curves A(t)and Dmin(t), and (b) the optimal curve,Dopt (t), denoted as the tightened string.

Another challenge that arises in the context of meeting hard deadline constraints is thatin a wireless setting, the channel is often time-varying. This variability in the channel conditionsrepresents both a challenge and an opportunity.The challenge is that if the channel is in a poor state asthe deadline approaches, it may be very difficult (or costly) to meet the deadline.On the other hand,the variations in channel quality gives rise to opportunistic scheduling, where the transmissionsare scheduled at times that the channel conditions are good, thus significantly reducing transmissionenergy. The idea of opportunistic scheduling is explored in the second half of Chapter 2, where weexplore both discrete-time and continuous time optimization methods.

Intuitively, in the presence of time-varying channels, one should wait for the channel to bein the best possible state to transmit. Indeed, such a problem has the flavor of an optimal stoppingtime, formulation,where the optimal solutionwould be a simple threshold on the channel conditions(i.e., if the channel is above a certain threshold, you would transmit, and otherwise you would wait).In the presence of a deadline, the transmitter cannot wait forever for the channel to be good, andthus, the threshold on the channel condition diminishes as the deadline approaches.

In Section 2.1.3 we explore the use of discrete-time dynamic programming techniques to solvethis problem. Of course, discrete-time dynamic programming suffers from the well-known curseof dimensionality problem, and may not be scalable to larger problem instances. An alternativeapproach, based on continuous-time optimization, is explored in Section 2.1.4, where we show thatthe optimal transmission rate can be obtained as a nearly closed-form expression.

8 1. INTRODUCTION

Having focused on transmission scheduling for a single user over a single link, we thenturn our attention to the case of multiple users. The presence of multiple users, who share thecommunication link, adds another interesting dimension to this problem. In particular,withmultipleusers, scheduling decisions must be made over both time slots as well as among the users. Over atime-varying channel one can schedule opportunistically by serving the user with a better channel,as we discuss in Section 2.2.1. Moreover, in a multiple access setting, it is possible to exploit thestructure of the multiple-access capacity region, where higher capacity can be achieved by havingusers transmit simultaneously. Thus, in Section 2.2.2 we discuss energy-efficient scheduling over amulti-access channel and introduce the FlowRight algorithm of [65] that uses an iterative methodto optimize energy consumption over time and across users.

Transmission Rate Adaptation under Average Delay ConstraintsIn contrast to real-time interactive applications, elastic applications, such as web browsing or filetransfers, do not impose a hard-deadline on the data. In these contexts, the data does not lose itsvalue if it is delayed, but obviously timely delivery is still essential.Thus, the quality-of-service metricof interest is average delay, and a wireless service provider may wish to design their system so thatthey can provide a guarantee on average delay.

We consider such formulations in Chapter 3, where the goal is to minimize the time averageenergy subject to a constraint on the average delay. In particular, we consider the case where dataarrives at random, as would be the case in many elastic applications, and is queued at the transmitteruntil it is transmitted. Rate allocation decisions must be made on-line, and the transmitter takesadvantage of his knowledge of channel and arrival statistics in making these rate allocation decisions.

Just as in Chapter 2, this problem can be formulated using a dynamic programming approach.In particular, we seek transmission policies that use the minimum average power P , subject to aconstraint on average delay D. For a given value of D, determining P(D) can be viewed as anaverage cost constrained Markov decision problem [3]. Such problems differ from standard Markovdecision problems in that in addition to optimizing a long-run objective (e.g., power), one mustalso satisfy a constraint that depends on second long-run cost (e.g., delay). Using techniques fromdynamic programming, we are able to derive some interesting structural properties of the optimalpolicy.

For example, for linear rate-power costs, the optimal transmission policy depends on the queuebacklog, via a critical backlog value, which depends on the channel and arrival state. The optimalpolicy tries to keep the queue size equal to this value. When the queue size is smaller than thecritical value, nothing is sent so that the queue size increases. When the queue exceeds this value,the optimal policy drains the queue so that it is exactly equal to this value. Hence, the policy isnon-decreasing in the buffer size for a fixed channel and arrival state. As we discussed in Chapter2, it is energy efficient for the transmitter to be opportunistic and wait for the channel to be in agood state. However, when delay enters the picture, the policy must balance between waiting for a


good state and increasing the delay. The critical backlog values can be viewed as a way of balancingbetween energy and delay. This insight partially extends to other rate-power costs as well.

Additional structural properties can be obtained when we consider asymptotic regimes interms of the delay constraints. In particular, we consider two regimes, when the average delay isallowed to be very large (i.e.,D ) and when the delay is kept very close to the minimum (i.e.,D = 1). In each case we discuss the limiting behavior of the required average power and furtherdiscuss the optimal rate at which this limit can be approached.When the delay is allowed to be verylarge, it follows that the optimal policy approaches the classical problem of maximizing the ergodiccapacity of a fading channel.The solution to this is the well-known water-filling power allocation.A class of buffer threshold policies, which only depend on the buffer size via a threshold are shownto achieve the optimal rate of convergence to this limit. In contrast, in the small delay regime, thelimiting power may be infinite or finite depending on the fading distribution near zero. In this case,a class of policies that only weakly depend on the channel via a threshold rule are shown to achievethe optimal convergence rates.

11

C H A PT E R 2

Transmission Rate Adaptationunder Deadline Constraints

Strict packet deadlines is an important class of quality-of-service constraints, wherein the data mustbe delivered to the destination within a fixed time interval. Such constraints arise naturally in manyreal-time communication andmonitoring scenarios.Examples of these include, online video stream-ing, real-time communication between remote participants, time-critical financial transactions, andreal-time environment monitoring.With the wide adoption of mobile computing devices, internet-based interactive services, and widespread sensor deployment, these application classes have seen atremendous growth in volume in recent years.

