TOWARDS A FRAMEWORK FOR EFFICIENT RESOURCE ALLOCATION IN WIRELESS NETWORKS: QUALITY-OF-SERVICE AND DISTRIBUTED DESIGN

Transcription

1 TOWARDS A FRAMEWORK FOR EFFICIENT RESOURCE ALLOCATION IN WIRELESS NETWORKS: QUALITY-OF-SERVICE AND DISTRIBUTED DESIGN DISSERTATION Presented in Partia Fufiment of the Requirements for the Degree of Doctor of Phiosophy in the Graduate Schoo of The Ohio State University By Bin Li, MS Graduate Program in Eectrica and Computer Engineering The Ohio State University 2014 Dissertation Committee: Atia Eryimaz, Advisor Ness B. Shroff Eyem Ekici

2 c Copyright by Bin Li 2014

3 ABSTRACT With the fast growing depoyment of smart mobie devices and increasingy demanding mutimedia appications, the future wireess networks must provide highquaity services to mobie users under resource-imited conditions. This necessitates the design of efficient and distributed agorithms with various key characteristics: high throughput, ow energy consumption, fast convergence, ow deay, and reguar service. Earier works extensivey study the first-order metrics, such as throughput, fairness, and energy consumption, and very few of them address the second-order metrics, such as convergence speed, deay and service reguarity critica for the growing time-sensitive and dynamic appications penetrating the wireess networks. Aso, it is important to consider the distributed impementations of theoreticay proven efficient agorithms. To that end, this dissertation mainy focuses on the foowing two aspects: (i) the performance and optimization of the second-order metrics; (ii) the distributed agorithm design. In the first part of this dissertation, we first deveop a cross-ayer agorithm that achieves the optima convergence speed at which the running average of the received service rates and the network utiity over a finite time horizon converges to their respective imits under the discrete transmission rate seections a typica feature of wireess networks. This resut is important in two aspects, in reveaing a previousyunknown imit on how fast the service rates can approach an optima point, and in providing a new agorithm that achieves the fastest possibe speed. Then, we focus ii

4 on the efficient agorithm design for overcoming unavoidabe temporary overoads. We deveop a nove queue reversa approach that reates the metrics in unstabe systems to the metrics in stabe systems, for which a rich set of toos and resuts exists. Furthermore, to support widey popuar rea-time appications, we deveop a scheduing agorithm that simutaneousy achieves maximum throughput, minimum deay in heavy-traffic regimes, and service reguarity guarantees. In the second part of this dissertation, we first expore the throughput imitations of randomized scheduers that are widey used in distributed impementations. This systematic study is important in heping network designers to use or avoid certain types of probabiistic scheduing strategies depending on the network topoogy. Then, we turn to the distributed design of an optima scheduing agorithm for serving both eastic and ineastic traffic over wireess fading channes. The corresponding resut provides one of the first promising means of effectivey handing changing conditions in distributed resource aocation agorithm design. Noting the high energy cost and operationa difficuty for a users to continuousy estimate the channe quaity before each transmission, we further design efficient and distributed joint channe probing and scheduing strategies under energy constraints. Overa, this thesis deveops methodoogies to study the metrics beyond the traditiona first-order requirements (e.g., throughput, fairness, average energy consumption, etc.), and to design distributed resource aocation agorithms, both of which enabe the incorporation of sharp quaity-of-service requirements into the efficient and distributed agorithm design. It aso opens an interesting and new avenue to the performance anaysis and optimization of the second or higher order metrics of compex networks, incuding smart power grids and coud computing. iii

5 To my parents, Jinjie Li and Jinzhao Chen, my sister, Juan Li, and my wife, Xiaozhen Wang. iv

6 ACKNOWLEDGMENTS I am sincerey gratefu to my advisor, Prof. Atia Eryimaz, for his invauabe guidance, generous support, and constant encouragement throughout my Ph.D. study. He taught me how to identify and attack probems essentia for the research. His broad knowedge, diigence and passion for research set up an exceent exampe for me and wi no doubt have a great impact on my future career. I aso woud ike to thank my doctora committee members, Prof. Ness B. Shroff and Prof. Eyem Ekici. My thanks aso go to Prof. C. Emre Koksa and Prof. Wei Zhang for serving in my candidacy committee. Their insightfu comments and hepfu suggestions improve the content of this thesis. Aso, I woud ike to thank a other professors at The Ohio State University who taught, heped or encouraged me during my Ph.D. study. I am aso gratefu to Prof. R. Srikant at The University of Iinois Urbana-Champaign for his hep and advice. I woud ike to express my great gratitude to my famiy, especiay my parents and my wife, Xiaozhen Wang, for their unconditiona ove, support and encouragement. Their care and understanding aowed me to pursue my dreams, and wi surey continue heping me in a my future endeavors. I woud ike to acknowedge a my current and former coeagues in the IPS ab Arun, Bo, Dongyue, Fangzhou, Gene, Irem, Jia, John, Justin, Karim, Ming, Onur, Ozan, Ozgur, Ruogu, Shengbo, Shuang, Swapna, Wenzhuo, Yang, Yousi, Zhoujia, v

7 Zizhan, and many others for many fruitfu conversations on both technica and nontechnica subjects. Aso, I woud ike to thank the ab secretary, Jeri, and the graduate academic counseor, Tricia, for their patient and hepfu attitude in answering my numerous questions. vi

