Arizona State University

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1 SCHEDULING AND POWER ALLOCATION TO OPTIMIZE SERVICE AND QUEUE-WAITING TIMES IN COGNITIVE RADIO UPLINKS By arxv: v1 [cs.it] 4 Jan 2016 Ahmed Emad Ewasha Commttee: Dr. Chan Tepedelenloğlu, Char Dr. Le Yng Dr. Danel Blss Dr. Olver Kosut A Comprehensve Examnaton Presented n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy Electrcal, Comupter and Energy Engneerng Arzona State Unversty January 13 th, 2016

2 TABLE OF CONTENTS Page Abstract v Chapters: 1. Introducton Cogntve Rado Transmsson Schemes Guaranteeng Qualty of Servce n Cogntve Rado Systems Servce Tme Queue-Watng Tme Delay Due To Servce Tme Overlay System Model Problem Statement and Proposed Soluton Solvng for Prmal Varables Solvng for Dual Varables Optmalty of the Proposed Soluton Generalzaton of Deadlne Constrants Offlne Soluton Onlne Power-and-Threshold Adaptaton Underlay System Multple Secondary Users Numercal Results Delay Due to Queue-Watng Tme Network Model Channel and Interference Model Queung Model Transmsson Process Problem Statement

3 3.2.1 Instantaneous Interference Constrant Average and Instantaneous Interference Constrant Proposed Power Allocaton and Schedulng Algorthm Frame-Based Polcy Satsfyng Delay Constrants Algorthm for Instantaneous Interference Constrant Algorthm for Average Interference Constrant Near-Optmal Low Complexty Algorthm for Average Interference Problem Achevable Rate Regon of the DOAC Performance Under Channel Estmaton Errors Smulaton Results Instantaneous Interference Average Interference Low-Complexty Algorthm Performance CSI Estmaton Errors Proposed Future Work Hard Deadlne Guarantees Throughput Optmalty Concluson Chapters: A. Proof of Lemma B. Proof of Theorem C. Proof of Lemma D. Proof of Theorem E. Exstence of The Servce Tme Moments F. Proof of Theorem Bblography

4 ABSTRACT Cogntve Rado (CR) s an emergng wreless communcaton paradgm to mprove the spectrum utlzaton. Cogntve Rado users, also known as secondary users (SUs), are allowed to transmt over the channels as long as they do not cause harmful nterference to the prmary users (PUs) who have lcensed access to those channels. The qualty of servce (QoS) assocated wth ths transmsson scheme mght deterorate f those channels are not utlzed effcently by these SUs. In addton, SUs located physcally close to PUs mght cause more harmful nterference than those who are far. Ths mght degrade the QoS of those SUs snce they wll be allocated the channel less frequently to protect the PUs. In ths report, we study the packet delay as a QoS metrc n CR systems. The packet delay s defned as the average tme spent by a packet n the queue watng for transmsson as well as that spent durng the transmsson process. The former s referred to as the queue watng tme whle the latter s the servce tme. In real-tme applcatons, the average delay of packets needs to be below a prespecfed threshold to guarantee an acceptable QoS. In ths work, we study the effect of both the schedulng and the power allocaton algorthms on the delay performance of the SUs. We study how these two parameters affect both the servce tme as well as the queue watng tme. To study the delay due to the servce tme we study the effect of multple channels on a sngle SU and, thus, gnore the schedulng problem. Specfcally, n a multchannel system where the channels are sensed sequentally, we study the tradeoff between throughput and delay. The problem s formulated as an optmal stoppng rule problem where t s requred to decde at whch channel the SU should stop sensng and begn transmsson. We provde a closed-form soluton for ths optmal stoppng problem v

5 and specfy the optmal amount of power that ths SU should be transmttng wth over ths channel. The algorthm trades off the servce tme versus the throughput to guarantee a maxmum throughput performance subject to a bound on the average servce tme. Ths tradeoff results from skppng lowqualty channels to seek the possblty of fndng hgh-qualty ones n the future at the expense of a hgher probablty of beng blocked from transmsson snce these future channels mght be busy. On the other hand, the queue watng tme s studed by consderng a mult-su sngle channel system. Specfcally, we study the effect of schedulng and power allocaton on the delay performance of all SUs n the system. We propose a delay optmal algorthm to ths problem that schedules the SUs to mnmze the delay whle protectng the PUs from harmful nterference. One of the contrbutons of ths algorthm s that t can provde dfferentated servce to the users even f ther channels are statstcally heterogeneous. In heterogeneous-channels system, users wth statstcally low channel qualty are expected to have worse delay performances. However, the proposed algorthm guarantees a prespecfed delay performance to each SU wthout volatng the PU s nterference constrant. Exstng schedulng algorthms do not provde such guarantees f the nterference channels are heterogeneous. Ths s because they are developed for conventonal non-cr wreless systems that neglect nterference snce channels are orthogonal. Fnally, we present two potental extensons to these studed problems. In the frst one, nstead of mposng a constrant on the average delay as assumed n ths report, we mpose strct deadlnes by whch the packets need to be transmtted. Problems wth strct deadlnes have ther applcatons n lve streamng and onlne gamng where packets are expected to reach ther destnaton before a prespecfed deadlne expres. If a packet msses ts deadlne t s dropped from the system and does not count towards the throughput. However, these applcatons can tolerate a small percentage out of the total packets mssng ther deadlnes. We have solved the sngle-su verson of ths problem n Chapter 2. Extensons to the mult-user case s an nterestng problem. v

6 The second extenson s to study the schedulng problem at hand amng at fndng schedulng algorthms that are throughput optmal and delay optmal at the same tme. Except for specal cases of our problem, the proposed schedulng and power allocaton polcy does not acheve the capacty regon. However, our prelmnary results show that ther exst throughput-optmal schedulng algorthms that are well studed n the lterature that can be developed to be delay optmal as well. v

7 CHAPTER 1 Introducton Cogntve Rado (CR) systems are emergng wreless communcaton systems that allow effcent spectrum utlzaton [1]. CRs refer to devces that coexst wth the lcensed spectrum owners called the prmary users (PUs), and that are capable of detectng ther presence. Once PU s actvty s detected on some frequency channel, the CR user refrans from any further transmsson on ths channel. Ths may result n servce dsconnecton for the CR user, thus degradng the qualty of servce (QoS). If the CR users have access to other channels, the QoS can be mproved by swtchng to another frequency channel nstead of completely stoppng transmsson. If not, then they should control ther transmsson power to avod harmful nterference to the PUs. Hence, CR users are requred to adjust ther transmsson power levels, and -thus- ther rates, accordng to the nterference level the PUs can tolerate. Ths adjustment could lead to severe degradaton n the QoS provded for the CR users, f not desgned carefully. 1.1 Cogntve Rado Transmsson Schemes There are two man transmsson schemes that CR systems may follow to coexst wth the PUs; the overlay and the underlay. In the overlay, CR users, also referred to as the secondary users (SUs), transmt ther sgnal only when the PUs are not usng the channel. In other words, the SUs look for the spectrum holes to transmt ther data as n Fg Hence, unlke conventonal rados, SUs s rados are equpped wth a spectrum sensor that s used to sense the spectrum before begnnng the transmsson phase. In ths sensng phase, the SUs lsten to all frequency channels to overhear the PUs transmsson 1

8 Channel 1 Channel 2 Channel 3... tme tme Prmary user s data tme Secondary user hoppng among spectrum holes Fgure 1.1: Spectrum holes are the locatons of the unused spectrum n tme and frequency. so as to decde whch channels are free from PUs and whch are not. Upon ths detecton process, the SU pcks up a channel, or more, out of the detected-free channels to transmt ts data over for a lmted amount of tme. Once the channel s occuped agan by the PU, the SU s expected to refran from transmsson over ths channel but allowed to use a dfferent channel after performng the sensng phase agan. A practcal spectrum sensor mght yeld wrong decsons, namely, t mght detect the presence of a PU on some channel although ths channel s actually free, or mght mss-detect the PU when t s usng the channel. These events are referred to as the false-alarm and mss-detecton events, respectvely. The hgher the false-alarm probablty the hgher the SU msses transmsson opportuntes and, thus, the lower the SU s throughput s. Smlarly, the hgher the probablty of mss-detecton the more the SU s packet colldes wth the PU s and leadng to a lower throughput snce collded packets are lost. Whle the false-alarm probablty affects the SU s throughput alone, the mss-detecton probablty affects both the SU and the PU. As the sensng phase duraton ncreases, these two probabltes decrease smultaneously. 2