Communicating data traffic over wireless links,with strict quality-of-service constraints, leadsto significant resource consumption in terms of bandwidth and energy expenditure [40].Optimizingresource utilization is thus important for modern wireless systems, albeit, challenging in the presenceof strict quality-of-service constraints. While, in general, there are many resources that need to beoptimized, in this chapter,we will focus onminimizing the transmission energy cost at the link-layer.Specifically, we will study how the technique of dynamic transmission rate control can be utilized toachieve the above objective, and explore several different mathematical formulations that highlighthow the convex power-rate relationship, time-varying channel conditions and bursty nature of trafficcan be exploited to minimize total transmission energy cost with packet deadlines and other quality-of-service constraints. We will also highlight the mathematical tools and techniques that can beapplied to solve these problems and develop new insights that will be useful in engineering practicalsolutions.

We begin in Section 2.1 with studying optimal rate control over a point-to-point wireless link.Specifically, Section 2.1.2 studies optimal rate control under a deterministic setting and serves as animportant step in developing many insights on how data arrivals and quality-of-service constraintsaffect optimal data transmission. Sections 2.1.3 and 2.1.4 study the problem of transmitting B unitsof data by deadline T under stochastic fading using discrete-time and continuous-time approachesrespectively. Sections 2.1.5 and 2.1.6 extend the results to consider multiple deadlines where wealso outline an illustrative online policy, which is motivated from the analytical results, to serve anarbitrary stream of packet arrivals with deadlines. Section 2.2 studiesmulti-user scenarios of dynamicrate control with deadlines constraints. Specifically, we consider two main set-ups: one involving atime-shared system with a single transmitter and multiple receivers, and the other, a multi-accessscenario involving multiple transmitters and a single receiver. These scenarios provide insights on

12 2. TRANSMISSIONRATEADAPTATIONUNDERDEADLINECONSTRAINTS

how to perform user-scheduling with rate control in amulti-user environment under packet deadlineconstraints. Finally, Section 2.3 summarizes the results presented in this chapter.

2.1 RATECONTROLOVERPOINT-TO-POINTWIRELESSLINK

Consider a point-to-point wireless link with a transmitter and a receiver. Data generated by higher-layer applications arrives to the transmitter queue with quality-of-service (QoS) constraints imposedby the application requirements.As the transmitter sends this data to the receiver, it has the capabilityto dynamically adjust its transmission rate which directly affects its transmission energy cost.Withthe goal of minimizing the total transmission energy cost, an important question that arises in thisbasic setting is how should the transmitter optimally adapt its rate in response to the quality-of-servicerequirements, arriving traffic and time-varying wireless channel conditions to minimize the transmissionenergy cost,while ensuring that theQoS constraints aremet?Our goal in this sectionwill be to address theabove question and develop new insights by presenting several different mathematical formulationsthat study different aspects of the above problem.

2.1.1 SYSTEMMODELIn a typical transmitter,data generated by an application is processed bymany layers in the networkingstack.These include the application layers, transport/IP layers, andMAC/data-link layers [58].Eachof these layers associate a context and the corresponding protocol specific information to the data.The exact functioning of these layers is dependent on the particular system implementation,however,from our perspective of designing optimal rate control algorithms it suffices to focus on the data-link layer the layer that handles how data is encoded and transmitted on the wireless channel;we abstract a transmitter system as simply a data queue. Within this abstraction, we will need tomodel three aspects: (1) data flow and QoS model which defines the flow of data in and out ofthe queue, and the quality-of-service constraints associated with it; (2) power-rate function model which defines how much transmission power is incurred when using a particular transmission rate;and (3) wireless channel model which defines the stochastic model for the time-varying wirelesschannel fluctuations. The general models for these three aspects are discussed in detail next, whichare then appropriately specialized in the subsequent sections.

Data Flow andQoSModelDataflow in a transmitter queue can be succinctly represented using a cumulative curvesmethodologybydefining three curves as givenbelow.These definitions are based on [73] and are presented verbatimhere.They are motivated from the concept of cumulative curves in network calculus theory [15; 23].

2.1. RATECONTROLOVERPOINT-TO-POINTWIRELESS LINK 13

data arrival curve

departure curve

min. departure curve

0Fig. (b)

time t

data

0

arrival curve

min. departurecurve

departure curve

Fig. (a)time t

Figure 2.1: Representation of cumulative curves arrival, departure, and minimum departure curves;(a) fluid arrival model, (b) packet arrival model.

Definition 2.1 (Arrival Curve) An arrival curveA(t), t 0, t R, is the total number of bits thathave arrived in time interval [0, t].

Definition 2.2 (Departure Curve) A departure curve D(t), t 0, t R, is the total number ofbits that have departed (served) in time [0, t].

Stated simply, the arrival curve specifies how data arrives to the transmitter queue and thedeparture curve specifies how data departs from the queue.

Broadly, there are two different mathematical models for data flow, namely, packet and fluidmodels. In a packet model, data is represented at discrete granularity of pre-defined packet sizes,while in a fluid model data is assumed to be infinitely divisible.The functionA(t) in a packet modelis generally a piecewise-constant function, whereas for a fluid model A(t) is a continuous function;more generally, in case of a mixed model, the arrival curve would be a combination of the two.

Since a departure curve represents the bits served by a transmitter, clearly, we require thatD(t) A(t) as the transmitter would not transmit more data than what has arrived to the queue bytime t ; this constraint is referred to as the causality constraint.

Next, we introduce a notion of a minimum departure curve to model the quality-of-servicerequirements. In general, quality-of-service constraints are application dependent and vary widelywith different application classes; hence, it would be impossible to have a universally applicablequality-of-servicemodel.Yet, the goal in the following definition is to present a simple,but reasonablygeneral, way to abstract the quality-of-service constraints within a cumulative curves framework.