8 VITA B.S. in Eectronic and Information Engineering, Xiamen University M.S. in Communication and Information System, Xiamen University 2009 Present Presidentia Feow, Graduate Research Associate, Graduate Teaching Associate Eectrica and Computer Engineering, The Ohio State University PUBLICATIONS Journa Papers: Bin Li, Ruogu Li, Atia Eryimaz, On the Optima Convergence Speed of Wireess Scheduing for Fair Resource Aocation, accepted by IEEE/ACM Transactions on Networking. Bin Li, Atia Eryimaz, Non-Derivative Agorithm Design for Efficient Routing over Unreiabe Stochastic Networks, Esevier s Performance Evauation, 71: 44-60, Bin Li, Atia Eryimaz, Optima Distributed Scheduing under Time-Varying Conditions: A Fast-CSMA Agorithm with Appications, IEEE Transactions on Wireess Communications, 12(7): , Bin Li, Atia Eryimaz, Exporing the Throughput Boundaries of Randomized Scheduers in Wireess Networks, IEEE/ACM Transactions on Networking, 20(4): ,2012. vii

9 Conference Papers: Bin Li, Atia Eryimaz, R. Srikant, Leandros Tassiuas, On Optima Routing in Overoaded Parae Queues, In Proc. IEEE conference on Decision and Contro (CDC), Forence, Itay, December, Bin Li, Ozgur Dakiic, Atia Eryimaz, Exporing the Tradeoff between Waiting Time and Service Cost in Non-Asymptotic Operating Regimes, In Asiomar Conference on Signas, Systems and Computers, Pacific Grove, Caifornia, November, Bin Li, Ruogu Li, Atia Eryimaz, Heavy-Traffic-Optima Scheduing Design with Reguar Service Guarantees in Wireess Networks. In ACM Internationa Symposium on Mobie Ad Hoc Networking and Computing (MOBIHOC), Bangaore, India, Juy, Bin Li, Atia Eryimaz, Ruogu Li, Wireess Scheduing for Utiity Maximization with Optima Convergence Speed, In Proc. IEEE Internationa Conference on Computer Communications (INFOCOM), Turin, Itay, Apri, Ruogu Li, Atia Eryimaz, Bin Li, Throughput-Optima Scheduing with Reguated Inter-Service Times, In Proc. IEEE Internationa Conference on Computer Communications (INFOCOM), Turin, Itay, Apri, Bin Li, Atia Eryimaz, Optima Constant Spitting for Efficient Routing over Unreiabe Networks, In Proc. IEEE conference on Decision and Contro (CDC), Maui, Hawaii, December, S. Lakshminarayana, Bin Li, M. Assaad, A. Eryimaz, M. Debbah, A Fast-CSMA Based Distributed Scheduing Agorithm under SINR Mode, In Proc. IEEE Internationa Symposium on Information Theory (ISIT), Cambridge, MA, Juy, Bin Li, Atia Eryimaz, Distributed Channe Probing for Efficient Transmission Scheduing over Wireess Fading Channes, In Proc. IEEE Internationa Conference on Computer Communications (INFOCOM) mini-conference, Orando, Forida, USA, March, Bin Li, Atia Eryimaz, A Fast-CSMA Agorithm for Deadine Constraint Scheduing over Wireess Fading Channes, In workshop on Resource Aocation and Cooperation in Wireess Networks (RAWNET), Princeton, NJ, May, Bin Li, Atia Eryimaz, On the Limitations of Randomization for Queue-Length- Based Scheduing in Wireess Networks, In Proc. IEEE Internationa Conference on Computer Communications (INFOCOM), Shanghai, China, Apri, viii

10 Bin Li, Atia Eryimaz, On the Boundaries of Randomization for Throughput-Optima Scheduing in Switches, In Proc. Aerton Conference on Communication, Contro, and Computing (Aerton), Monticeo, IL, Sept FIELDS OF STUDY Major Fied: Eectrica and Computer Engineering Speciaization: Networking ix

11 TABLE OF CONTENTS Abstract Dedication Acknowedgments Vita ii iii v vii List of Figures xv CHAPTER PAGE 1 Introduction Resource Aocation for Time-Sensitive Appications Distributed Resource Aocation Design Network Mode I Resource Aocation for Time-Sensitive Appications 11 2 Efficient Resource Aocation with Optima Convergence Speed Probem Formuation A Motivating Exampe Convergence Speed in Rate Deviation A Lower Bound on the Expectation of Rate Deviation A Rate Deviation Optima Poicy Convergence Speed in Utiity Benefit An Upper Bound on the Utiity Benefit Utiity Benefit Optimaity of the RDO Agorithm Utiity Benefit Optimaity of the Dua Agorithm Simuation Resuts Non-fading Scenario Fading Scenario Summary x

12 3 Agorithm Design with Optima Convergence Speed in Overoaded Systems Probem Formuation Lower Bound Anaysis Queue Reversa Theorem A Lower Bound on the Cumuative Unused Service Overoad Anaysis of RR and JSQ poicies Lower and Upper Bounds under the RR Poicy Optimaity of the JSQ Poicy in Symmetric Conditions Simuation Resuts The Impact of Overoad Leve ɛ on Mean Unused Services The Impact of Server Number L on Mean Unused Service Summary Resource Aocation Agorithm for Achieving Maximum Throughput, Heavy-Traffic Optimaity, and Service Reguarity Guarantee Probem Formuation Reguar Service Scheduer Service Reguarity Performance Anaysis Lower Bound Anaysis Upper Bound Anaysis Tradeoff Between Mean Deay and Service Reguarity Heavy-Traffic Optimaity Anaysis Simuation Resuts Throughput Performance Service Reguarity Performance Heavy-Traffic Performance Summary II Distributed Resource Aocation Agorithm Design 80 5 Limitations of Randomization for Distributed Resource Aocation Probem Formuation Overview of Main Resuts Sufficient Conditions f-throughput-optimaity of the RSOF Scheduer Throughput-Optimaity of RMOF and RFOS Scheduers Necessary Conditions Simuation Resuts Throughput Performance Deay Performance Summary xi