9 sec T seconds 1 2. Sensng Phase Transmsson Phase Fgure 1.2: The sensng phase s used to sense M channels to detect the presence of the PU. The SU starts transmttng ts data n the transmsson phase on one of the free channels. However, ncreasng the sensng phase duraton comes at the expense of the transmsson phase duraton thus decreasng the throughput. Ths tradeoff has been studed extensvely n the lterature [2]. In the underlay scheme, the SU s allowed to transmt over any frequency channel at any tme as long as the PU can tolerate the nterference caused by ths transmsson. Ths tolerable level s referred to as the nterference temperature as dctated by the Federal Communcatons Commsson (FCC) [3]. In order to guarantee ths protecton for the PU, the SU has to adjust ts transmsson power accordng to the gan of the channel to the prmary recever referred to as the nterference channel. The knowledge of ths gan nstantaneously s essental at the SU s transmtter. Whle ths channel knowledge mght be nfeasble n CR systems that assume no cooperaton between the PU and the SU, n some scenaros the SU mght be able to overhear the plots sent by the prmary recever when t s actng as a transmtter f the PU s usng a tme dvson duplex scheme. In both cases, the overlay and the underlay, the SU mght nterfere wth the PU. Ths n turn dctates that the SU should adopt ts channel access scheme n such a way that ths nterference s tolerable so that the PU s qualty of servce (QoS) s not degraded. Wth that beng sad, we mght expect that the SUs located physcally closer to the PUs mght suffer larger degradaton n ther QoS compared to those that are far because closer SUs transmt wth smaller amounts of power. Ths problem does not appear n conventonal non CR cellular systems snce frequency channels tend to be orthogonal n 3

10 non CR systems. In other words, n non CR systems, all users are allocated the channels va some scheduler that guarantees those users do not nterfere wth each other. Whle n CR systems, snce SUs nterfere wth PUs, we need to develop schedulng and power control algorthms that prevent harmful nterference to PUs, as well as guaranteeng acceptable QoS for the SUs. 1.2 Guaranteeng Qualty of Servce n Cogntve Rado Systems Snce CR users operate n an nterference lmted envronment, they are expected to experence lower QoS than n conventonal systems. However, the QoS provded needs to fall wthn the acceptable level that vares wth the applcaton. For example, the average delay of a packet n onlne streamng s requred to not more than 300ms whle that n onlne gamng should not exceed 50ms. However, these two applcatons mght tolerate small losses n ther transmtted packets whch s not the case wth some other applcatons as fle sharng and emal applcatons that, on the other hand, mght tolerate packet delays. The QoS can nclude, but s not lmted to, throughput, delay, bt-error-rate, nterference caused to the PU. Out of these metrcs the most two major ones are the throughput and the delay that have ganed strong attenton n the lterature recently [4]. The throughput metrc s defned as the average amount of packets (or bts) per channel-use that can be delvered n the SU s network wthout volatng the PU s nterference constrants. On the other hand, the delay refers to as the amount of tme elapsed from the nstant a packet jons the SU s buffer untl t s successfully and fully transmtted to ts ntended recever. A hgher throughput s usually acheved by the effcent power allocaton algorthms whle better delay performances are usually acheved by effcent schedulng of users. The problem of schedulng and/or power control has been wdely studed n the lterature (see [5 11], and references theren). These works am at optmzng the throughput, provdng delay guarantees and/or guaranteeng protecton from nterference. In real-tme applcatons, such as audo/vdeo conference calls, one of the most mportant QoS metrcs s the delay metrc. The delay s defned as the average 4

11 amount of tme a packet spends n the system startng from the nstant t arrves to the buffer untl t s completely transmtted. In real-tme applcatons, packets are expected to arrve at the destnaton before a prespecfed deadlne [12]. Thus, the average packet delay needs to be as small as possble to prevent jtter and to guarantee acceptable QoS for these applcatons [10, 11]. There are two dfferent factors that cause delay n data networks. The frst s the servce tme whch s the amount of tme requred to transmt ths packet. The second s the queue-watng tme whch s the tme spent by a packet n the queue watng for ts transmsson to begn. The sum of both yelds the delay. Thus, n order to optmze over the delay we should study both factors. 1.3 Servce Tme The servce tme s affected by the amount of resources allocated to the packet at the tme of transmsson. Resources mght nclude power, channel bandwdth, codng rate and transmsson tme. Several works have been studed to address how to optmally allocate these resources over tme and users. However, from a practcal mplementaton pont of vew, the most challengng resource s channel bandwdth. Ths s because ncreasng the bandwdth requres allocatng multple channels to a user whch mght requre the user to be equpped wth hgh cost transmtters (recevers) capable of transmttng (recevng) over multple channels smultaneously. On the other hand, allocatng a sngle fxed channel to a user s not optmal. The problem of channel allocaton n mult-channel CR systems has ganed attenton n recent works due to the challenges assocated wth the sensng and access mechansms n a multchannel CR system. Practcal hardware constrants on the SUs transcevers may prevent them from sensng multple channels smultaneously to detect the state of these channels (free/busy). Ths leads the SU to sense the channels sequentally, then decde whch channel should be used for transmsson [13,14]. In a tme slotted system f sequental channel sensng s employed, the SU senses the channels one at a tme and stops sensng when a channel s found free. But due to the ndependent fadng among channels, the SU s allowed to 5

12 skp a free channel f ts qualty, measured by ts power gan, s low and sense another channel seekng the possblty of a hgher future gan. Otherwse, f the gan s hgh, the SU stops at ths free channel to begn transmsson. The queston of when to stop sensng can be formulated as an optmal stoppng rule problem [14 17]. In [15] the authors present the optmal stoppng rule for ths problem n a non-cr system. The work n [14] develops an algorthm to fnd the optmal order by whch channels are to be sequentally sensed n a CR scenaro, whereas [16] studes the case where the SUs are allowed to transmt on multple contguous channels smultaneously. The authors presented the optmal stoppng rule for ths problem n a non-fadng wreless channel. Transmssons on multple channels smultaneously may be a strong assumpton for low-cost transcevers especally when they cannot sense multple channels smultaneously. In general, f a perfect sensng mechansm s adopted, the SU wll not cause nterference to the PU snce the former transmts only on spectrum holes (referred to as an overlay system). Nevertheless, f the sensng mechansm s mperfect, or f the SU s system s an underlay one (where the SU uses the channels as long as the nterference to the PU s tolerable), the transmtted power needs to be controlled to prevent harmful nterference to the PU. References [18] and [5] consder power control and show that the optmal power control strategy s a water-fllng approach under some nterference constran mposed on the SU transmtter. Yet, all of the above work studes sngle channel systems whch cannot be extended to multple channels n a straghtforward manner. A multuser CR system was consdered n [19] n a tme slotted system. To allocate the frequency channel to one of the SUs, the authors proposed a contenton mechansm that does not depend on the SUs channel gans, thus neglectng the advantage of multuser dversty. A major challenge n a multchannel system s the sequental nature of the sensng where the SU needs to take a decson to stop and begn transmsson, or contnue sensng based on the nformaton t has so far. Ths decson needs to trade-off between watng for a potentally hgher throughput and takng advantage of the current free channel. Moreover, 6

13 f transmsson takes place on a gven channel, the SU needs to decde the amount of power transmtted to maxmze ts throughput gven some average nterference and average power constrants. In Chapter 2, we model the overlay and underlay scenaros of a mult-channel CR system that are sensed sequentally. The problem s solved for a sngle SU frst then we dscuss extensons to a mult-su scenaro. For the sngle SU case, the problem s formulated as a jont optmal-stoppng-rule and powercontrol problem wth the goal of maxmzng the SU s throughput subject to average power and average nterference constrants. Ths formulaton results n ncreasng the expected servce tme of the SU s packets. The expected servce tme s the average number of tme slots that pass whle the SU attempts to fnd a free channel, before successfully transmttng a packet. The ncrease n the servce tme s due to skppng free channels, due to ther poor gan, hopng to fnd a future channel of suffcently hgh gan. If no channels havng a satsfactory gan were found, the SU wll not be able to transmt ts packet, and wll have to wat for longer tme to fnd a satsfactory channel. Ths ncrease n servce tme ncreases the queung delay. Thus, we solve the problem subject to a bound on the expected servce tme whch controls the delay. In the multple SUs case, we show that the soluton to the sngle SU problem can be appled drectly to the mult-su system wth a mnor modfcaton. We also show that the complexty of the soluton decreases when the system has a large number of SUs. To the best of our knowledge, ths s the frst work to study the jont power-control and optmalstoppng-rule problem n a mult channel CR system. Our contrbuton n ths work s the formulaton of a jont power-control and optmal-stoppng-rule problem that also ncorporates a delay constrant and present a low complexty soluton n the presence of nterference/collson constrant from the SU to the PU due to the mperfect sensng mechansm. The prelmnary results n [20] consder an overlay framework for sngle user case whle neglectng sensng errors. But n ths work, we also study the problem n the underlay scenaro where nterference s allowed from the secondary transmtter (ST) to the prmary recever (PR) and extend to multple SU case. We also generalze the soluton to the mult-su case when the number of SUs s large. We dscuss the applcablty of our formulaton n 7