Definition 2.3 (Minimum Departure Curve) Given an arrival curve A(t), a minimum departurecurve Dmin(t) is a function such that Dmin(t) A(t),t 0, and is defined as the cumulativeminimum number of bits that if departed by time t would satisfy the quality-of-service requirements.


Thus, a minimum departure curve abstracts the quality-of-service constraints as a cumulativecurve.Figure 2.1 shows a pictorial representation of these various cumulative curves.The arrival curveA(t) is always the topmost curve, while the minimum departure curve Dmin(t) is the bottommostcurve representing the quality-of-service constraints. A departure curve D(t) lies between the A(t)and Dmin(t) curves.

The function Dmin(t) should be viewed as a constraint function which imposes constraintson how the data should be transmitted; i.e., any departure curve D(t) must satisfy D(t) Dmin(t).Clearly, the exact nature of this constraint depends on the underlying quality-of-service requirementswhich finally determines the form of the minimum departure curve. To illustrate the generality ofthe above approach in modeling various quality-of-service constraints, we next discuss two examplesof common quality-of-service constraints and construct their respective minimum departure curves.Note that in the definition of Dmin(t), the implicitly assumed service discipline is hidden since welook at the data flow only in a cumulative sense.We highlight this in the examples below.

(a) Deadline constrainttime

data

A(t)

Dmin(t) = A(t-d)d

(b) Buffer constrainttime

data

A(t)

Dmin(t) = max[A(t)-B,0]

B

Figure 2.2: Examples showing how a minimum departure curve models quality-of-service con-straints: (a) packet deadline constraint of d, and (b) buffer constraint of B.

Deadline Constraint : Consider a stream of packet arrivals with a uniform deadline constraintof d; i.e., each packet must depart within a time window of d from its arrival instant. Figure 2.2(a)depicts the arrival curve A(t) of such an arrival stream. Define a minimum departure curve asDmin(t) = 0, t [0, d) andDmin(t) = A(t d), t d.Following a first-come-first-served servicediscipline such that the departure curve satisfiesDmin(t) D(t) A(t), t , it can be seen that thedeadline constraints will be satisfied. Thus, in this case,Dmin(t) is simply a time-shifted version ofA(t) as shown in Figure 2.2(a).This example can be easily extended to the case of variable deadlineconstraints where the order of arriving data and deadlines is preserved (i.e., early arriving data hasearlier deadlines) [73].

Buffer Constraint : Consider a stream of packet arrivals and a constraint on data transmissionwhich requires that the transmitter queue size does not exceed B,t 0. In practice, such a con-


straint could arise either due to a limited buffer-size at the transmitter (and no packet dropping)or it could be an indirect quality-of-service constraint which ensures that the average packet de-lays are bounded for an arrival stream. Given an arrival curve A(t) and a departure curve D(t), thequeue size is given as b(t) = A(t) D(t). Since, we require b(t) B, any departure curve must sat-isfy D(t) max[A(t) B, 0]. Thus, as shown in Figure 2.2(b), following a first-come-first-servedservice discipline, the minimum departure curve must be Dmin(t) = max[A(t) B, 0]. One cansimilarly generalize this case to a time varying buffer constraint B(t).

Power-Rate FunctionModelAt the data-link layer of a transmitter, data bits are processed, encoded, modulated, and convertedinto analog signals for transmission over a wireless channel. Modern wireless devices are equippedwith rate-adaptive capabilities [39; 45] which allows the transmitter to adjust the transmission rateover time. This is achieved in various ways which include adjusting the power level, symbol rate,coding scheme, constellation size, and any combination of these approaches. In most transmitter-receiver systems, for a certain bit-error probability, the incurred transmission power (energy per unittime) is an increasing function of the rate chosen.While the exact relationship between transmissionpower and the data rate depends on the particular implementation of the transmitter and the receiversystem, a general property of convexity applies for most systems under this property, transmissionpower is a convex function of the rate.

Let P(t) denote the required transmission power to reliably transmit at rate r(t) at time t .We assume the following power-rate relationship,

P(t) = g(r(t), c(t)) (2.1)

where the function g(r, c) is a convex increasing function with respect to the rate and g(r, c) 0for r 0. The relationship in (2.1) is a general transmission model for most encoding schemes andis a standard mathematical model widely studied in the literature in various forms [5; 12; 21; 29;41; 49; 52; 65]. As appropriate, we will further specialize the above power-rate relationship in thesubsequent sections.

Wireless ChannelModelAs is well-known,wireless signals are electromagnetic radiation from a transmitter to a receiver.Overthe open air, these signals suffer from several well-known phenomena such as path-loss, shadowing,multi-path fading, etc., due to changes in the environment and node mobility, that result in time-varying signal strength fluctuations.The effect of these variations is generally reflected in the discretebaseband signal at the receiver as a time-varying fading coefficient [64], and is modeled naturally asa stochastic process.

We consider broadly two models for channel fading: i.i.d. andMarkov channel fading. In i.i.d.channel fading the channel state evolves as a discrete-time stochastic process with the channel statec(t) being i.i.d. random variables at different time slots.


For theMarkov channel model,we consider a first-order, continuous-time, discrete state spaceMarkov model [6; 72; 75]. Let C(t) represent the channel stochastic process with state space C.Thevarious states c C are denoted as c1, c2, . . . , cm. Let cc denote the channel transition rate fromstate c to c.The sum transition rate at which the channel transitions from a state c is,c = c =c cc.After transitioning to state c, the expected time that C(t) spends in state c is 1/c which can beviewed as the coherence time of the channel in state c.