13 6 Optima Distributed Scheduing Design under Time-Varying Conditions The Principe of Fast-CSMA design Scenario 1: Scheduing Eastic Traffic with CSI FCSMA Agorithm Impementation Simuation Resuts Scenario 2: Scheduing Eastic Traffic without CSI FCSMA Agorithm Impementation Simuation Resuts Scenario 3: Scheduing Ineastic Traffic with CSI Basic Setup FCSMA Agorithm Impementation Simuation Resuts Scenario 4: Scheduing Ineastic Traffic without CSI FCSMA Agorithm Impementation Simuation Resuts Practica Impementation Suggestions Summary Efficient Distributed Channe Probing and Scheduing Design Probem Formuation A Motivating Scenario Optima Centraized Probing and Transmission Characterization of the Capacity Region An Optima Joint Probing and Transmission Agorithm Sequentia Greedy Probing Poicy and Anaysis A Sequentia Greedy Probing Agorithm Optimaity of the SGP Agorithm for Symmetric Channes The Modified SGP Poicy and Anaysis Distributed Impementation with Fast-CSMA Simuation Resuts The Impact of Iterative Steps The Impact of Using Deayed Queue Length Information The Performance of Greedy Probing Agorithms Summary Concusions and Future Works Appendix A: Proofs for Chapter A.1 Proof of Lemma A.2 Proof of Proposition A.3 Proof of Lemma A.4 Proof of Lemma xii

14 A.5 Proof of Proposition A.6 Proof of Proposition A.7 Proof of Proposition A.8 Proof of Proposition A.9 Proof of Lemma A Appendix B: Proofs for Chapter B.1 Proof of Inequaity (4.2.4) B.2 Proof of Proposition B.3 Proof of Inequaity (B.2.1) B.4 Proof of Lemma B B.5 Proof of Inequaity (B.2.20) B.6 Proof of Lemma B.7 Proof of Proposition B.8 Proof of Equation (B.7.5) B.9 Detaied Heavy-Traffic Anaysis B.9.1 State-Space Coapse B.9.2 Proof of Heavy-Traffic Optimaity B.10Proof of Lemma B B.11Proof of Lemma B B.12Proof of Proposition B B.13Proof of Lemma B B.14Proof of Lemma B B.15Proof of Inequaity (B.9.11) B.16Proof of Inequaity (B.9.13) Appendix C: Proofs for Chapter C.1 Properties of Functiona Casses C.2 Proof of Lemma C.3 Proof of Inequaity (5.3.4) Appendix D: Proofs for Chapter D.1 Proof of Lemma D.2 Proof of Proposition Appendix E: Proofs for Chapter E.1 Proof of Proposition E.2 Proof of Lemma E.3 Proof of Lemma E.4 Proof of Proposition E.5 Some Properties of Function f(e, e) E.6 Proof of Basic Iterative Equation xiii

15 E.7 Proof of f(d, d) f(b, d) in Lemma E.8 Proof of Lemma E.9 Proof of Lemma E.10Proof of Lemma E.11Proof of Proposition E.12Proof of Proposition Bibiography xiv

16 LIST OF FIGURES FIGURE PAGE 2.1 Reationship between r (ɛ) and r The convergence speed of an i.i.d Bernoui sequence The reationship between R (ɛ) and R The operation of the RDO Agorithm at ink The utiity benefit of an agorithm in cass P Dua Agorithm performance with varying ɛ Variants of the RDO Agorithm Routing to parae queues (a) Forward queue; (b) Reverse queue Lower bounding system The impact of ɛ on the expected cumuative unused service The impact of L on the expected cumuative unused service Deay and service reguarity performance of the RSG Agorithm Geometric structure of capacity region Throughput performance of the RSG Agorithm Tradeoff between mean queue ength and service reguarity Heavy-traffic performance in the symmetric case Asymmetric arrivas in the asymmetric case The reationship between casses A, B and C xv

17 5.2 Throughput performance of the RSOF Scheduer Throughput performance of the RFOS Scheduer Throughput performance vaidation of the randomized scheduers Deay performance comparison of the randomized scheduers with different functiona forms (a) Markov chain for a CSMA agorithm (b) Markov chain for a FC- SMA agorithm Performance of FCSMA for scheduing eastic traffic with CSI Performance comparison between FCSMA and QCSMA Performance of FCSMA for scheduing eastic traffic without CSI Performance of FCSMA for scheduing ineastic traffic with CSI Performance of FCSMA for scheduing ineastic traffic without CSI Performance comparison between FCSMA and its discrete-time version Maximum throughput under different number of users Throughput performance of RP poicy The directed graph G = (X, E) when L = Impact of iterative steps Impact of using deayed queue ength information Impact of asymmetric channe statistics Impact of asymmetric channe rates A.1 An exampe when r (ɛ) = 1 2, S {0, 1} and c = A.2 The reationship between L =1 U (r(δ) )r (δ) and L =1 U (r )r B.1 Markov Chain {X[t]} t E.1 The reations among a sets xvi