14 typcal delay-constraned scenaros where packets arrve smultaneously and have a same deadlne. We show that the proposed algorthm can be used to solve ths problem offlne, to maxmze the throughput and meet the deadlne constrant at the same tme. Moreover, we propose an onlne algorthm that gves hgher throughput compared to the offlne approach whle meetng the deadlne constrant. 1.4 Queue-Watng Tme Unlke the servce tme, the delay due to queue-watng tme s affected by the schedulng algorthm. The more frequently a user s allocated the channel for transmsson, the less ts queue-watng tme s, but the more the queue-watng tmes for the other users are. Delay due to the queue-watng tme s also well studed recently n the lterature and schedulng algorthms have been proposed to guarantee small delay for users n conventonal systems [21 23]. In [21], the authors study the jont schedulngand-power-allocaton problem n the presence of an average power constrant. Although n [21] the proposed algorthm offers an acceptable delay performance, all users are assumed to transmt wth the same power. A power allocaton and routng algorthm s proposed n [23] to maxmze the capacty regon under an nstantaneous power constrant. Whle the authors show an upper bound on the average delay, ths delay performance s not guaranteed to be optmal. Although queung theory, that was orgnally developed to model packets at a server, can be appled to wreless channels, the challenges are dfferent. One of the man challenges s the fadng nature of the wreless channel that changes from a slot to another. Fadng requres adaptng the user s power and/or rate accordng to the channel s fadng coeffcent. The dea of power and/or rate adaptaton based on the channel condton does not have an analogy n server problems and, thus, s absent n the aforementoned references. Instead, exstng works treat wreless channels as on-off fadng channels and do not consder multple fadng levels. Among the relevant references that consder a more general fadng channel model are [23], whch was dscussed above, [24, 25] where the optmzaton over the schedulng 8

15 algorthm was out of the scope of ther work, and [26] that neglects the nterference constrant snce t consders a non CR system. In contrast wth [6 9, 27] that do not optmze the queung delay, the problem of mnmzng the sum of SUs average delays s consdered n ths work. The proposed algorthm guarantees a bound on the nstantaneous nterference to the PUs, a guarantee that s absent n [21, 23]. Based on Lyapunov optmzaton technques [21], an algorthm that dynamcally schedules the SUs as well as optmally controllng ther transmsson power s presented. The contrbutons n ths work are: ) Proposng a jont power-control and schedulng algorthm that s optmal wth respect to the average delay of the SUs n an nterference-lmted system; ) Showng that the proposed algorthm can provde dfferentated servce to the dfferent SUs based on ther heterogeneous QoS requrements. Moreover, the complexty of the algorthm s shown to be polynomal n tme snce t s equvalent to that of sortng a vector of N numbers, where N s the number of SUs n the system. 9

16 CHAPTER 2 Delay Due To Servce Tme In ths chapter we study the delay resultng from the servce tme of packets and neglect the delay resultng from the watng tme n the queues. We treat the cogntve rado system as a sngle secondary user (SU) accessng a mult-channel system. The man problem studed n ths chapter s the tradeoff between the servce tme and the throughput. We assume the SU senses the channels sequentally to detect the presence of the prmary user (PU), and stops ts search to access a channel f t offers a sgnfcantly hgh throughput. The tradeoff exsts because stoppng at early-sensed channels gves low average servce tme but, at the same tme, gves low throughput snce early channels mght have low gans. The jont optmal stoppng rule and power control problem s formulated as a throughput maxmzaton problem wth an average servce tme and power constrant. We note that n ths chapter we use the word delay to refer to the servce tme. 2.1 Overlay System Model Consder a PU network that has a lcensed access to M orthogonal frequency channels. Tme s slotted wth a tme slot duraton of T seconds. The SU s network conssts of a sngle ST (SU and ST wll be used nterchangeably) attemptng to send real-tme data to ts ntended secondary recever (SR) through one of the channels lcensed to the PU. Before a tme slot begns, the SU s assumed to have ordered the channels accordng to some sequence (we note that the method of orderng the channels s outsde the scope of ths work. The reader s referred to [14] for further detals about channel orderng), 10

17 labeled 1,..., M. The set of channels s denoted by M = {1,..., M}. Before the SU attempts to transmt ts packet over channel, t senses ths channel to determne ts avalablty state whch s descrbed by a Bernoull random varable b wth parameter θ (θ s called the avalablty probablty of channel ). If b = 0 (whch happens wth probablty θ ), then channel s free and the SU may transmt over t untl the on-gong tme slot ends. If b = 1, channel s busy, and the SU proceeds to sense channel +1. Channel avalabltes are statstcally ndependent across frequency channels and across tme slots. We assume that the SU has lmted capabltes n the sense that no two channels can be sensed smultaneously. Ths may be the case when consderng rados havng a sngle sensng module wth a fxed bandwdth, so that t can be tuned to only one frequency channel at a tme. The reader s referred to [28], [29] and [30] for detaled nformaton on advanced spectrum sensng technques. Therefore, at the begnnng of a gven tme slot, the SU selects a channel, say channel 1, senses t for τ seconds (τ T/M), and f t s free, the SU transmts on ths channel f ts channel gan s hgh enough 1. Otherwse, the SU skps ths channel and senses channel 2, and so on untl t fnds a free channel. If all channels are busy (.e. the PU has transmsson actvtes on all M channels) then ths tme slot wll be consdered as blocked. In ths case, the SU wats for the followng tme slot and begns sensng followng the same channel sensng sequence. As the sensng duraton ncreases, the transmsson phase duraton decreases whch decreases the throughput. But we cannot arbtrarly decrease the value of τ snce ths decreases the relablty of the sensng outcome. Ths trade-off has been studed extensvely n the lterature, e.g. [31], [32]. In ths work we study the mpact of sequental channel sensng on the throughput rather than the sensng duraton on the throughput. Hence we assume that τ s a fxed parameter and s not optmzed over. For detals on the trade-off between throughput and sensng duraton n ths sequental sensng problem the reader s referred to [2]. 1 How hgh s hgh s gong to be explaned later 11

18 seconds.. Sensng Phase Transmsson Phase Fgure 2.1: Sensng and transmsson phases n one tme slot. The SU senses each channel for τ seconds, determnes ts state, then probes the gan f the channel s found free. The sensng phase ends f the probed gan γ > γ th (), n whch case k =. Hence, k s a random varable that depends on the channel states and gans. The fadng channel between ST and SR s assumed to be flat fadng wth ndependent, dentcally dstrbuted (..d.) channel gans across the M channels. To acheve hgher data rates, the SU adapts ts data rate accordng to the nstantaneous power gan of the channel before begnnng transmsson on ths channel. To do ths, once the SU fnds a free channel, say channel, the gan γ s probed. The data rate wll be proportonal to log(1 + P 1, (γ )γ ), where P 1, (γ ) s the power transmtted by the SU at channel as a functon of the nstantaneous gan [33]. Fg. 2.1 shows a potental scenaro where the SU senses k channels, skps the frst k 1, and uses the k th channel for transmsson untl the end of ths on-gong tme slot. In ths scenaro the SU stops at the k th channel, for some k M. Stoppng at channel depends on two factors: 1) the avalablty of channel b, and 2) the nstantaneous channel gan γ. Clearly, b and γ are random varables that change from one tme slot to another. Hence, k, that depends on these two factors, s a random varable. More specfcally, t depends on the states [b 1,..., b M ] along wth the gans of each channel [γ 1,..., γ M ]. To understand why, consder that the SU senses channel, fnds t free and probes ts gan γ. If γ s found to be low, then the SU skps channel (although free) and senses channel + 1. Ths s to take advantage of the possblty that γ j γ for j >. On the other hand, f γ s suffcently large, the SU stops at channel and begns transmsson. In that latter case k =. Defnng the two random vectors b = [b 1,..., b M ] T and γ = [γ 1,..., γ M ] T, k s a determnstc functon of b and γ. 12

19 We defne the stoppng rule by defnng a threshold γ th () to whch each γ s compared when the th channel s found free. If γ γ th (), channel s consdered to have a hgh gan and hence the SU stops and transmts at channel. Otherwse, channel s skpped and channel + 1 sensed. In the extreme case when γ th () = 0, the SU wll not skp channel f t s found free. Increasng γ th () allows the SU to skp channel whenever γ < γ th (), to search for a better channel, thus potentally ncreasng the throughput. Settng γ th () too large allows channel to be skpped even f γ s hgh. Ths consttutes the trade-off n choosng the thresholds γ th (). The optmal values of γ th () M, determne the optmal stoppng rule. Let P 1, (γ) denote the power transmtted at the th channel when the nstantaneous channel gan s γ, f channel was chosen for transmsson. Snce the SU can transmt on one channel at a tme, the power transmtted at any tme slot at channel s P 1, (γ )1 ( = k ), where 1 ( = k ) = 1 f = k and 0 otherwse. Defne c 1 τ T as the fracton of the tme slot remanng for the SU s transmsson f the SU transmts on the th channel n the sensng sequence. The average power constrant s E γ,b [c k P k (γ k )] P avg, where the expectaton s wth respect to the random vectors γ and b. We wll henceforth drop the subscrpt from the expected value operator E. Ths expectaton can be calculated recursvely from S (Γ th (), P 1, ) = θ c γ th () P 1, (γ)f γ (γ) dγ + [ 1 θ Fγ (γ th ()) ] S +1 (Γ th ( + 1), P +1 ), (2.1) M, where P 1, [P 1, (γ),..., P 1,M (γ)] T and Γ th () [γ th (),..., γ th (M)] T are the vectors of the power functons and thresholds respectvely, wth S M+1 (Γ th (M +1), P M+1 ) 0, f γ (γ) s the Probablty Densty Functon (PDF) of the gan γ of channel, and F γ (x) f x γ (γ) dγ s the complementary cumulatve dstrbuton functon. The frst term n (2.1) s the average power transmtted at channel gven that channel s chosen for transmsson (.e. gven that k = ). The second term represents the case where channel s skpped and channel + 1 s sensed. It can be shown that S 1 (Γ th (1), P 1,1 ) = E [c k P k (γ)]. Moreover, we wll also drop the ndex from the subscrpt of f γ (γ) and F γ (γ) snce channels suffer..d. fadng. Although we have only ncluded an average power constrant n our 13