Let = supc c, and define a random variable Z(c) as,

Z(c)={c/c, with prob. cc/, c = c1, with prob. 1 c/

(2.2)

Using Z(c), the channel process evolution can be succinctly described as follows:The channel processC(t) evolves as a Markov process where, given a particular channel state c, there is an exponentiallydistributed time duration with rate after which the channel state changes. The new state is a randomvariable which is given as C = Z(c)c. This is a standard uniformization technique which gives astochastically identical scenario and does not result in any loss in the generality of the process.Given a channel state ci , the values taken by the random variable Z(ci) are denoted as {zij }, wherezij = cj /ci , and the probability that Z(ci) = zij is denoted as pij . If there is no transition fromstate ci to cj , then, pij = 0.

2.1.2 DETERMINISTIC SETTINGWITHOUTCHANNELFADINGLet us start with a deterministic settingwhere the arrival curve is deterministic and known in advance,and there is no channel fading. In terms of the mathematical model, therefore,A(t) is known for allt and P(t) = g(r(t)); i.e., the power-rate function g() does not change with time.This setup servestwo purposes first, being mathematically tractable, it helps develop many insights on how dataarrivals and QoS constraints affect transmission rate control and, second, it models static scenarioswhere data arrivals can be predicted in advance and channel fading is negligible on the time-scaleof data transmission. The results presented in this section are based on [70; 73].

Consider an arrival curve A(t) over time interval [0, T ]. As discussed in Section 2.1.1, basedon the QoS constraints, one can construct the minimum departure curve Dmin(t). Thus, A(t) andDmin(t) are given curves over time interval [0, T ].The goal of the transmitter is to transmit the data,which arrives to its queue according to A(t), and minimize the total transmission energy cost whilesatisfying the constraints imposed by Dmin(t). An admissible departure curve D(t), which specifieshowdata is transmitted, is one that satisfies both the causality and theQoS constraints; i.e.,Dmin(t) D(t) A(t), t [0, T ]. Note that a departure curve explicitly defines a transmission policy (i.e.,at time t transmit at rateD(t)); thus, we will use the terms departure curve and transmission policyinterchangeably. The energy minimization problem is to obtain the minimum-energy admissible


departure curve and can be expressed mathematically as follows:

minD(t)

E(D(t)) = T

0g(D(t))dt (2.3)

subject to Dmin(t) D(t) A(t), t [0, T ]Without loss of generality, we can assume thatDmin(0) = 0,D(0) = 0, andDmin(T ) = A(T ).Thelast equality simply states that the queue must be empty by time T , otherwise any excess data whichhas no quality-of-service constraints can be simply discarded from consideration.

Referring to Figure 2.1, the energy minimization problem can be understood as finding acumulative curve that lies between given curves A(t) and Dmin(t) (admissibility requirements) andhas the lowest total energy cost (performance goal).

TheBT -problemAs a warm up, let us first begin with the following example: at time t = 0, a transmitter has B unitsof data which must be transmitted to the receiver by deadline T using minimum energy; there areno new arrivals during time t (0, T ). We refer to this as the BT -problem.

The two curves A(t) and Dmin(t) for this problem are straightforward. The arrival curveA(t) = B, t [0, T ] as there are no new arrivals beyond an initial amount B of data. Theminimum departure curve isDmin(t) = 0, t [0, T ) andDmin(T ) = B, since there is no minimumdata transmission requirement until the deadline when all the data must have been transmitted.Figure 2.3 depicts the various cumulative curves for the BT -problem including a few admissibledeparture curves.

time

data

A(t)

Dmin(t)

D(t)

T0

B

Dopt(t)

Figure 2.3: Cumulative curves,A(t),D(t) and Dmin(t) for the BT -problem.

If we restrict attention to constant-rate transmission policies, choosing a rate lower thanB/T will violate the deadline constraint. Choosing a rate higher than B/T will result in a fastertransmission; however, even though data transmission takes less time, the total energy cost is higher


since the power-rate function is convex in the rate which means that transmitting at a higher raterequires much higher increase in the transmission power (i.e., power increases faster than linear).Thus, intuitively, for constant-rate transmissions a rateB/T minimizes the energy cost.The questionthat now arises is: what happens if we also allow policies with non-constant transmission rates? Aswe discuss next, the above argument generalizes to non-constant transmission policies; namely, theoptimal transmission policy for the BT -problem is to transmit at a constant rate of B/T , i.e.,

Dopt (t) = BtT

, t [0, T ], (2.4)where Dopt (t) denotes the optimal departure curve.

To see why the above is true, consider the integral version of Jensens inequality [36] as givenbelow:

Let f (t), p(t) be two functions defined for a t b such that f (t) and p(t) > 0.Let (u) be a convex function defined on the interval u ; then

( baf (t)p(t)dt bap(t)dt

) ba(f (t))p(t)dt b

ap(t)dt

(2.5)

with strict inequality if () is strictly convex and a = b, = .Now consider an admissible departure curveD(t) and substitute the following in (2.5):p(t) =

1, () = g(), f () = D(), a = 0 and b = T . Thus, we get,

g

( T0 D

(t)dt T0 dt

)

T0 g(D

(t))dt T0 dt

(2.6)

g

(D(T ) D(0)

T

)T

T0

g(D(t))dt (2.7)

g (B/T ) T T

0g(D(t))dt (2.8)

In (2.8) above, the left-hand side gives the total energy cost of a constant-rate transmission policy(with rate B/T ), while the right-hand side is the total cost of an admissible departure curve; theoptimality claims thus follow from the inequality in (2.8).

Note that when g() is strictly convex, the inequality in (2.8) is strict for any admissibledeparture curve; hence, in this case the constant-rate policy at rate B/T is also the unique optimalsolution.

While not surprising to see that a transmission policy with rate B/T minimizes the energycost, when viewed in terms of the cumulative curves, an interesting geometric visualization emergesof this optimal solution.Consider Figure 2.3 and suppose that a string is tied at one end to the origin(0, 0) while the other end is passed through the point (T , B). Now if we pull this string tight, itstrajectory, which in the simple BT -case is a straight line, gives the optimal departure curve as givenin (2.4).