18 CHAPTER 1 INTRODUCTION Wireess networks typicay have restrictive resources (e.g., time, frequency, space and power) and thus a network users need to share these precious resources. This necessitates the design of efficient resource aocation agorithms that determine when and how to aocate imited resources for network users. Traditiona resource aocation agorithms (e.g., [91, 92, 89, 37, 59, 54, 88, 16, 14, 55, 69, 70, 4, 34, 67], see [87, 21, 66] for an overview) aim to the optimization of first-order metrics such as throughput, fairness, average energy consumption, etc. These works pay an important roe in deveoping systematic approaches for resource aocation agorithm designs with ongterm efficiency guarantees, most notaby Lyapunov-drift-minimization-based design (e.g., [91, 70, 67, 66]) and optimization-based design (e.g., [37, 59, 54, 88, 16, 56, 87]). Yet, optimizing these first-order metrics is not a sufficient objective for increasingy many network appications, which are time-sensitive and possess highy dynamic user behavior. In addition, wireess networks may experience temporary overoads during peak demand hours or at hotspots serving arge number of users, which requires the resource aocation agorithms to quicky respond to these unavoidabe transient phases of overoads. A these features require the design of resource aocation agorithms with fast convergence characteristic. Previous works either focus on the convergence aspect of agorithm designs (e.g., [37, 59, 16, 14, 55, 69]), or the 1

19 convergence speed of particuar agorithms (e.g., [18, 30, 93, 57]) in the stabe networked systems. There does not exist an extensive study of agorithm design with optima convergence speed. This motivates us to systematicay anayze and design agorithms in terms of their convergence speed in both underoaded and overoaded networked systems. Moreover, to support rea-time appications, resource aocation agorithms not ony achieve high throughput or satisfy fairness criteria, but aso meet the requirements of Quaity-of-Service (QoS) incuding packet deivery ratio (or throughput), deay, and service reguarity. Recenty, a arge body of works focus on the design of agorithms that improve various aspects of the QoS, especiay on the deay performance of the agorithms. For exampe, some works consider designing agorithms with ow end-to-end deay performance, such as [7, 97, 95]. Constant deay bounds (e.g. [68]) and deivery ratio requirements for deadine-constrained traffic (e.g. [27, 28, 29, 31, 44]) are some of the other QoS metrics considered in the iterature. However, none of these works simutaneousy addresses the throughput, deay and service reguarity critica metrics for rea-time appications. This motivates us to consider the scheduing design that achieves maximum throughput, minimum deay and best service reguarity. In addition, ow-compexity impementation of theoreticay efficient resource aocation agorithms is strongy desirabe in the presence of many users appearing in rea-word networked systems, and has been a topic of extensive research activity (e.g., [90, 55, 13, 63, 94, 25, 53, 35, 13, 43]). One such thread eads to the deveopment of a cass of evoutionary randomized agorithms (aso named pick and compare agorithms) with throughput-optimaity characteristics (see [90, 13, 84]). Another thread eads to the deveopment of distributed but suboptima randomized/greedy strategies (see [55, 35, 6]). Reativey recenty, another exciting thread of resuts have emerged 2

20 that can guarantee throughput-optimaity by cevery utiizing queue-ength information in the context of carrier sense mutipe access (CSMA) (e.g., [60, 33, 80, 72]). Therefore, randomized agorithms have the potentia to possess exceent network performance, and more importanty, they can be impemented distributivey. Whie randomization aows fexibiities in the distributed impementation of agorithms, it causes inefficient operation and may be hurtfu if it is not performed carefuy. Despite the presence of a variety of earier works (e.g., [33, 72]) on the design and anaysis of particuar randomized scheduers, there does not exist an extensive study on the imitations of randomization in scheduing designs. This motivates us to deveop a common framework for the modeing and anaysis of randomized scheduers. Furthermore, it is important to consider the distributed scheduing agorithm design under time-varying conditions and imited energy constraints, under which the most network appications operate and existing distributed agorithms (e.g., [33, 72]) do not work efficienty. In this thesis, we address a these chaenging issues by deveoping a rigorous theoretica foundation for efficient and distributed agorithm design supporting diverse appications in wireess networks. In particuar, our dissertation research focuses on the foowing two main aspects: (i) efficient resource aocation agorithm design for time-sensitive appications; (ii) distributed resource aocation agorithm design. The resuts presented in this thesis have been pubished (or submitted) in [45, 49, 46, 52, 48, 47, 39, 41, 43, 40, 44, 42, 38]. Next, we briefy summarize our main contributions. 1.1 Resource Aocation for Time-Sensitive Appications In this research effort, we first focus on the convergence speed of cross-ayer agorithms in wireess networks, which are dominated by the dynamics of incoming and outgoing 3

21 users as we as the time-sensitive appications. However, the design of controers with fast convergence speed in most wireess networks is compicated by two natura constraints: (i) interference constraints eading to discrete ink scheduing choices; and (ii) a finite set of choices for the transmission rate seection over the schedued inks. The atter constraint is caused by both digita communication (e.g., moduation, coding, etc.) and hardware design principes. For exampe, in IEEE b standard, there are ony four transmission rates: 1Mpbs, 2Mpbs, 5.5Mpbs and 11Mbps. Previous works focus either on the design and anaysis of poicies with optima ong-term behavior (e.g., [37, 59, 16, 14, 55, 69]), or on the design of distributed Interior Point (e.g., [18]) and Newton s methods (e.g., [30, 93, 57]) for fast convergence in wired networks. They do not incorporate an important feature of wireess networks, namey the discreteness in the transmission rate seections. We tacke this chaenge by expicity incorporating such discrete constraints to understand their impact on the convergence speed at which the running average of the received service rates and the network utiity converges to their respective imits. By providing universa bounds on the convergence speed of any scheme, we estabish the imits of convergence speed under discrete scheduing and transmission rate constraints. Using this bound, we deveop an agorithm that achieves the optima convergence speed in both metrics. Somewhat surprisingy, we aso show that even a first-order method such as the weknown dua agorithm can achieve the aforementioned optima convergence speed in terms of its utiity vaue. These resuts are important in two aspects: (i) it reveas a previousy-unknown imit on how fast the service rates can approach an optima point; (ii) it provides a new agorithm that achieves the fastest possibe speed. As such, it provides the means of quicky serving the demands of dynamicay arriving and eaving users, as in many wireess mobie networks. 4