20 problem, we wll modfy, after solvng the problem, the soluton to nclude an nstantaneous power constrant as well. The SU s average throughput s defned as E[c k log(1 + P k (γ k )γ k )]. Smlar to the average power, we denote the expected throughput as U 1 (Γ th (1), P 1,1 ) whch can be derved usng the followng recursve formula U (Γ th (),P 1, ) = θ c log (1 + P 1, (γ)γ) f γ (γ) dγ+ γ th () [ 1 θ Fγ (γ th ()) ] U +1 (Γ th ( + 1), P +1 ) (2.2) M, wth U M+1 (, ) 0. U 1 (Γ th (1), P 1,1 ) represents the expected data rate of the SU as a functon of the threshold vector Γ th (1) and the power functon vector P 1,1. If the SU skps all channels, ether due to beng busy, due to ther low gan or due to a combnaton of both, then the current tme slot s sad to be blocked. The SU has to wat for the followng tme slot to begn searchng for a free channel agan. Ths results n a delay n servng (transmttng) the SU s packet. Defne the delay D as the number of tme slots the SU consumes before successfully transmttng a packet. That s, D 1 s a random varable that represents the number of consecutvely blocked tme slots. In real-tme applcatons, there may exst some average delay requrement D max on the packets that must not be exceeded. Snce the avalablty of each channel s ndependent across tme slots, D follows a geometrc dstrbuton havng E[D] = (Pr[Success]) 1 where Pr[Success] = 1 Pr[Blockng]. In other words, Pr[Success] s the probablty that the SU fnds a free channel wth hgh enough gan so that t does not skp all M channels n a tme slot. It s gven by Pr[Success] p 1 (Γ th (1)) whch can be calculated recursvely usng the followng equaton p (Γ th ()) = θ Fγ (γ th ()) + [ 1 θ Fγ (γ th ()) ] p +1 (Γ th ( + 1)), (2.3) M, where p M+1 0. Here, p (Γ th ()) s the probablty of transmsson on channel, + 1,..., or M. 14

21 2.2 Problem Statement and Proposed Soluton From equaton (2.2) we see that the SU s expected throughput U 1 depends on the threshold vector Γ th (1) and the power vector P 1,1. The goal s to fnd the optmum values of Γ th (1) R M and functons P 1,1 that maxmze U 1 subject to an average power constrant and an expected packet delay constrant. The delay constrant can be wrtten as E[D] D max or, equvalently, p 1 (Γ th (1)) 1/ D max. Mathematcally, the problem becomes maxmze U 1 (Γ th (1), P 1,1 ) subject to S 1 (Γ th (1), P 1,1 ) P avg p 1 (Γ th (1)) D 1 max varables Γ th (1), P 1,1, where the frst constrant represents the average power constrant, whle the second s a bound on the average packet delay. We allow the power P 1, to be an arbtrary functon of γ and optmze over ths functon to maxmze the throughput subject to average power and delay constrants. Even though (2.4) s not proven to be convex, we provde closed-form expressons for the optmal thresholds and power-functons vector. To ths end, we frst calculate the Lagrangan assocated wth (2.4). Let λ P and λ D be the dual varables assocated wth the constrants n problem (2.4). The Lagrangan for (2.4) becomes (2.4) L (Γ th (1), P 1,1, λ P, λ D ) = U 1 (Γ th (1), P 1,1 ) λ P (S 1 (Γ th (1), P 1,1 ) P avg ) + λ D ( p 1 (Γ th (1)) 1 D max ). (2.5) Dfferentatng (2.5) wth respect to each of the prmal varables P 1, (γ) and γ th () and equatng the resultng dervatves to zero, we obtan the KKT equatons below whch are necessary condtons for 15

22 optmalty [34], [35]: ( 1 P1,(γ) = 1 ) +, γ > γ λ th (), (2.6) P γ ( ( 1 log ) ) + ( 1 λ P γth () γth () λ P 1 ) + λ P γth ) () = U +1 λ P S +1 λ D (1 p +1, (2.7) c S1 P avg, p 1 1, λ D P 0, λ D 0, (2.8) max λ P (S1 P avg ) = 0, (2.9) ( λ D p 1 1 ) = 0, (2.10) D max M. We use U +1 U +1 ( Γ th ( + 1), P +1) whle S +1 S +1 ( Γ th ( + 1), P +1) and p +1 p +1 (Γ th ( + 1)) for brevty n the sequel. We note that U M+1 (, ) = S M+1 (, ) = p M+1 ( ) 0 by defnton. We observe that these KKT equatons nvolve the prmal (Γ th (1) and P 1) and the dual (λ P and λ D ) varables. Our approach s to fnd a closed-form expresson for the prmal varables n terms of the dual varables, then propose a low-complexty algorthm to obtan the soluton for the dual varables. The optmalty of ths approach s dscussed at the end of ths secton (n Secton 2.2.3) where we show that, loosely speakng, the KKT equatons provde a unque soluton to the prmal-dual varables. Hence, based on ths unque soluton, and on the fact that the KKT equatons are necessary condtons for the optmal soluton, then ths soluton s not only necessary but suffcent as well, and hence optmal Solvng for Prmal Varables Equaton (2.6) s a water-fllng strategy wth a slght modfcaton due to havng the condton γ > γ th (). Ths condton comes from the sequental sensng of the channels whch s absent n the classc water-fllng strategy [33]. Equaton (2.6) gves a closed-form soluton for P 1,1. On the other hand, the entres of the vector Γ th (1) are found va the set of equatons (2.7). Note that equaton (2.7) ndeed forms a set of M equatons, each solves for one of the γth (), M. We refer to ths set as 16

23 the threshold-fndng equatons. For a gven value of, solvng for γth () requres the knowledge of only γth ( + 1) through γ th (M), and does not requre knowng γ th (1) through γ th ( 1). Thus, these M equatons can be solved usng back-substtuton startng from γth (M). To solve for γ th (), we use the fact that γ th () λ P that s proven n the followng lemma. Lemma 1. The optmal soluton of problem (2.4) satsfes γ th () λ P M. Proof. See Appendx A for proof. The ntuton behnd Lemma 1 s as follows. If, for some channel, γth () < λ P was possble, and the nstantaneous gan γ happened to fall n the range [γth (), λ P ) at a gven tme slot, then the SU wll not skp channel snce γ > γ th (). But the power transmtted on channel s P 1,(γ ) = (1/λ P 1/γ ) + = 0 snce γ < λ P. Ths means that the SU wll nether skp nor transmt on channel, whch does not make sense from the SU s throughput perspectve. To overcome ths event, the SU needs to set γ th () at least as large as λ P so that whenever γ < λ P, the SU skps channel rather than transmttng wth zero power. Lemma 1 allows us to remove the ( ) + sgn n equaton (2.7) when solvng for γth (). Rewrtng (2.7) we get λ P γ th ( ) λ exp P () γth () = ( exp U +1 λ P S +1 λ D (1 p +1 1 c ) ), M, (2.11) Equaton (2.11) s now on the form W exp(w ) = c, whose soluton s W = W 0 (c), where W 0 (x) s the prncple branch of the Lambert W functon [36] and s gven by W 0 (x) = ( n) n 1 n=1 x n. The only n! soluton to (2.11) whch satsfes Lemma 1 s gven for M by γ th () = W 0 ( exp λ P )). (2.12) ( (U +1 λ P S +1 λ D(1 p +1)) + c 1 17