General Arrival andMinimumDeparture CurvesConsider now the case of general arrival and minimum departure curves. As mentioned earlier, ourgoal is to find D(t), such that Dmin(t) D(t) A(t), t [0, T ] with the least energy cost. Wesaw previously in the BT -case that the optimal solution can be visualized as a stretched string.Thisappealing visualization, in fact, generalizes and holds for the case of generalA(t) andDmin(t) curves.

Optimal Departure Curve as Stretched String [73]: Given the arrival and minimum departurecurves,A(t) and Dmin(t), respectively, consider a string with one end tied at the origin and the other endpassed through the point (T ,Dmin(T )). Let the string be constrained to lie between A(t) and Dmin(t);i.e., these two curves form hard boundaries for the string. Now, if we make the string tight, its trajectory isthe optimal departure curve.

A(t)

Dmin(t)

string

A(t)

Dmin(t)Dopt(t)

0 T 0 TFig. (a) Fig. (b)

Figure 2.4: Optimal departure curve depicted as a taut string: (a) string lying between the curves A(t)and Dmin(t), and (b) the optimal curve,Dopt (t), denoted as the tightened string.

Figure 2.4 illustrates the stretched-string visualization where A(t), Dmin(t) and the corre-sponding Dopt (t) curves are depicted. As shown in the figure, Dopt (t) is a stretched string lyingbetween the A(t) and Dmin(t) curves. Note that, as opposed to the BT -problem, now the optimalcurve is not simply a single rate transmission, but instead has intervals of constant rate transmissions.Furthermore, as depicted in [73], there could also be portions of continuously varying transmissionrates depending on the shape of A(t) and Dmin(t) curves.

To understandwhy the above description gives the optimal policy,we first discuss the necessarycondition for an admissible departure curve to be energy optimal as presented in [73] and referredto there as the optimality criterion; it is presented verbatim below.

Optimality Criterion: Let D(t) be an admissible departure curve and L(t) be a straight linesegment over [a, b] that joins points D(a) and D(b), 0 a < b T . If L(t) satisfies Dmin(t)


L(t) A(t) and L(t) D(t), the new departure curve Dnew(t) constructed as,

Dnew(t) = D(t), t [0, a)= L(t), t [a, b]= D(t), t (b, T ]

satisfies E(Dnew(t)) E(D(t)), where the inequality is strict if g(.) is strictly convex.In simple terms, the optimality criterion states that replacing a portion of an admissible

departure curve with an admissible straight line (if one exists) results in a lower energy cost.In light of the result for the BT -problem, the optimality criterion is fairly intuitive, since

along a departure curve, moving from point D(a) to D(b) is equal to transmitting (D(b) D(a))units of data in time (b a), and if this transmission can be done at constant rate (i.e., followinga straight line L(t)) without violating the admissibility constraints, it will have a lower energy cost.Thus, for any such departure curve we can obtain a lower energy departure curve by changing thetransmission policy to a constant-rate transmission over t [a, b) at rate (D(b) D(a))/(b a).A departure curve will be said to satisfy the optimality criterion if it does not have any two points thatcan be joined by a distinct admissible straight line.

Interestingly, for a linear power-rate function, i.e.,P = Kr , whereK > 0 is a constant, it canbe easily seen from (2.3) that all admissible departure curves have the same energy cost. Thus, inthis case, transmission rate-adaptation is meaningless as it does not affect the total energy cost ofdata transmission and any policy that satisfies the causality and the QoS constraints suffices.

It can be shown that a departure curve that satisfies the optimality criterion is unique, andif g() is strictly convex, such a departure curve is, in fact, the unique energy-minimizing curve.We refer the reader to [73] for a proof of the uniqueness property. Thus, we see that, under strictconvexity of the power-rate function, an admissible departure curve that satisfies the optimalitycriterion gives the optimal solution.

Now, it is easy to see that a stretched-string, as discussed earlier, gives the optimal departurecurve. Because in a taut condition a string cannot be made tighter between any two points along itstrajectory, which means that the optimality criterion must be satisfied along such a curve, otherwiseany such portion of the curve can be replaced by a straight line, making the string tighter, leadingto a contradiction.

Besides having lowest energy cost, the optimal departure curve also has other interestingproperties as discussed in detail in [73]. These include (a) minimal maximum power, where theoptimal departure curve has the smallest maximum-instantaneous-power cost (i.e., the maximumvalue of the transmission power incurred over the entire departure curve is minimized by usingthe optimal departure curve), or the smallest maximum-instantaneous-transmission-rate, amongall admissible departure curves; and (b) shortest length, where the optimal departure curve has theshortest length among all admissible departure curves. The first property is useful when the energyminimization problem in (2.3) also has a maximum power constraint. In that case, either (2.3) with


the additional maximum power constraint has no solution or the optimal policy is the same as thestretched string solution.

For illustration, we next discuss two specific examples for which the optimal departure curvecan be explicitly obtained using the stretched-string visualization. These examples are presented todevelop intuition and provide insights on how a stretched-string visualization can be utilized toobtain concrete rate-adaptation policies.

ExamplesThe first example that we consider is a generalization of the BT -problem where a transmitter queuehas data with multiple deadlines, while the second example considers the case where data arrives tothe transmitter queue and all of it must be served by a common deadline T .

Example 1 (Buffer with multi-deadline packets): Consider a transmitter queue with M datapackets with the j th packet having bj units of data and a deadline tj , j = 1, ..,M .Without loss ofgenerality, let t1 < t2 < < tM . The packets in the queue are served in the earliest-deadline-firstorder and the goal is to minimize the total transmission energy cost.This problem models scenarioswhere a transmitter wants to empty its queue under deadline constraints.