22 Then, we consider the convergence speed of efficient agorithms in overoaded systems. Severa interesting works (e.g., [22, 8, 86, 50]) have anayzed the performance of we-known poicies in overoaded conditions. In particuar, these works have studied the performance and optimization of the metric of queue overfow rates, and the reated metric of the departure rates of served packets. One drawback of the currenty used performance metrics of overfow rate and departure rate is that, being based on ong-term time averages, they may not be abe to differentiate between poicies in terms of their convergence speeds to the same imit. Noting the compexity in the performance anaysis of overoaded systems, we consider the convergence speed anaysis in a simpe cassic probem of routing random arrivas to parae queues with random services in overoaded regimes. We propose and anayze the metric of Cumuative Unused Service (CUS) over time to anayze the performance of routing poicies in overoaded systems. This metric not ony measures the amount of underutiization in the muti-server system over time, but aso captures the speed at which the running-average of the departure rates converges to their imiting vaue. The proposed CUS metric is difficut to anayze in both stabe and unstabe systems due to its non-stationary nature. To tacke this chaenge, we estabish a nove queue reversa resut that equates the expected cumuative unused service in the unstabe system to the expected queue-ength of a reated (in fact, reversed) stabe system. With this connection, we can obtain the mean cumuative unused service metric by studying the mean queue-ength of a stabe Markov chain, for which a rich set of toos and resuts exists. Using this resut for a singe-server queue, we obtain a ower bound on the expected unused service in the parae queueing system for any feasibe routing poicy. We then compare this ower bound to the performance of two simpe routing poicies: Randomized and Join-the-Shortest-Queue (JSQ) routing. Through numerica studies, we revea two interesting properties of the JSQ poicy: (i) 5

23 the JSQ poicy achieves the optima convergence speed in ighty overoaded regimes, i.e., when the arriva rate approaches the tota service rates; (ii) the convergence speed under the JSQ poicy is independent of number of servers compared to the derived ow bound on the convergence speed under any feasibe poicy. To support rea-time appications, we then consider the design of scheduing strategies that maximize system throughput, minimize mean deay, and provide reguar service for a users. We deveop a parametric cass of maximum-weight type scheduing agorithms, caed Reguar Service Guarantee (RSG) Agorithm, where each user s weight consists of its own queue-ength and a counter that tracks the time since the ast service. The RSG Agorithm not ony is throughput-optima, but aso achieves a tradeoff between the service reguarity performance and the mean deay under the RSG Agorithm, i.e., the service reguarity performance of the RSG Agorithm improves at the cost of increasing mean deay. This further motivates us to investigate whether satisfactory service reguarity and ow mean-deay can be simutaneousy achieved by the RSG Agorithm by carefuy seecting its design parameter. To that end, we show that the RSG Agorithm can minimize the tota mean queue-ength to estabish mean deay optimaity under heaviy-oaded conditions as ong as its design parameter scaes sowy with the network oad. To the best of our knowedge, this is the first work that provides reguar service whie aso achieving maximum throughput and heavy-traffic optimaity in mean queue-engths. 1.2 Distributed Resource Aocation Design Distributed agorithm design is strongy desirabe for impementing theoreticay proven efficient agorithms in practica networks. Randomization is a powerfu and 6

24 pervasive strategy for deveoping distributed scheduing agorithms in interferenceimited wireess networks. Yet, despite the presence of a variety of earier works on the design and anaysis of particuar randomized scheduers (e.g., [33, 72]), there does not exist an extensive study of the imitations of randomization on the efficient scheduing in wireess networks. To that end, we deveop a common framework for the modeing and anaysis of randomized scheduers. In particuar, we revea that the performance of randomized scheduers may especiay be sensitive to the network topoogy and the functiona form used in assigning priority to network users. Then, we estabish necessary and sufficient conditions on the network topoogy and the functiona forms for maximum throughput of randomized scheduers. This extensive understanding of the imitations of randomization is important in reveaing the vunerabiities and strengths of a wide range of scheduing strategies, and it equips network designers with the machinery for determining the efficient scheduing rues for the network they wi operate. After understanding the imitations of randomization for distributed agorithms, we focus on the distributed scheduing agorithm design under time-varying conditions, under which the most network appications operate. Recenty, ow-compexity and distributed Carrier Sense Mutipe Access (CSMA)-based scheduing agorithms (e.g., [33, 73, 23, 81]) have attracted extensive research interests due to their throughputoptima characteristics in genera network topoogies. However, these agorithms are not we-suited for serving deadine-constrained traffic, such as those generated by voice or video streaming appications, over time-varying channes due to the arge convergence time of the underying system dynamics. This motivates us to attack the probem of distributed scheduing for both eastic and ineastic traffic over timevarying channes. Specificay, we propose a Fast-CSMA (FCSMA) agorithm that 7