24 Hence, Γ th (1) and P 1 are found va equatons (2.12) and (2.6) respectvely whch are one-to-one mappngs from the dual varables (λ P, λ D ). And f we had an nstantaneous power constrant P 1,(γ) P max, we could wrte down the Lagrangan and solve for P 1, (γ). The detals are smlar to the case wthout an nstantaneous power constrant and are, thus, omtted for brevty. The reader s referred to [5] for a smlar proof. The expresson for P 1,(γ) s gven by P 1,(γ) = { ( 1 λ P 1 γ P max ) + f 1 1 < P λ P γ max otherwse. (2.13) Snce the optmal prmal varables are explct functons of the optmal dual varables, once the optmal dual varables are found, the optmal prmal varables are found and the optmzaton problem s solved. We now dscuss how to solve for these dual varables Solvng for Dual Varables The optmum dual varable λ P must satsfy equaton (2.9). Thus f λ P > 0, then we need S 1 P avg = 0. Ths equaton can be solved usng any sutable root-fndng algorthm. Hence, we propose Algorthm 1 that uses bsecton [37]. In each teraton n, the algorthm calculates S1 gven that λ P = λ (n) P, and gven some fxed λ D, compares t to P avg to update λ (n+1) P accordngly. The algorthm termnates when S 1 = P avg,.e. λ (n) P = λ P. The superorty of ths algorthm over the exhaustve search s due to the use of the bsecton algorthm that does not go over all the search space of λ P. In order for the bsecton to converge, there must exst a sngle soluton for equaton S 1 = P avg. Ths s proven n Theorem 1. Theorem 1. S 1 s decreasng n λ P [0, ) gven some fxed λ D 0. Moreover, the optmal value λ P satsfyng S 1 = P avg s upper bounded by λ max P Proof. See Appendx B for the proof. M =1 θ c /P avg. We note that Algorthm 1 can be systematcally modfed to call any other root-fndng algorthm (e.g. the secant algorthm [37] that converges faster than the bsecton algorthm). Now, to search for λ D, we state the followng lemma. 18

25 Algorthm 1 Fndng λ P gven some λ D 1: Intalze n 1, λ mn P 0, λ max P 2: whle S1 P avg > ɛ do 3: Calculate S1 gven that λ P = λ(n) P. Call t S(n). 4: f S (n) P avg > 0 then 5: λ mn P 6: else 7: λ max P 8: end f 9: λ (n+1) P = λ (n) P 10: n n : end whle 12: λ P λ(n) P = λ (n) P ( λ mn P + ) λmax P /2 M =1 θ c /P avg, λ (1) P ( λ mn P + ) λmax P /2 Lemma 2. The optmum value λ D that solves problem (2.4) satsfes 0 λ D < λmax D, where wth t gven by ( ( mn ( λ max P λ max P M =2 1 j=2 (1 θ j) θ. λ max D ( ))) ( ( 1 1, F γ θ 1 / Dmax F 1 γ log (γ/λ max P ) f γ(γ) dγ c 1 [log (t) t + 1] + U2 max 1 p max 2 )) 1 θ 1 Dmax (2.14) and U max 2 s an upper bound on U 2 and s ) ( M =2 θ c ), whle p max 2 s an upper bound on p 2 and s gven by Proof. See Appendx C. Lemma 2 gves an upper bound on λ D. Ths bound decreases the search space of λ D drastcally nstead of searchng over R. Thus the soluton of problem (2.4) can be summarzed on 3 steps: 1) Fx λ D [0, λmax D ) and fnd the correspondng optmum λ P usng Algorthm 1. 2) Substtute the par (λ P, λ D ) n equatons (2.6) and (2.12) to get the power and threshold functons, then evaluate U 1 from (2.2). 3) Repeat steps 1 and 2 for other values of λ D untl reachng the optmum λ D that satsfes p 1 = 1/ D max. If there are multple λ D s satsfyng p 1 = 1/ D max, then the optmum one s the one that gves the hghest U1. Although the order by whch the channels are sensed s assumed fxed, the proposed algorthm can be modfed to optmze over the sensng order by a relatvely low complexty sortng algorthm. 19

26 Partcularly, the dynamc programmng proposed n [14] can be called by Algorthm 1 to order the channels. The complexty of the sortng algorthm alone s O(2 M ) compared to the O(M!) of the exhaustve search to sort the M channels. The modfcaton to our proposed algorthm would be n step 3 of Algorthm 1, where S 1 would be optmzed over the number of channels (as well as Γ th (1)) Optmalty of the Proposed Soluton Snce the problem n (2.4) s not proven to be convex, the KKT condtons provde only necessary condtons for optmalty and need not be suffcent [38]. Ths means that there mght exst multple solutons (.e. multple solutons for the prmal and/or dual varables) satsfyng the KKT condtons, at least one of whch s optmal. But snce Theorem 1 proves that there exsts one unque soluton to λ P gven λ D, then Γ th (1) and P 1 are unque as well (from equatons (2.6) and (2.12)) gven some λ D. Hence, by sweepng λ D over [0, λmax D ), we have a unque soluton satsfyng the KKT condtons, whch means that the KKT condtons are suffcent as well and our approach s optmal for problem (2.4). 2.3 Generalzaton of Deadlne Constrants In the overlay and underlay schemes dscussed thus far, we were assumng that each packet has a hard deadlne of one tme slot. If a packet s not delvered as soon as t arrves at the ST, then t s dropped from the system. But n real-tme applcatons, data arrves at the ST s buffer on a frame-byframe structure. Meanng multple packets (consttutng the same frame) arrve smultaneously rather than one at a tme. A frame conssts of a fxed number of packets, and each packet fts nto exactly one tme slot of duraton T. Each frame has ts own deadlne and thus, packets belongng to the same frame have the same deadlne [39]. Ths deadlne represents the maxmum number of tme slots that the packets, belongng to the same frame, need to be transmtted by, on average. In ths secton we solve ths problem for the overlay scenaro. The soluton presented n Secton 2.2 can be thought of as a specal case of the problem presented n ths secton where the deadlne was equal to 1 tme slot and each frame conssts of one packet. We show that the soluton presented n 20

27 Secton 2.2 can be used to solve ths generalzed problem n an offlne fashon (.e. before attemptng to transmt any packet of the frame). Moreover, we propose an onlne update algorthm that updates the thresholds and power functons each tme slot and show that ths outperforms the offlne soluton Offlne Soluton Assume that each frame conssts of K packets and that the entre frame has a deadlne of t f tme slots (t f > K). If the SU does not succeed n transmttng the K packets before the t f tme slots, then the whole frame s consdered wasted. Snce nstantaneous channel gans and PU s actvtes are ndependent across tme slots, the probablty that the SU succeeds n transmttng the frame n t f tme slots or less s gven by P frame (K, t f ) = t f n=k ( ) tf p n (1 p) t f n n (2.15) where p s the probablty of transmttng a packet on some channel n a sngle tme slot and s gven by (2.3) or (2.21) f the SU s system was overlay or underlay respectvely. P frame (K, t f ) represents the probablty of fndng K or more free tme slots out of a total of t f tme slots. In order to guarantee some QoS for the real-tme data the SU needs to keep the probablty of successful frame transmsson above a mnmum value denoted r mn, that s P frame r mn. Hence the problem becomes a throughput maxmzaton problem subject to some average power and QoS constrants as follows maxmze U 1 (Γ th (1), P 1,1 ) subject to S 1 (Γ th (1), P 1,1 ) P avg P frame (K, t f ) r mn varables Γ th (1), P 1,1. (2.16) Ths s the optmzaton problem assumng an overlay system snce we used equatons (2.2) and (2.1) for the throughput and power, respectvely. It can also be modfed systematcally to the case of an underlay system. Snce there exsts a one-to-one mappng between P frame (K, t f ) and p, then there exsts a value for D max such that the nequalty p 1/ D max s equvalent to the QoS nequalty P frame (K, t f ) r mn. That s, we can replace nequalty P frame (K, t f ) r mn by p 1/ D max for some D max that depends on r mn, K and t f that are known a pror. Consequently, problem (2.16) s reduced to the smpler, 21

28 yet equvalent, sngle-tme-slot problem (2.4) and the SU can solve for P 1 and Γ th (1) vectors followng the approach proposed n Secton 2.2. The SU solves ths problem offlne (.e. before the begnnng of the frame transmsson) and uses ths soluton each tme slot of the t f tme slots. Wth ths offlne scheme, the SU wll be able to meet the QoS and the average power constrant requrements as well as maxmzng ts throughput Onlne Power-and-Threshold Adaptaton In problem (2.4), we have seen that as 1/ D max decreases, the system becomes less strngent n terms of the delay constrant. Ths results n an ncrease n the average throughput U 1. Wth ths n mnd, let us assume, n the generalzed delay model, that at tme slot 1 the SU succeeds n transmttng a packet. Thus, at tme slot 2 the SU has K 1 remanng packets to be transmtted n t f 1 tme slots. And from the propertes of equaton (2.15), P frame (K 1, t f 1) > P frame (K, t f ). Ths means that the system becomes less strngent n terms of the QoS constrant after a successful packet transmsson. Ths advantage appears n the form of hgher throughput. To see how we can make use of ths advantage, defne P frame (K(t), t f t + 1) as P frame (K(t), t f t + 1) = t f t+1 ( ) tf t + 1 (p(t)) n (1 p(t)) t f t+1 n, (2.17) n n=k(t) where K(t) s the remanng number of packets before tme slot t {1,..., t f } and p(t) s the probablty of successful transmsson at tme slot t. At each tme slot t {1,...t f }, the SU modfes the QoS constrant to be P frame (K(t), t f t + 1) r mn nstead of P frame (K, t f ) r mn, that was used n the offlne adaptaton, and solve the followng problem to obtan the power and threshold vectors. maxmze U 1 (Γ th (1), P 1,1 ) subject to S 1 (Γ th (1), P 1,1 ) P avg P frame (K(t), t f t + 1) r mn varables Γ th (1), P 1,1, (2.18) When the delay constrant n (2.18) s replaced by ts equvalent constrant p 1/ D max, the resultng problem can be solved usng the overlay approach 22