Let Bj = jl=1 bl , which denotes the total amount of data in the first j packets. Since thereare no new data arrivals, the arrival curve A(t) is constant and equal to BM , while the minimumdeparture curve Dmin(t) is piecewise constant with jumps at times tj corresponding to the packetdeadlines as these represent the times by which a certain amount of data must have been transmitted.The two curves are shown in Figure 2.5(a). From the stretched-string visualization, one can see thatthe optimal departure curve is piecewise linear with the slopes of the linear segments monotonicallydecreasing in time. Figure 2.5(b) shows the construction of the first linear segment. In general, thesegments are constructed recursively as follows.

A(t)

t3(a)

0 t1 t2

s1Dmin(t)

B1

BM Dopt(t)

B2

tM t3(b)

0 t1 t2

B1

BM

B2

segment with max.slope,s1 = max (Bj / tj)

tM

s2

s1

Figure 2.5: (a) Cumulative curvesA(t),Dmin(t) andDopt (t) for Example 1: buffer with multi-deadlinepackets; (b) the first segment of Dopt (t) constructed as the segment with slope s1 = maxj

(Bjtj

).


Begin at the origin (0, 0) and look at line-segments that join the points (0, 0) and (tj , Bj )(jump points ofDmin(t)). From these segments, choose the one with the largest slope. Let denote

the maximizing index; i.e., = arg max(Bjtj

). The first segment of Dopt (t) is then constant-rate

transmission with rate Bt

over t = [0, t ). Starting at t , shift the origin to the new point (t , B)and repeat the process to obtain the next segment. The slopes {s1, .., sq} of these linear segmentsare thus given as,

sm = maxj{lm,..,M}

(Bj B(lm1)tj t(lm1)

)(2.9)

lm+1 = 1 + arg maxj{lm,..,M}

(Bj B(lm1)tj t(lm1)

)(2.10)

where l1 = 1, t0 = 0, B0 = 0 with initial m = 1. The iteration stops at lm+1 = M + 1.Example 2 (Arrivals with a common deadline) [52]: Consider a sequence of N packets arriving

to the transmitter in time interval [0, T ) with the constraint that all the packets must depart by timeT (common deadline). This problem models a scenario where an application generates data whichmust be transmitted to a receiver at the end of a common time window; for example, a sensor nodecollecting measurements that must be transmitted to the collection/fusion node within a fixed timeduration.

0

AN

s1

s2

Dopt(t)Dmin(t)

A3

t1 t2 t3 tN=T

A(t)

A1A2

0

AN

segment withmin. slope

s1 = min (Ai / ti)

A3

t1 t2 t3 tN=T

A1A2

Figure 2.6: Curves A(t),Dmin(t), and Dopt (t) for Example 2, and a depiction of the construction ofthe first segment of Dopt (t).

Without loss of generality, let the first packet arrive at time 0 and the rest at times {t i}N1i=1 . LetAi denote the cumulative amount of data in the first i packets which represents the total data arrivedto the queue just before time t i . The curves A(t) and Dmin(t) for this problem are straightforwardto construct and depicted in Figure 2.6 (where, tN = T ). An interesting point to note here is that,while the problem descriptions in Examples 1 and 2 are different, there is an apparent similarity


in the two problems. For example, the cumulative curves picture for the two scenarios are rotatedversions of each other. From the stretched-string visualization, the optimal departure curve consistsof piecewise-linear segments as in Example 1, but now with increasing slopes as shown in Figure 2.6.These linear segments can be constructed in a similar fashion as in Example 1 and their slopes aregiven as,

sm = mini{lm,..,N}

(Ai A(lm1)t i t (lm1)

)(2.11)

lm+1 = 1 + arg mini{lm,..,N}

(Ai A(lm1)t i t (lm1)

)(2.12)

where l1 = 1, t0 = 0, A0 = 0 with initial m = 1. The iteration stops at lm+1 = N + 1.In summary, the cumulative curves approach for formulating the transmission rate control

problem provides a powerful framework for studying energy minimization problems. As shown inthe scenarios studied in this section, it provides an intuitive graphical visualization of the problemand the respective optimal solution, and thus helps develop new insights into the problem.

2.1.3 STOCHASTICBT -PROBLEM:DISCRETE-TIMEAPPROACHThe optimal rate adaptation policy, as shown in Section 2.1.2, adapts the transmission rate in responseto the data arrivals and the quality-of-service constraints as imposed by the respective cumulativecurves.Over awireless channel, there is yet another factor that comes into play namely, time-varyingchannel conditions. For a given transmission rate, the transmission power incurred is a function of thechannel gain. If the channel gain is large (i.e., good channel condition) the incurred transmissionpower is lower than the case when the channel gain is small (i.e., bad channel condition). Bymeasuring the channel gain using pilot-signal techniques [64], a transmitter can make a reasonablyaccurate estimate of the channel condition at the next time-step, and opportunistically exploit thisadditional channel state information to adapt the transmission rate and further minimize energycost.

To focus on studying the effect of time-varying channel conditions on dynamic rate adaptation,we restrict our attention to the canonical BT -problem of transmitting B units of data by deadlineT , but now over a time-varying channel; thus, in this section, we do not consider data arrivals whichare treated later in Sections 2.1.5 and 2.1.6.We present two approaches: (1) discrete-time dynamicprogramming approach, and (2) continuous-time stochastic optimal control approach. These twoapproaches give many new insights on the problem and also provide several solutions under varyingassumptions.To begin, we first focus on the discrete-time dynamic programming approach and thenconsider the continuous-time framework in Section 2.1.4. The technical details in this section arebased on [28; 42; 43].