25 converges much faster than the existing CSMA agorithms and thus yieds significant advantages for time-sensitive appications. This resut provides one of the first promising means of effectivey handing changing conditions in distributed resource aocation agorithm design. Yet, a these distributed agorithms presume the knowedge of channe state information (CSI) at the beginning of each transmission. We note that it is highy energy-consuming and operationay difficut for a network users to continuousy acquire CSI before each data transmission decision. This further motivates us to investigate the question of whether and how throughput gains can sti be achieved with significant reductions in channe probing requirements and without centraized coordination amongst the competing users. Earier works in the design of joint probing and transmission strategies (e.g., [24, 51, 9, 74]) are not suitabe for distributed operation in arge-scae networks, since they assume centraized controers that utiize the whoe system state information. We tacke this chaenge by first providing an optima centraized joint probing and transmission agorithm under the probing constraints. Noting the difficuties in the impementation of the centraized soution, we then deveop a nove Sequentia Greedy Probing (SGP) agorithm by using the maximum-minimums identity, which is naturay we-suited for physica impementation and distributed operation. The resuting SGP agorithm can achieve expicit performance guarantees that are tight in certain regimes of interest. We further discuss the distributed impementation of the greedy soution by using the Fast-CSMA technique. 1.3 Network Mode Here, we introduce the network mode used throughout this thesis. We consider a wireess network with a set L = {1, 2,..., L} of inks, where a ink represents a pair of 8

26 a transmitter and a receiver that are within the transmission range of each other. We assume that the system operates in sotted time with normaized sots t {1, 2,...}. Due to the interference-imited nature of wireess transmissions, the success or faiure of a transmission over a ink depends on whether an interfering ink is aso active in the same sot, which is caed the ink-based confict mode. We ca a set of inks that can be active simutaneousy as a feasibe schedue and denote it as S[t] = (S [t]) L =1, where S [t] = 1 if the ink is schedued in sot t and S [t] = 0, otherwise. We use S to denote the set of a feasibe schedues. We capture the channe fading over ink via a non-negative random variabe C [t], with C [t] C max,, t, for some C max <, which measures the maximum amount of service avaiabe in sot t, if the ink is schedued. We assume that C[t] = (C [t]) L =1, t 0, are independenty and identicay distributed (i.i.d.) over time. Let S (c) {Sc : S S} denote the set of feasibe rate vectors when the channe is in state c, where ab = (a b ) L =1 denotes the component-wise product of two vectors a and b. Then, the capacity region is defined as R c Pr{C[t] = c} CH{S (c) }, (1.3.1) where CH{A} denotes a convex hu of the set A, and the summation is a Minkowski addition of sets. We assume a per-ink traffic mode, where A [t] denotes the number of packets arriving at ink in sot t that are independenty distributed over inks, and i.i.d. over time with finite mean λ > 0, and A [t] A max,, t, for some A max <. Accordingy, a queue is maintained for each ink with Q [t] denoting its queue ength at the beginning of time sot t. Then, the evoution of queue is described as foows: Q [t + 1] = (Q [t] + A [t] C [t]s [t]) +,, (1.3.2) 9

27 where (x) + = max{x, 0}. We say that the queue is strongy stabe (see [21]) if it satisfies im sup T 1 T T E[Q [t]] <. (1.3.3) t=1 We ca system stabe if a queues are strongy stabe. We consider the poicies under which the system evoves as a Markov Chain. We ca an agorithm throughputoptima if it makes a queues strongy stabe for any arriva rate vector λ = (λ ) L =1 that ies stricty within the capacity region. The rest of the dissertation is organized as foows: In Chapter 2, we study the design of cross-ayer agorithms with optima convergence speed in wireess networks. In Chapter 3, we turn our attention to the convergence speed anaysis of routing agorithms in overoaded parae queueing systems. In Chapter 4, we consider the scheduing design for achieving maximum throughput, heavy-traffic optimaity and service reguarity guarantee. In Chapter 5, we expore the throughput imitations of randomized scheduers in wireess networks. In Chapter 6, we turn to the optima scheduing design for time-varying appications. We further design distributed joint probing and scheduing agorithms under the imited probing rates in Chapter 7. We then concude in Chapter 8. 10

28 Part I Resource Aocation for Time-Sensitive Appications 11

29 CHAPTER 2 EFFICIENT RESOURCE ALLOCATION WITH OPTIMAL CONVERGENCE SPEED In this chapter, we consider the convergence speed of cross-ayer agorithms in wireess networks, which are dominated by the dynamics of incoming and outgoing users as we as the time-sensitive appications. As we discussed in Chapter 1, the design of controers with fast convergence speed in most wireess networks is compicated by two natura constraints: (i) interference constraints eading to discrete ink scheduing choices; and (ii) a finite set of choices for the transmission rate seection over the schedued inks. Previous works mainy focus on the design and anaysis of poicies with optima imiting behavior. A arge body of works (e.g. [37, 59, 16, 14, 55, 69]) has utiized dua and prima-dua methods to deveop cross-ayer poicies with ong-term optimaity guarantees. Such soutions are amenabe to distributed impementation due to their natura decomposition into oosey couped components. However, being first-order methods, they suffer from the sow convergence speed shared by a such methods (e.g. [71, 5, 1]). This speed deficiency of dua methods has recenty spurred an exciting thread of research activity in the design of distributed Interior Point (e.g. [18]) and Newton s (e.g. [30, 93, 57]) methods for network utiity maximization. However, these works 12