29 proposed n Secton 2.2 wthout much ncrease n computatonal complexty snce the power functons and thresholds are gven n closed-form expressons. Wth ths onlne adaptaton, the average throughput U 1 ncreases whle stll satsfyng the QoS constrant. 2.4 Underlay System In the overlay system, the SU tres to locate the free channels at each tme slot to access these spectrum holes wthout nterferng wth the PUs. Recently, the FCC has allowed the SUs to nterfere wth the PU s network as long as ths nterference does not harm the PUs [40]. If the nterference from the SU measured at the PU s recever s below the tolerable level, then the nterference s deemed acceptable. In order to model the nterference at the PR, we assume that the SU uses a channel sensng technque that produces the suffcent statstc z at channel [41,42]. The SU s assumed to know the dstrbuton of z gven channel s free and busy, namely f z b (z b = 0) and f z b (z b = 1) respectvely. For brevty, we omt the subscrpt from b whenever t s clear from the context. The value of z ndcates how confdent the SU s n the presence of the PU at channel. Thus the SU stops at channel accordng to how lkely busy t s and how much data rate t wll gan from ths channel (.e. accordng to z and γ respectvely). Hence, when the SU senses channel to acqure z, the channel gan γ s probed and compared to some functon γ th (, z ); f γ γ th (, z ) transmsson occurs on channel, otherwse, channel s skpped and + 1 s sensed. Potentally, γ th (, z ) s a functon n the statstc z. Ths means that, at channel, for each possble value that z mght take, there s a correspondng threshold γ th (, z). Before formulatng the problem we note that ths model can capture the overlay wth sensng errors model as a specal case where f z b (z b = 1) = (1 P MD )δ(z z b ) + P MD δ(z z f ) whle f z b (z b = 0) = P FA δ(z z b ) + (1 P FA )δ(z z f ), where P MD and P FA are the probabltes of mssed-detecton and false-alarm respectvely, whle δ(z) s the Drac delta functon, and z b and z f that represent the values that z takes when the channel s busy and free, respectvely. Hence, the nterference 23

30 constrant, whch wll soon be descrbed, can be modfed to a detecton probablty constrant and/or a false alarm probablty constrant. from The SU s expected throughput s gven by U 1 (Γ th (1, z), P 1 ) whch can be calculated recursvely U (Γ th (, z), P ) = c log(1 + P γ th (,z) (γ) γ)f γ (γ) dγf z (z) dz+ p skp U +1 (Γ th ( + 1, z), P +1 ), M, (2.19) where U M+1 (Γ th (M + 1, z), P M+1 ) 0, Γ th (, z) [γ th (, z),..., γ th (M, z)] T, f z (z) θ f z b (z b = 0)+ (1 θ )f z b (z b = 1) s the PDF of the random varable z and p skp γth (,z) f 0 γ (γ) dγf z (z) dz. The frst term n (2.19) s the SU s throughput at channel averaged over all realzatons of z and that of γ γ th (, z). The second term s the average throughput when the SU skps channel due to ts low gan. Also, let the average nterference from the SU s transmtter to the PU s recever, aggregated over all M channels, be I 1 (Γ th (1, z), P 1 ). Ths represents the total nterference affectng the PU s network due to the exstence of the SU. The SU s responsble for guaranteeng that ths nterference does not exceed a threshold I avg dctated by the PU s network. I 1 (Γ th (1, z), P 1 ) can be derved usng the followng recursve formula I (Γ th (, z), P ) = (1 θ ) c P γ th (,z) (γ) f γ (γ) dγf z b (z b = 1) dz +p skp I +1 (Γ th ( + 1, z), P +1 ), M, (2.20) where I M+1 (Γ th (M + 1, z), P M+1 ) 0. Ths nterference model s based on the assumpton that the channel gan from the SU s transmtter to the PU s recever s known at the SU s transmtter. Ths s the case for recprocal channels when the PR acts as a transmtter and transmts tranng data to ts ntended prmary transmtter (when t s actng as a recever) [43]. The ST overhears ths tranng data and estmates the channel from tself to the PR. Moreover, the gan at each channel from the ST to the PR s assumed unty for presentaton smplcty. Ths could be extended easly to the case of non-unty-gan by multplyng the frst term n (2.20) by the gan from the ST to the PR at channel. Fnally, p 1 (Γ th (1, z)) s the probablty of a successful transmsson n the current tme slot and can be 24

31 calculated usng p (Γ th (, z)) = p skp f γ th (,z) γ(γ) dγf z (z) dz+ p +1 (Γ th ( + 1, z)), M, p M+1 (Γ th (M + 1, z)) 0. Gven ths background, the problem s maxmze U 1 (Γ th (1, z), P 1 ) subject to I 1 (Γ th (1, z), P 1 ) I avg p 1 (Γ th (1, z)) D 1 max varables Γ th (1, z), P 1, (2.21) (2.22) Let λ I and λ D be the Lagrange multplers assocated wth the nterference and delay constrants of problem (2.22), respectvely. Problem (2.22) s more challengng compared to the overlay case. Ths s because, unlke n (2.4), the thresholds n (2.22) are functons rather than constants. The KKT condtons for (2.22) are gven by ( P 1 (γ) = λ I Pr [b = 1 z] 1 ) +, M. (2.23) γ γ th (, z) = λ I Pr [b = 1 z] )), M, (2.24) W 0 ( exp ( (U +1 λ I I +1 λ D(1 p +1)) + c 1 n addton to the prmal feasblty, dual feasblty and the complementary slackness equatons gven n (2.8), (2.9) and (2.10), where U +1 U 1 (Γ th (1, z), P 1 (γ)), I +1 I 1 (Γ th (1, z), P 1 (γ)) and p +1 p 1 (Γ th (1, z)) whle Pr [b = 1 z] s the condtonal probablty that channel s busy gven z and s gven by Pr [b = 1 z] = (1 θ ) f z b (z b = 1). (2.25) f z (z) Note that P (γ) s ncreasng n γ and s upper bounded by the term 1/ (λ I Pr [b = 1 z]). Hence, as Pr [b = 1 z] ncreases, the SU s maxmum power becomes more lmted,.e. the maxmum power decreases. Ths s because the PU s more lkely to be occupyng channel. Thus the power transmtted from the SU should decrease n order to protect the PU. Algorthm 1 can also be used to fnd λ I. Only a sngle modfcaton s requred n the algorthm whch s that S 1 would be replaced by I 1. Thus the soluton of problem (2.22) can be summarzed on 3 steps: 25

32 1) Fx λ D R+ and fnd the correspondng optmum λ I usng the modfed verson of Algorthm 1. 2) Substtute the par (λ I, λ D ) n equatons (2.23) and (2.24) to get the power and threshold functons, then evaluate U1 from (2.19). 3) Repeat steps 1 and 2 for other values of λ D untl reachng the optmum λ D that satsfes p 1 = 1/ D max and f there are multple λ D s satsfyng p 1 = 1/ D max, then the optmum one s the one that gves the hghest U 1. Ths approach yelds the optmal soluton. Next, Theorem 2 asserts the monotoncty of I 1 n λ I whch allows usng the bsecton to fnd λ I gven some fxed λ D. Theorem 2. I 1 s decreasng n λ I [0, ) gven some fxed λ D 0. Proof. We dfferentate I1 wth respect to λ I gven that P (γ) and γth (, z) are gven by equatons (2.23) and (2.24) respectvely, then show that ths dervatve s negatve. The proof s omtted snce t follows the same lnes of Theorem 1. Although the nterference power constrant s suffcent for the problem to prevent the power functons from gong to nfnty, n some applcatons one may have an addtonal power constrant on the SUs. Hence, problem (2.22) can be modfed to ntroduce an average power constrant that s gven by S 1 (Γ th (1, z), P 1 ) P avg where S (Γ th (, z), P ) = c P γ th (,z) (γ) f γ (γ) dγf z (z) dz +p skp S +1 (Γ th ( + 1, z), P +1 ). (2.26) It can be easly shown that the soluton to the modfed problem s smlar to that presented n equatons (2.23) and (2.24) whch s ( P 1 (γ) = λ P + λ I Pr [b = 1 z] 1 ) +, (2.27) γ γ th (, z) = (λ P + λ I Pr [b = 1 z]) )), (2.28) W 0 ( exp ( (U +1 λ I I +1 λ P S +1 λ D(1 p +1)) + c 1 M where S S (Γ th (, z), P (γ)). Ths soluton s more general snce t takes nto account both the average nterference and the average power constrant besdes the delay constrant. Moreover, t 26

