Delay-Bounded Packet Scheduling of Bursty Traffic over Wireless Channels

Transcription

1 IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 Delay-Bounded Packet cheduling o Bursty Traic over Wireless Channels Dinesh Rajan, Ashutosh abharwal and Behnaam Aazhang Abstract In this paper, we study minimal power transmission o bursty sources over wireless channels with constraints on mean queuing delay. The power minimizing schedulers adapt power and rate o transmission based on the queue and channel state. We show that packet scheduling based on queue state can be used to trade queuing delay with transmission power, even on additive white Gaussian noise channels. Our extensive simulations show that small increases in average delay can lead to substantial savings in transmission power, thereby providing another avenue or mobile devices to save on battery power. We propose a lowcomplexity scheduler that has near optimal perormance. We also construct a variable rate QAM based transmission scheme to show the beneits o the proposed ormulation in a practical communication system. Power optimal schedulers with absolute packet delay constraints are also studied and their perormance is evaluated via simulations. Index Terms Packet scheduling, power control, queuing delay, traic regulation, wireless channels. I. INTRODUCTION The current and uture wireless systems will support a multitude o services with a wide range o delay, rate and reliability requirements. The task o delivering wide variety o services is complicated by the hostile time-varying nature o wireless channels. The limited battery resources at the mobile devices add yet another dimension in the challenge o reliable content delivery. Thus, it is imperative that methods or wireless transmission should be designed to achieve target delay and throughput with minimal power consumption. Recognizing that transmission power is one o the major battery consumers in a mobile device, in this paper, we study the tradeo between transmission power and communication delay in wireless channels. Delaying communication based on channel conditions to save transmission power is commonly used in wireless systems. The transmission scheme which maximizes long-term throughput transmits more power and inormation in good channel states, and less in poor conditions []. Thus, it delays some parts o input traic to wait or good channel states, buying more utility or the available power resources to achieve the maximal long-term throughput. The inormation theoretic concept o power and rate control is relected in the INFOTATION [2] based architecture, where mobile nodes Dinesh Rajan is with the Department o Electrical Engineering, outhern Methodist University, Dallas, TX. Ashutosh abharwal and Behnaam Aazhang are with the Department o Electrical and Computer Engineering, Rice University, Houston, TX. This work was supported in part by Nokia Corporation, and by NF under grant 339. In a typical TDMA phone, approximately 6% o the battery consumption can be attributed to transmission RF ampliier. transmit data only when they are close to base-stations. By waiting or extremely good conditions, more or less guaranteed by the proximity to base-stations, mobile devices ensure that transmission power is most eiciently used. But the wait to reach close to base-stations (oten sparsely placed) can result in large transmission delays. In this paper, we address the problem o scheduling transmission o packet data over a time-slotted single-user wireless link. We incorporate queuing delay 2 as one o our main design constraint. With a bound on mean queuing delay, our objective is to minimize the average transmission power. The inclusion o explicit delay constraint is motivated by the need to support applications with dierent delay-sensitivities. The solution to the proposed ormulation leads to methods which perorm power and rate control, even in additive white Gaussian channels (AWGN), i the traic is bursty. Note that, much like [], rate and power adaptation is done based on the Gaussian channel coding theorem [3]; the relation between instantaneous rate and power is The ollowing example clariies how the convex relation between power and rate leads to reduced power requirements as the delay increases. Consider the transmission o an On-O source over an AWGN channel with noise variance. The On-O source produces packets at a constant rate, packets/timeslot, in the On state and no packets in the O state. Let the source arrivals be independent and identically distributed across time. First, consider a scheme in which all packets in the system are transmitted as soon as they arrive, the average power required to guarantee error ree reception " # %& () %& * equals (assuming that packets require transmission at bits/sec/hz). Now, consider an alternate transmission scheme % which schedules packets as % ollows. When there are or more packets available, o those packets are transmitted (assuming is an even integer) and the remaining packets are % buered or uture transmission. I the queue has less than packets, all packets are buered. In the second transmission scheme, the average power required,+ is no less than, which is. lower than due to convexity o exponential unction. The power savings in the second scheme can be attributed to increased packet - 2 There are many components that aect the total transmission delay. In this paper we only consider queuing delays. The delay due to encoding, propagation and decoding are assumed to be small and nearly constant, and hence neglected.