30 do not incorporate two aforementioned features of wireess networks, namey the discreteness in the scheduing and transmission rate seections. We expicity incorporate these intrinsic characteristics of wireess networks in our anaysis and agorithm design. To the best of our knowedge, this is the first work that systematicay anayzes and designs agorithms in terms of their converge speed in wireess networks with such discrete constraints. Next, we ist our main contributions, aong with references on where they appear in this chapter. We show that the convergence speed 1 at which the running average of the received service rates (cf. Section 2.3.1) and their utiity (cf. Section 2.3.1) over ( ) 1 T time sots cannot be faster than Ω. This fundamenta imitation on the T convergence speed is caused by the discrete nature of the aowabe transmission rates under the operation of any stabiizing and asymptoticay optima fow contro and scheduing poicy. We deveop a generic agorithm that can work with a range of fow rate controers, and achieves the optima convergence speed in both rate (cf. Section 2.3.2) and utiity (cf. Section 2.4.2) metrics. Somewhat surprisingy, we aso show that even a first-order method such as the we-known dua agorithm can achieve the aforementioned optima convergence speed in terms of its utiity vaue (cf. Section 2.4.3). These resuts coectivey revea that, under wireess networks subject to discrete scheduing and rate constraints, the convergence speed of cross-ayer agorithms is dominated by the convergence speed of the scheduing component, and not the fow rate controer. As such, the speed improvements in the fow rate convergence, 1 The foowing standard notations are used to describe the rates of growth of two rea-vaued sequences {a n } and {b n }:a n = O(b n ) if c > 0 such that a n c b n ; a n = Ω(b n ) if b n = O(a n ). 13

31 unfortunatey, cannot extend to the received service rates or utiities in wireess networks. On the bright side, however, with carefu design we can achieve the optima convergence speed under such constraints. 2.1 Probem Formuation We consider a muti-hop fading wireess network with L inks. Due to moduation, coding, as we as other practica constraints, each ink has to transmit at one of a finite set of rates 2. We use R[t] = (R [t]) L =1 to denote the service rate vector offered to the inks in sot t, which must be seected from a feasibe set of transmission rates. We note that the capacity region R is a poyhedron due to the discreteness of the transmission rate choices, and hence can be written as R = {y 0 : Hy b}, where y R L and H is some non-negative matrix. Note that H has L coumns and the number of rows in H is equa to the dimension of the vector b associated with the number of interference constraints. To capture the heterogeneous and potentiay inter-dependent preferences of users, we define a utiity function U : R L + R + that measures the tota network utiity when ink receives an average service rate of r, where r = (r ) L =1. We assume U(r) to be a stricty concave function that is non-decreasing in each coordinate. The objective of Network Utiity Maximization (NUM), then, is to design a congestion contro and scheduing agorithm such that the average service rate vector r soves the foowing optimization probem: 2 For exampe, IEEE a standard uses OFDM transmission technique and can support rates in Mega bits per second seected from the finite set {6, 9, 12, 18, 24, 36, 48, 54}; In CDMA2000 1xEV-DO specification, the forward ink transmission rate in kio bits per second is chosen from the finite set {38.4, 76.8, 153.6, 307.2, 614.4, 921.6, , , }. 14

32 Definition (Network Utiity Maximization (NUM)) max U(r) (2.1.1) r=(r ) L =1 Subject to r R, (2.1.2) where R is defined in (1.3.1). The strict concavity of U( ) together with the convexity of R guarantees the uniqueness of the soution of NUM, which is denoted as r = (r ) L =1. Aso, due to the non-decreasing nature of U( ), r must ie on the boundary of R. It is important to note that r is the optima average offered service rate to the inks. The purpose of the fow rate controer, however, is to determine the optima injection rate of traffic into the network whie maintaining network stabiity. Reca that Q [t] denotes the queue-ength at ink L at the beginning of sot t. Let G [t] denote the amount of injected data into Queue- in sot t under a given fow rate controer, and reca that R [t] denotes the service rate offered to ink in sot t under a given scheduer. Then, the evoution of Q can be expressed as Q [t + 1] = (Q [t] + G [t] R [t]) +, t 1. In this chapter, we are interested in the convergence speed of a broad cass of joint fow rate contro and scheduing poicies P that are both stabiizing and asymptoticay rate optima. To define this cass of poicies abstracty, we introduce the parameter ɛ > 0 as a generic term to characterize the performance of the joint poicy under specific design choices. Accordingy, the average injection rate 3 of a given poicy under parameter ɛ is r (ɛ). Simiary, we wi use the superscript (ɛ) over (Q [t]), (G [t]), (R [t]), etc. to express the queue-engths, injections, offered service 3 For each parameter ɛ > 0, the system is stabe and has a steady-state distribution. Thus, the ong-term average injection rate is we-defined. 15

33 rates, etc. under the poicy with parameter ɛ. The stabiity condition requires that r (ɛ) is stricty within the capacity region R for a ɛ > 0, and the asymptotic rate optimaity condition requires that im ɛ 0 r (ɛ) = r, i.e., the asymptoticay optima poicy achieves the optima service rate vector in the imit. Figure 2.1 shows the reationship between r (ɛ) and r in a two-dimensiona case. Figure 2.1: Reationship between r (ɛ) and r Thus, the parameter ɛ captures the coseness of the injection rate to the optima service rate r under the cass of joint poicies parametrized by ɛ. We note that this abstraction incudes a wide range of joint contro and scheduing poicies in the iterature. For exampe, in the we-known subgradient-based designs (e.g., [37, 59, 16, 14, 55]) the generic term ɛ maps to the particuar design parameter that corresponds to the step-size on the subgradient iteration. The stabiity condition of the joint fow rate contro and scheduing poicies in P impies that the running average of departures over time must aso converge to r (ɛ). 16