33 allows for the case where the power constrant s nactve whch happens f the PU has a strct average nterference constrant. In ths case the optmum soluton would result n λ P = 0 makng equatons (2.27) and (2.28) dentcal to equatons (2.23) and (2.24) respectvely. 2.5 Multple Secondary Users In ths secton, we show how our sngle SU framework can be extended to multple SUs n a multuser dversty framework wthout ncrease n the complexty of the algorthm. We wll show that when the number of SUs s hgh, wth slght modfcatons to the defntons of the throughput, power and probablty of success, the sngle SU optmzaton problem n (2.4) (or (2.22)) can capture the mult-su scenaro. Moreover, the proposed soluton for the overlay model stll works for the mult-su scenaro. Fnally, at the end of ths secton, we show that the proposed algorthm provdes a throughput-optmal and delay-optmal soluton wth even a lower complexty for fndng the thresholds compared to the sngle SU case, f the number of SUs s large. Consder a CR network wth L SUs assocated wth a centralzed secondary base staton (BS) n a downlnk overlay scenaro. Before descrbng the system model, we would lke to note that when we say that channel wll be sensed, ths means that each user wll ndependently sense channel and feedback the sensng outcome to the BS to make a global decson. Although we neglect sensng errors n ths secton, the analyss wll work smlarly n the presence of sensng errors by usng the underlay model. At the begnnng of each tme slot the L SUs sense channel 1. If t s free, each SU observes t free wth no errors and probes the nstantaneous channel gan and feeds t back to the BS. The BS compares the maxmum receved channel gan among the L receved channel gans to γ th (1). Channel 1 s assgned to the user havng the maxmum channel gan f ths maxmum gan s hgher than γ th (1), whle the remanng L 1 users contnue to sense channel 2. On the other hand f the maxmum channel gan s less than γ th (1), channel 1 s skpped and channel 2 s sensed by all L users. Unlke the case n the sngle SU scenaro where only a sngle channel s clamed per tme slot, n ths mult-su system, the BS 27

34 can allocate more than one channel n one tme slot such that each SU s not allocated more than one channel and each channel s not allocated to more than one SU. Based on ths scheme, the expected per-su throughput U L 1 s calculated from U l = θ c l γ th () θ Fl (γ th ()) log (1 + P 1, (γ)γ) f l (γ) dγ+ ( 1 1 ) U l 1 +1 l + ( 1 θ Fl (γ th ()) ) U+1 l (2.29) M and l {L + 1,..., L} wth ntalzaton UM+1 l = 0. Here f l(γ) represents the densty of the maxmum gan among l..d. users gans, whle F l (γ) s ts complementary cumulatve dstrbuton functon. We study the case where L M, thus when a channel s allocated to a user we can assume that the remanng number of users s stll L. Thus we approxmate l wth L l {L,..., L} and M. Smlar to the the throughput derved n (2.29), we could wrte the exact expressons for the per-su average power and per-su probablty of transmsson. And snce L M, we can approxmate S l wth S L and p l wth p L, l {L + 1,..., L} and M. The per-su expected throughput U1 L, the average power S L 1 and the probablty of transmsson p L 1 can be derved from U L (Γ th (), P 1, ) = θ c L S L (Γ th (), P 1, ) = θ c L γ th () [ 1 θ F L (γ th ()) L γ th () [ 1 θ F L (γ th ()) L log (1 + P 1, (γ)γ) f L (γ) dγ+ ] U L +1 (Γ th ( + 1), P +1 ) (2.30) P 1, (γ)f L (γ) dγ+ ] S L +1(Γ th ( + 1), P +1 ), (2.31) p L (Γ th ()) = θ L F L (γ th ())+ [ 1 θ F ] L (γ th ()) p L L +1(Γ th ( + 1)), (2.32) M, respectvely, wth UM+1 L = SL M+1 = pl M+1 = 0. To formulate the mult-su optmzaton problem, we replace U 1, S 1 and p 1 n (2.4) wth U L 1, S L 1 and p L 1 derved n equatons (2.30), (2.31) and (2.32), respectvely. Takng the Lagrangan and followng the same procedure as n Secton 2.2, we reach at the 28

35 soluton for P1, and γth () as gven by equatons (2.6) and (2.12) respectvely. Hence, equatons (2.6) and (2.12) represent the optmal soluton for the mult-su scenaro. The detals are omtted snce they follow those of the sngle SU case dscussed n Secton 2.2. Next we show that ths soluton s optmal wth respect to the delay as well as the throughput when L s large. We show ths by studyng the system after gnorng the delay constrant and show that the resultng soluton of ths system (whch s what we refer to as the unconstraned problem) s a delay optmal one as well. The soluton of the unconstraned problem s gven by settng λ D = 0 n (2.12) arrvng at γ th () λ D =0 = λ P ( )). (2.33) L (U+1 W 0 ( exp λ P SL +1) + c 1 M. As the number of SUs ncreases, the per-user expected throughput U L 1 decreases snce these users share the total throughput. Moreover, U L decreases as well M decreasng the value of γ th () (from equaton (2.33) untl reachng ts mnmum (.e. γth () = λ P ) (the rght-hand-sde of (2.33) s mnmum when ts denomnator s as much negatve as possble. That s, when W 0 (x) = 1 snce W 0 (x) 1, x R) as L. From (2.32), t can be easly shown that p L 1 (Γ th (1)) s monotoncally decreasng n γ th () M. Thus the mnmum possble average delay (correspondng to the maxmum p L 1 (Γ th (1))) occurs when γ th () s at ts mnmum possble value for all M. Consequently, havng γ th () = λ P means that the system s at the optmum delay pont. That s, the unconstraned problem cannot acheve any smaller delay wth an addtonal delay constrant. Hence, the mult-su problem, that s formulated by addng a delay constrant to the unconstraned problem, acheves the optmum delay performance when L s asymptotcally large. Recall that the overall complexty of soluton for the sngle SU case s due to three factors: 1) evaluatng the Lambert W functon n Algorthm 1, 2) the bsecton algorthm n Algorthm 1 and 3) the search over λ D. On the other hand, the complexty of soluton for the mult-su case decreases asymptotcally (as L ). Ths s because of two reasons: 1) When L M, γth () λ P M. 29

36 Whch means that we wll not have to evaluate the Lambert W functon n (2.12) but nstead we set γ th () = λ P, snce L M. 2) When γ th () = λ P there wll be no need to fnd λ D snce the delay s mnmum (we recall that n the sngle SU case, we need to calculate λ D to substtute t n (2.12) to evaluate γ th (), but n the mult-su case γ th () = λ P ). 2.6 Numercal Results We show the performance of the proposed soluton for the overlay and underlay scenaros. The slot duraton s taken to be unty (.e. all tme measurements are taken relatve to the tme slot duraton), whle τ = 0.05T. Here, we use M = 10 channels that suffer..d. Raylegh fadng. The avalablty probablty s taken as θ = 0.05 throughout the smulatons. The power gan γ s exponentally dstrbuted as f γ (γ) = exp (γ/ γ) / γ where γ s the average channel gan and s set to be 1 unless otherwse specfed. Fg. 2.2 plots the expected throughput U1 for the overlay scenaro after solvng problem (2.4). U1 s plotted usng equaton (2.2) that represents the average number of bts transmtted dvded by the average tme requred to transmt those bts, takng nto account the tme wasted due to the blocked tme slots. We plot U1 wth D max = 1.02T and wth D max = (.e. neglectng the delay constrant). We refer to the former problem as constraned problem, whle to the latter as unconstraned problem. We also compare the performance to the non optmum stoppng rule case (No-OSR) where the SU transmts at the frst avalable channel. We expect the No-OSR case to have the best delay and the worst throughput performances. We can see that the unconstraned problem has the best throughput amongst all constraned problems. Examnng the constraned problem for dfferent sensng orders of the channels, we observe that when the channels are sorted n an ascendng order of θ, the throughput s hgher. Ths s because a channel has a hgher chance of beng skpped f put at the begnnng of the order compared to the case f put at the end of the order. Ths s a property of the problem no matter how the channels are 30

37 ordered,.e. ths property holds even f all channels have equal values of θ. Hence, t s more favorable to put the hgh qualty channels at the end of the sensng order so that they are not put n a poston of beng frequently skpped. However, ths s not necessarly optmum order, whch s out of the scope of ths work and s left as a future work for ths delay-constraned optmzaton problem. We also plot the expected throughput of a smple stoppng rule that we call K-out-of-M scheme, where we choose the hghest K channels n avalablty probablty and gnore the remanng channels as f they do not exst n the system. The SU senses those K channels sequentally, probes the gan of each free channel, f any, and transmts on the channel wth the hghest gan. Ths scheme has a constant fracton Kτ/T of tme wasted each slot. Yet t has the advantage of choosng the best channel among multple avalable ones. In Fg. 2.2 we can see that the degradaton of the throughput when K = 5 compared to the optmal stoppng rule scheme. The reason s two-fold: 1) Due to the constant wasted fracton of tme, and 2) Ignorng the remanng channels that could potentally be free wth a hgh gan f they were consdered as opposed to the constraned problem. The delay s shown n Fg. 2.3 for both the constraned and the unconstraned problems. We see that the unconstraned problem suffers around 6% ncrease n the delay, at P avg = 10, compared to the constraned one. Studyng the system performance under low average channel gan s essental. A low average channel gan represents a SU s channel beng n a permanent deep fade or f there s a relatvely hgh nterference level at the secondary recever. Fg. 2.4 shows γth () versus the γ. At low γ, the throughput s expected to be small, hence γ th () s close to ts mnmum value λ P so that even f γ s relatvely small, should not be skpped. In other words, at low average channel gan, the expected throughput s small, thus a relatvely low nstantaneous gan wll be satsfactory for stoppng at channel. Whle when the average channel gan ncreases, γth () should ncrease so that only hgh nstantaneous gans should lead to stoppng at channel. In both cases, hgh and low γ there stll s a trade-off between choosng a hgh versus a low value of γ th (). 31