2 2 IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 delay in the system. The above simple example illustrates the average power savings due to power and rate control combined with queuing based communication delay. In the sequel, we ormalize the joint power and rate control schemes, which we generically label as schedulers, and derive minimal power methods with delay bounds. The power gain rom additional communication delay was independently recognized in [4], and lazy packet scheduling was proposed to reduce average power consumption. Though related in essence, our ormulation diers considerably rom the work in [4]. In [4], optimal o-line and near optimal online schedulers were proposed using both inite and ininite horizon optimization. An indirect bound on packet delay was imposed by requiring the schedulers transmit all packets arriving in time interval / 3254 to depart no later than 2. Thus, the scheduler lushes the queue every 2 seconds. The emphasis in [4] was to demonstrate that additional delay can help save power, hence no speciic constraints on target delay were imposed. chedulers satisying explicitly imposed packet delay bounds were derived in [5], where the authors ound minimal power transmission schemes or a two-state Gilbert-Elliot channels. Though the work in [5] is closest to our ormulation, the authors used a more optimistic relation between power and rate - the power was assumed to depend linearly on the rate. As will become evident rom the subsequent development in this paper, the gain due to additional delay in [5] is only due to channel time-variation, and not the source burstiness. In [6], the authors propose dynamic power control policies in time-varying channels under energy and three dierent delay constraints; the optimal policies are based on thresholding received signal strength and residual battery energy. Again, the work in [6] notes the impact o increasing delay on energy eiciency but it diers considerably rom our ormulation o the delay-constrained transmission. Power savings in ading channels with additional delay has also been investigated in [7], []. The main contributions in this paper are as ollows: 6 For AWGN channels, we completely characterize the achievable delay-power region or the set o randomized stationary schedulers which make their transmission decisions based on queue state. A randomized scheduler can take one o several possible actions or a given system state with inite probability, and generalizes deterministic schedulers which take the same action or a given system state. As the irst important result, we show that or delaypower characterization, deterministic schedulers orm a basis o the set o randomized schedulers. Throughout this paper, the analysis is perormed or inite buer size systems with no permissible buer overlow. As another application o the proposed ormulation, we derive optimal delay-bounded schedulers or transmission o constant rate traic over inite-state ading channels. Depending on the delay constraint, the scheduling decisions exhibit a mix o time water-illing [] and outage minimizing power control [9]. I the delay bound is small, the scheduler transmits packets in poor channel conditions like the outage power control while the scheduler uses only good channel states or large delays. 6 The optimal power minimizing scheduler requires a dynamic programming based optimization, which is computationally cumbersome or large systems (sources with large bursts or systems with large buer sizes). Based on the empirically observed properties o the optimal schedulers, we propose a single-parameter scheduler, labeled log-linear scheduler, with near optimal perormance. The empirically observed near-optimal perormance o the log-linear scheduler is used to derive an approximate relation between mean queuing delay and average transmit power. The approximate relation is ound to accurately predict the perormance o optimal schedulers or moderate to high delays. 6 We show that similar results can be obtained i the mean queuing delay bound is replaced by an absolute delay bound such that no packet suers more than a prespeciied maximum delay. 6 To demonstrate the practical application o the proposed scheduling, we construct a convolutionally coded variable rate QAM system, and show that small additional delay can lead to substantial power savings. In variable rate QAM systems where the transmission rate and power is controlled by the transmitter, the receiver is unaware o the transmitted rate. We show that the transmission rate inormation (required or appropriate decoding) can be derived without requiring any protocol inormation. All results in this paper assume independent and identically distributed packet arrivals in each slot or simplicity. The analysis can be extended to Markov arrival processes which are more appropriate models or multimedia applications like video transmission. The main results in the paper can be attributed to inclusion o inite delay constraints along with a bursty traic model. I delay is not a consideration, adequate queuing delay can completely remove all burstiness and standard inormation theoretic analysis applies []. The importance o incorporating traic model in the design o wireless communication systems has been well recognized; or insightul reviews, see [], [2]. The eect o queuing delay on probability o error is studied in [3] using a simpliied Gaussian multi-access channel model. In this paper, we study only single user systems and schedule packets over time. cheduling is more commonly reerred to selecting packets when there are multiple input lows [4]; e.g., irst come irst serve (FCF), last come irst serve (LCF), earliest deadline irst (EDF) and weighted air queuing (WFQ). There is extensive work on scheduling over wireless channels [5] [9]. Again, the main idea is to eiciently use system resources, oten with an aim o air division o resources. One o the interesting results in this paper is that eicient scheduling o packets helps even in a single user single low system. In an actual wireless network, the proposed packet scheduler can be combined with a multi-low service discipline, like WFQ, to achieve airness with eicient power usage. The single user problem provides methods or traic regulation in circuit-switched TDMA systems, and the multiuser problem orms the basis or scheduling in CDMA systems. In our related work [2], we ocus on the multi-user