Dynamic Programming FormulationA natural approach to formulate the dynamic rate control problem under channel fading is usingstochastic dynamic programming.For theBT -problemof transmittingB units of data by deadlineT ,the mathematical formulation can be expressed as follows. Let t = 1, 2, , T denote the discrete-time index corresponding to a time-slotted system of data transmission. Let xt be the amount ofdata remaining to be sent and rt be the chosen transmission rate at time t .Without loss of generality,assuming the actual duration of a time slot to be unity, the amount of data transmitted in time slot tis also rt .Thus, xt evolves as xt+1 = xt rt . Denote the channel quality at time t as ct .The amountof energy consumed per time slot is then Pt = g(rt , ct ) (note that in the time-slotted model, powerand energy are interchangeable when measured per time slot).The objective is to find a transmissionpolicy that minimizes the expected energy cost; i.e.,

minrt

E

[T

t=1g(rt , ct )

](2.13)

subject to 0 rt xt ,t (2.14)T

t=1rt B (2.15)

The first constraint ensures that the amount of data transmitted per slot is non-negative and less thanthe amount remaining, while the second constraint ensures that the deadline constraint is satisfied.

While the above dynamic program has been formulated for the BT -problem, it is clear thatit can be generalized to the case of an arbitrary arrival and minimum departure curve. In particular,the constraints in (2.14) and (2.15) will need to be generalized based on the respective arrivaland minimum departure curves. In its full generality, it could be solved numerically using severalexisting techniques of solving constrained stochastic dynamic programs; however, the complexityof obtaining a numerical solution increases significantly and may even become intractable withincreasing generalization of the above dynamic program.

One of the natural specializations that can be made to the dynamic program in (2.13) is toconsider an i.i.d. channel model; i.e., assume that ct is independent and identically distributed intime. Such a model is a good approximation in scenarios where the time slot duration is on the orderof coherence time of the wireless channel, in which case the channel gain is i.i.d. over time. Underthis scenario, the recursive Bellman equation for obtaining the optimal solution of the BT -problemcan be obtained as follows.

Let Jt (x, c) denote the value function (cost-to-go) for time slot t , which represents theexpected energy cost incurred starting from time slot t with x amounts of data left and channel statec. Using Bellmans principle [13], we obtain the following recursion for t T 1 [28]:

Jt (x, c) = min0rx

[g(r, c) + E[Jt+1(x r, ct+1)]

](2.16)

and for t = T (the last time slot), JT (x, c) = g(x, c) (i.e., all the remaining data is transmitted tomeet the deadline constraint). For the i.i.d. channel model, we can define an expected value function


Jt (x) as Jt (x) = E[Jt (x, ct )]. The Bellman equation in (2.16) then becomes,Jt (x, c) = min

0rx[g(r, c) + Jt+1(x r)

]. (2.17)

Equation (2.17) can be intuitively understood as follows: at time slot t , total cost incurred is theenergy cost g(r, c) of transmitting r units of data under channel condition c, plus the average optimalenergy cost of transmitting the remaining x r units of data from time slot t + 1 onward; these arethe terms on the right-hand side of (2.17). Minimizing this total cost over all allowed choices of rgives the optimal cost at time slot t .

The recursive equation in (2.17) can be solved numerically by first starting with time-step Tfor which JT (x, c) = g(x, c), and then progressively solving for t = T 1, T 2, , 1; furtherdescription of such numerical solutions can be found in [28]. Additionally, in several special casesan optimal solution for (2.17) can be obtained. We next focus our attention on such special caseswith the goal of highlighting the insights that can be obtained from them.

Linear Power-Rate Function with Power LimitConsider the special case of a linear power-rate function where g(r, c) = r/c, but with a maximumpower limit of P per time slot. Thus, while the power required to transmit at rate r is linear in rfor a given channel state c, the maximum rate at which data can be transmitted is Pc. The linearityof the power-rate function implies that, in the absence of a maximum power limit, any amount ofremaining data can be transmitted in a single time slot while incurring the same energy cost per bit.Thismeans that there is no need to spread the data transmission in time; instead, the entire amount ofremaining data can be optimally transmitted in a single time slot.The energy minimization problemthen reduces to one of selecting the best such time slot which can be cast as an optimal stoppingproblem [13].

Under a more general maximum power limit constraint, data transmission has to be spreadin time, and in this case the dynamic programming recursion in (2.17) becomes [28],

Jt (x, c) = min0rmin(x,P c)

{rc

+ Jt+1(x r)}

(2.18)

As outlined in detail in [28], an optimal solution for this case can be obtained as follows.Define a function t (c) as,

t (c) = argminu0

[Jt+1(u) u

c

](2.19)

Then, the optimal data transmission policy is a threshold based policy and takes the following form:

rt ={

0, if xt t (ct )min(xt t (ct ), P ct ), if t (ct ) < xt

(2.20)

The quantity t (ct ) defines a threshold such that if the amount of remaining data is belowthis threshold the optimal transmission rate is zero (i.e., no data is transmitted), whereas above that


threshold a transmission rate of (xt t (ct )) is selected (subject to the maximum allowed rate ofPct ). As given in (2.19), the threshold t (ct ) depends on the current channel state ct with thethreshold being lower for good channel states. Thus, the form of the above transmission policymeans that in some time slots it would be beneficial to save the transmission of data for a later time.In [28], the authors further describe a special case of channel states where the thresholds t (ct ) canbe explicitly computed. In this special case, the channel states ct take values in integer multiples ofa constant cmin; the value function then is shown to be piecewise-linear and can be computed inclosed-form.

Optimal Policy in Certain Limit CasesWhile the previous section simplified the recursion in (2.17) by considering a specific form of thepower-rate function g(r, c) (namely, linear),we now consider a different kind of simplification wherewe relax the constraints in the dynamic program. Such relaxations will help us obtain the optimalpolicy in an analytical form which will be shown to coincide with the optimal solution of the un-relaxed problem under various asymptotic regimes. Thus, by studying such relaxations and theirrelationships with the limit cases, we will obtain new insights on the structure of the optimal policywhich could then be used to develop heuristic solutions in practice for the general problem. Thetechnical details in this section are based on [42; 43].