38 5 4.5 Expected Throughput (Nats/channel use) Unconstraned Random Constraned Ascendng Constraned Random Constraned Descendng No OSR K-out-of-M Average Power (P avg ) Fgure 2.2: The expected throughput for the overlay scenaro for four cases: 1) Proposed constraned problem: wth average delay constrant for three channel orderng possbltes (ascendng orderng of channel avalablty probabltes, descendng orderng, and random orderng), 2) Unconstraned problem that gnores the delay constrant, 3) No optmum stoppng rule (No-OSR) where the SU transmts at the frst free channel and 4) K-out-of-M scheme where the SU assumes the system has only K = 5 channels and gnores the remanng M K channels. 32

39 Unconstraned Random Constraned Descend Constraned Random Constraned Ascend No OSR Expected Delay (# Tme Slots) Average Power (P avg ) Fgure 2.3: The expected delay for the overlay scenaro for problem (2.4). The unconstraned problem can suffer arbtrary hgh delay. The constraned problem has a guaranteed average delay for all orderng strateges. The No-OSR scenaro, on the other hand, has the best delay performance snce the SU uses the frst free channel. 33

40 Optmum Threshold value γ th () =1 =2 =3 =4 =5 =6 =7 =8 =9 =10 λ P Average Channel Gan ( γ) Fgure 2.4: The gap between the optmum threshold γth () and ts mnmum value λ P ncreases as the average gan ncreases. Ths s because as γ ncreases, U +1 ncreases as well. Hence γth () ncreases so that only suffcently hgh nstantaneous gans should lead to stoppng at channel. 34

41 The sensng channel (.e. the channel between the PT and ST over whch the ST overhears the PT actvty) s modeled as AWGN wth unt varance. The dstrbutons of the energy detector output z (average energy of N samples sampled from ths sensng channel) under the free and busy hypotheses are the Ch-square and a Noncentral Ch-square dstrbutons gven by ( N f z b (z b = 0) = σ 2 ) N z N 1 (N 1)! exp f z b (z b = 1) = ( ) N ( z ) N 1 ( 2 N (z + E) exp σ 2 E σ 2 ( Nz σ 2 ) ( IN 1 Bes ), (2.34) 2N ) Ez, (2.35) σ 2 where σ 2, whch s set to 1, s the varance of the Gaussan nose of the energy detector, E s the amount of energy receved by the ST due to the actvty of the PT and s taken as E = 2σ 2 throughout the smulatons, whle I Bes (x) s the modfed Bessel functon of the frst knd and th order, and N = 10. The man problem we are addressng n ths chapter s the optmal stoppng rule that dctates for the SU when to stop sensng and start transmttng. As we have seen, ths s dentfed by the threshold vector Γ th (1, z). If the SU does not consder the optmal stoppng rule problem and rather transmts as soon as t detects a free channel, then t wll be wastng future opportuntes of possbly hgher throughput. Hence, we expect a degradaton n the throughput. We plot the two scenaros n Fg. 2.5 for the underlay system wth no delay constrant. Throughout ths chapter, we use bold fonts for vectors and astersk to denote that x s the optmal value of x; all logarthms are natural, whle the expected value operator s denoted E[ ] and s taken wth respect to all the random varables n ts argument. Fnally, we use (x) + max(x, 0) and R to denote the set of the real numbers. For the multple SU scenaro, the numercal analyss were run for the case of L = 30 SUs whle M = 10 channels. We assumed the fadng channels are..d. among users and among frequency channels. Each channel s exponentally dstrbuted wth unty average channel gan. And snce L s large, the dstrbuton of the maxmum gan among L random gans converges n dstrbuton to 35

42 Underlay Expected Throughput (Nats/channel use) Wth Optmal Stoppng Rule Wthout Optmal Stoppng Rule PU s Interference Threshold I avg Fgure 2.5: The underlay expected throughput versus the average nterference threshold I avg. Two scenaros are shown: wth and wthout the optmal stoppng rule formulaton. In the latter, the SU transmts as soon as a channel s found free. 36

43 the Gumbel dstrbuton [44] havng a cumulatve dstrbuton functon of exp ( exp ( γ/ γ)). The per-user throughput U L 1 s plotted n Fg. 2.6 where the throughput of the delay-constraned and of the unconstraned optmzaton problems concde. Ths s because when L M, the soluton of the unconstraned problem s delay optmal as well. Hence, addng a delay constrant does not sacrfce the throughput, when L s large. Moreover, the delay performance shown n Fg. 2.7 shows that the delay does not change wth and wthout consderng the average delay constrant snce the system s delay- (and throughput-) optmal already. We have smulated the system for the onlne algorthm of Secton 2.3 for K(1) = 2 packets and t f = 4 tme slots. We smulated the system at r mn = 0.95 whch means that the QoS of the SU requres that at least 95% of the frames to be successfully transmtted. Fg. 2.8 shows the mprovement n the throughput of the onlne over the offlne adaptaton. Ths s because the SU adapts the power and thresholds at each tme slot to allocate the remanng resources (.e. remanng tme slots) accordng to the remanng number of packets and the desred QoS. Ths comes at the expense of re-solvng the problem at each tme slot (.e. t f tmes more). 37

44 Expected Throughput per SU(Nats/channel use) Constraned Unconstraned Average Power (P avg ) Fgure 2.6: Per user throughput of the system at L = 30 SUs. The throughput of the constraned and unconstraned problem concde snce the system s throughput (and delay) optmal. 38

45 Expected Delay per SU(Tme Slots) Constraned Problem Unconstraned Problem Average Power (P avg ) Fgure 2.7: The average delay seen by each user n the system at L = 30 SUs. The delay of the constraned and unconstraned problems concde snce the system s delay (and throughput) optmal. 39

46 Expected Throughput (Nats/channel use) Onlne Adaptaton Offlne Adaptaton Average Power (P avg ) Fgure 2.8: The performance of the onlne adaptaton algorthm for the general delay case. 40

47 CHAPTER 3 Delay Due to Queue-Watng Tme In ths chapter we study the delay resultng from the servce tme as well as that from queue-watng tme. The servce tme s affected by the power transmtted by the SU, whle the queue-watng tme s affected by the schedulng algorthm. We propose a delay-optmal schedulng-and-power-allocaton algorthm that guarantees bounds on the SUs delays whle causng an acceptable nterference to the PUs. Ths algorthm s useful to provde far delay guarantees to the SUs when delay farness cannot be acheved due to the heterogenety n SUs channel statstcs. 3.1 Network Model We assume a CR system consstng of a sngle secondary base staton (BS) servng N secondary users (SUs) ndexed by the set N {1, N} (Fg. 3.1). We are consderng the uplnk phase where each SU has ts own queue buffer for packets that need to be sent to the BS. The SUs share a sngle frequency channel wth a sngle PU that has lcensed access to ths channel. The CR system operates n an underlay fashon where the PU s usng the channel contnuously at all tmes. SUs are allowed to transmt as long as they do not cause harmful nterference to the PU. In ths work, we consder two dfferent scenaros where the nterference can be consdered as harmful. The frst s an nstantaneous nterference constrant where the nterference receved by the PU at any gven slot should not exceed a prespecfed threshold I nst, whle the second s an average nterference constrant where the nterference receved by the PU averaged over a large duraton of tme should not exceed a prespecfed threshold 41

48 The mage part wth relatonshp ID rid2 was not found n the fle. The mage part wth relatonshp ID rid2 was not found n the fle. SU1 PU BS SU2 Fgure 3.1: The CR system consdered s an uplnk one wth N SUs (n ths fgure N = 2) communcatng wth ther BS. There exsts an nterference lnk between each SU and the exstng PU. The PU s assumed to be usng the channel contnuously.. tme I avg. Moreover, n order for the secondary BS to be able to decode the receved sgnal, no more than one SU at a tme slot s to be assgned the channel for transmsson Channel and Interference Model We assume a tme slotted structure where each slot s of duraton T seconds, and equal to the coherence tme of the channel. The channel between SU and the BS s block fadng wth nstantaneous power gan γ (t), at tme slot t, followng the probablty mass functon f γ (γ) wth mean γ and..d. across tme slots, and γ max s the maxmum gan γ (t) could take. The channel gan s also ndependent across SUs but not necessary dentcally dstrbuted allowng heterogenety among users. SUs use a rate adaptaton scheme based on the channel gan γ (t). The transmsson rate of SU at tme slot t s R (t) ( = log 1 + P (t) γ (t) ) bts, (3.1) 42