34 Since the running average of departures 4 up to time T is the rea measure of received service unti that time, we are interested in its convergence speed to r (ɛ). To be more precise, for the poicy with parameter ɛ we use D (ɛ) [t] min(r (ɛ) [t], Q (ɛ) [t]) to denote the departures in sot t for ink L, and define its running average unti T 1 as d (ɛ) [T ] 1 T T t=1 D (ɛ) [t], L, (2.1.3) and use d (ɛ) [T ] (d (ɛ) [T ]). Next, we introduce the metrics of interest in our study of convergence speed, both in the running average departure rate and its corresponding utiity vaue. Definition (Metrics of Interest) For any poicy in P with parameter ɛ, we define the rate deviation φ(d (ɛ) [T ], r (ɛ) ) between d (ɛ) [T ] and r (ɛ) at time T as φ(d (ɛ) [T ], r (ɛ) ) d (ɛ) [T ] r (ɛ), (2.1.4) and the utiity benefit received unti time T as U(d (ɛ) [T ]), where y is the 2 norm of the vector y. In the rest of chapter, we wi: (i) provide an exampe showing the fundamenta speed imitation exerted by the discrete choice of transmission rates (cf. Section 2.2); (ii) estabish fundamenta imits on the speed at which E[φ(d (ɛ) [T ], r (ɛ) )] converges to zero as T increases (cf. Section 2.3.1); (iii) deveop joint fow contro and scheduing poicy with provaby optima convergence speed in terms of rate deviation (cf. Section 2.3.2); (iv) derive fundamenta imits on the speed at which the utiity benefit converges to the optima utiity vaue of NUM when sources of randomness are eiminated (cf. Section 2.4.1); (v) show that our proposed agorithm, as we as the 4 Due to the discreteness of transmission rate choices, it is unikey that the departure rate in sot T converges to r (ɛ), as T increases. 17

35 we-known dua agorithm, achieves the optima convergence speed in terms of utiity benefit (cf. Section 2.4.2, 2.4.3); and finay (vi) provide the detaied comparison between the proposed agorithms and the traditiona dua agorithm in terms of the convergence speed and the average deay through simuations (cf. Section 2.5). 2.2 A Motivating Exampe In this section, we study a simpe exampe to see how the convergence speed of a sequence is imited by the discreteness of its eements. In particuar, we consider the convergence speed of any zero-one sequence converging to 0.5. For any zero-one sequence {D[t] : D[t] {0, 1}} t 0, we have 1 T D[t] 0.5 T = 1 T 2T 2 D[t] T. (2.2.1) Noting that both 2 t=1 t=1 T D[t] and T are integers, we have t=1 φ(d[t ], 0.5) = = 1 T T D[t] 0.5 t=1 1 2T, T if T 2 D[t]; = 0, if T = 2 t=1 T D[t]. t=1 (2.2.2) Hence, the subsequence {φ(d[t k ], 0.5) : T k is odd} is aways ower-bounded by 1 2T k and thus the convergence speed of any zero-one sequence cannot be faster than ( ) 1 Ω. To vaidate this resut, we consider an independenty and identicay distributed (i.i.d.) Bernoui random sequence with mean 0.5. Figure 2.2 shows one T reaization of this random sequence. From this figure, we can see that φ(d[t ], 0.5) hits 0 for some T, and is aways non-zero when T is odd. The subsequence {φ(d[t k ], 0.5) : T k is odd} is aways ower-bounded by 1 2T k. 18

36 Rate deviation φ(d[t], 0.5) φ(d[t k ], 0.5) 1 2T Time step T Figure 2.2: The convergence speed of an i.i.d Bernoui sequence. This simpe exampe suggests that the discreteness in the choice of eements in the sequence exerts a fundamenta imitation on the speed with which its running average over time can approach its imit. In what foows, we wi show that this observation indeed hods even in the wider context of a muti-hop fading wireess network with a finite seection of transmission rates. 2.3 Convergence Speed in Rate Deviation In this section, we study the optima convergence speed in terms of rate deviation over wireess fading channes. To that end, we first give the fundamenta ower bound on the expected rate deviation for any agorithm. Then, we provide an agorithm that can achieve this ower bound and estabish the optimaity of the proposed agorithm. 19

37 2.3.1 A Lower Bound on the Expectation of Rate Deviation In this subsection, we show that for any poicy in P, the convergence speed of expected ( ) 1 rate deviation is Ω. To that end, we need the foowing integer assumption on T the transmission rate, which measures the number of packets that can be transmitted in one time sot. Assumption The service rate R for each ink L is seected from a finite and nonnegative-integer-vaued set {B,1, B,2,..., B,K }, where 0 B,1 < B,2 <... < B,K and K is some positive integer. Next, we give the foowing key emma, which wi aso be usefu in the ater section. Lemma Let I {a 1, a 2,..., a K }, where 0 a 1 < a 2 <... < a K and K is some positive integer. If r (a i, a i+1 ) for some i = 1,..., K 1, then for any sequence ( { r ai {I[t] : I[t] I} t 1, there exists a constant c r 0, min, a }) i+1 r such T that if I[t] r T c r T, then t=1 1 T +1 I[t] r T + 1 t=1 Remark: Note that K can be as arge as. Proof. See Appendix A.1 for the proof. c r T + 1. (2.3.1) Proposition Under Assumption 2.3.1, for any poicy in P with parameter ɛ, if r (ɛ) is not a vector with a integer-vaued coordinates, then convergence speed of the ( ) 1 expected rate deviation to zero is Ω, i.e., there exists a stricty positive constant T c and a positive integer-vaued increasing sequence {T k } k=1 such that φ(d (ɛ) [T k ], r (ɛ) ) c T k, k 1, (2.3.2) 20