As before, xt denotes the amount of data remaining to be sent, ct the channel quality, andrt the transmission rate at time t . The channel state ct is independent and identically distributed(i.i.d.) in time and the power-rate function g(rt , ct ) is taken as an exponential function of the form:g(rt , ct ) = ert 1ct .Case 1: large B and finite T

Suppose that B is large relative to a fixed deadline constraint T . In this case, it is easy to seeintuitively that all the slots will be utilized for data transmission and the boundary constraints ofrt 0 and rt xt will not be met with equality for most of the time slots. Thus, we can expect theoptimal solution to resemble the solution of a relaxed energy minimization problem which does notimpose the above constraints on rt .

Specifically, consider the dynamic program in (2.13) and remove the constraint (2.14) from it.Then, under this relaxation and for an exponential power-rate function, the dynamic programmingrecursion in (2.17) becomes [43],

Jt (xt , ct ) = minrt

[ert 1

ct+ Jt+1(xt rt )

], t T 1 (2.21)

where the constraint on the choice of rt has been removed.For the last time slot,JT (xT , cT ) = exT 1cT .The above recursion can be solved analytically to yield an optimal policy which is referred in [43] asthe Boundary-relaxed transmission policy, and is given by,

rt (xt , ct ) = xtT t + 1 +

T tT t + 1 log (ctG(1, 2, , Tt )) (2.22)


where G(1, , Tt ) = (Ttk=1 k)1/Tt (geometric mean), and 1, 2, , Tt are defined as:m =

(E[(1/c)1/m

])m,m = 1, 2, (2.23)

The relationship in (2.22) can be explained intuitively as follows. The optimal rate consistsof two terms, where the first term, xt

Tt+1 , is the constant rate needed to empty the buffer by thedeadline in the remaining T t + 1 time slots. The second term is positive or negative dependingon whether ct is greater than or less than 1/G(1, , Tt ). Thus, 1/G(1, , Tt ) representsa channel threshold such that if the channel quality is better than this threshold, more data beyondthe constant rate is transmitted, while if the channel quality is worse than the threshold, less data istransmitted. Thus, the optimal policy starts with the minimum rate xt

Tt+1 and then adapts its ratebased on the current channel condition.

Naturally, the solution in (2.22) is not guaranteed to satisfy the constraint 0 rt xt inevery time slot; however, a feasible solution can be constructed from (2.22) by limiting the solutionto within the range [0, xt ]; i.e., rt is truncated on the lower end at 0 and on the higher end at xt .Intuitively, such a truncated policy should coincide with the optimal solution for large B keeping Tfixed, since for large B the number of bits allocated for transmission in any slot will highly likely bestrictly positive. In fact, this argument can be made mathematically rigorous as done in [43].

Thus, we see that as B becomes large, the boundary-relaxed policy given in (2.22), truncatedto within [0, xt ], provides a good approximation to the optimal policy.

Case 2: small B and finite T

Consider now the case when B is small relative to the deadline constraint T . In this case,intuitively, one would expect the optimal solution to wait for a time slot with the best channelquality and empty the buffer in that slot. The thresholds on whether to wait or transmit woulddepend on how close the current slot is to the deadline and the statistics of the channel process.

Specifically, consider the energy minimization problem in (2.13) and suppose now that theboundary constraint in (2.14) is modified to a stronger constraint which requires that either rt = 0or rt = xt ; i.e., in any slot, either no data is transmitted or all the remaining data is transmitted atonce. Under this constraint and for an exponential power-rate function, the dynamic programmingrecursion in (2.17) becomes [43],

Jt (xt , ct ) = minrt{0,xt }

[ert 1

ct+ Jt+1(xt rt )

], t T 1 (2.24)

Equivalently, the above dynamic program is an optimal stopping problem whose solution is a se-quential threshold policy, which is referred in [43] as the One-shot transmission policy, and isgiven by,

rt ={B, first t such that ct > 1/t0, otherwise

(2.25)


where 1/t is the channel threshold and is computed recursively as follows:

t =

, t = TE[1/c], t = T 1E[min(1/c, t+1)], t = T 2, , 1

(2.26)

It is easy to see that the thresholds 1/t progressively decrease as the deadline comes closer; hence,closer to the deadline the optimal policy holds off the transmission with decreasing probability.

The one-shot transmission policy described above tries to find the best time slot to sendthe data subject to the deadline constraint. Intuitively, if the data amount B is small, sucha policy should be close to the optimal as there would be no need to spread the data trans-mission over time. Thus, one would expect the one-shot transmission policy to converge to theoptimal solution asB 0.This is shown formally in [43] and we refer the reader to [43] for a proof.

Case 3: large B and large T

Consider now the case when both the data amount B and the deadline T are large, but theirratio is finite. In this case, intuitively, the deadline constraint relaxes to an average rate constraintwhere the transmitter has to achieve a long-term transmission rate of B/T with the lowest energycost.

Specifically, suppose first that we wish to find a transmission policy that minimizes the averageenergy cost subject to an average rate constraint; i.e., we wish to find an optimal policy that adaptsto the channel state and achieves a transmission rate of r with minimum energy. Such a policy canbe obtained by solving the following optimization problem:

minr(c)

E

[er(c) 1

c

](2.27)

subject to E[r(c)] r , r(c) 0 (2.28)In the above problem, we wish to find a mapping of the channel states c to the transmission ratesr(c) that minimizes the average energy cost. The optimal solution to this is the standard water-filling citegoldsmithbook solution and is given by [43],

rerg(c) ={

log(c/erg), c erg,0, otherwise

(2.29)

where erg serves as a channel threshold and must be chosen to satisfy the long-term transmissionrate of r ;

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Energy-Efficient Scheduling under Delay Constraints for Wireless Networks

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