3 R = : ; RAJAN ET.AL.: DELAY-BOUNDED PACKET CHEDULING OF BURTY TRAFFIC OVER WIRELE CHANNEL 3 problem. 7 In this paper, we ocus on schedulers which do not drop or loose packets, and seek to minimize their average power utilization. I the minimal power scheduler requires more power than available, then the transmitter is orced to drop packets. A power-limited outage based ormulation is studied in [2] [23], where we show that additional delay helps reduce probability o outage. The rest o the paper is organized as ollows. In ection II we introduce the system model and set up the scheduling problem o interest. We provide a technique to ind power eicient schedulers under average delay constraints and discuss some o their properties in ection III. cheduling under absolute delay constraints is addressed in ection IV. We demonstrate scheduling gains in a simple practical system in ection V. We conclude in ection VII. II. PRELIMINARIE AND PROBLEM FORMULATION In this section, we introduce the time-slotted system model used in the paper. We deine the queue, source and channel model in ection II-A. Our emphasis in this paper is on design o schedulers which guarantee inormation theoretic reliability, discussed in ection II-B. Finally, a class o reliable schedulers, labeled zero outage schedulers is presented and characterized in ection II-C. A. ystem Model We consider a single user time-slotted system with an input buer, shown in Figure ; the period o each time-slot is assumed to be 29 seconds. The source produces packets at an average rate o : packets/time-slot. In time-slot ;, the source produces <>= packets, each o size? bits, where <>= has a <>= with a inite support, / BC A BDE4, i.e., the largest number o packets that arrive in a time-slot is D. The arrival process F*<G=,H is assumed to be independent and identically distributed rom one time-slot to another. The arriving packets are queued in the input buer, which can store a maximum o I packets. The number o queued packets in the buer at the beginning o the ;KJML time-slot are denoted by N =. The scheduler chooses O = packets or transmission at the beginning o the time-slot, and uses power = or transmission. ince the length o the time-slot is ixed, the rate o transmission is varied based on the selected number o packets, O =. Denoting the transmitted signal by PQ=, the received signal is given by E = PQ= UT = () where T = is the complex circularly symmetric additive white Gaussian noise with zero mean and variance. The channel gain = is assumed to be constant over the period o a time-slot, i.e., the coherence interval o the channel is the same as the length o the time-slot. Two channel models are considered: In the irst model, =WV, thus the channel model () reduces to a Gaussian channel. In the second model, = is assumed to be a inite state Markov channel, i.e., = orms a Markov chain. Both the transmitter and receiver are assumed to have perect knowledge o the channel gain =. Fig.. chematic o ystem model The transmitted signal P = is a unction o the number o packets transmitted O =, the channel coding, modulation and waveorm used. We use inormation theoretically optimal coding and modulation to obtain P =, which serves two purposes. First, we can get closed orm relationship between O = and power = needed or reliable transmission. econdly, the power we obtain serves as a universal lower bound on power required by any scheduler designed or a speciic coding and modulation scheme. At the transmitter, the buer update is given by N =GX ZY\[^] N = < = O = I The average packet delay is related to the average buer length via Little s theorem [24] as ollows, g ` abdce _ (2) F"N=,H (3) where : F>< = H is the average packet arrival rate. We assume that all packets that arrive in time-slot ; can be h* transmitted only in time-slot or later. A natural constraint on O = is that ji O = i Nk=, i.e., we cannot transmit more packets than available in the queue. The smallest average delay in the system is achieved when all buered packets are transmitted in each time-slot, i.e., O = Nk=, which implies that ` abdcl m FdN=H n, assuming that I iod. In other words, the buer is lushed at all time instants, and all packets are transmitted in the slot immediately ater which they arrive. A scheduler is a mapping rom the current buer state N= and channel state = to the number o packets transmitted O = and transmit power = (ee ection II-C or more details.). We will restrict our attention to a class o schedulers which do not drop any packet and achieve a predetermined minimum reliability or every transmitted packet; a scheduler which does not lose any packets is labeled as a zero outage scheduler. The objective is to characterize the achievable delay-power region which is deined as ollows. ` Deinition (Delay-power region): A point on the average delay versus average transmit power plane is achievable i there exists a zero outage ` scheduler which achieves an average delay no greater than and average power no greater than. The delay-power region can be characterized by inding the minimal average transmit power required or any delay, over the class o all randomized zero outage schedulers p (ormally

4 ? = = = - Ÿ š œ ½ Æ = Æ» I I ˆ = = ½ 4 IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 deined in ection II-C). Formally, lq `sr t Yu[^] v wyx3z {~},w [^Y = F*,=,H (4) ` r where is the average packet delay bound and q ` r* ` r no greater than. Given the solution q ` r* ` r such that ƒ q `sr is the minimum average power required to achieve average delays, all points belong to the delay-power region. the average transmit power d - Ž kx ªŽ «is. The minimum distance between signal points is given by or all, hence these constellations achieve approximately the same bit error rate. Thus, increasing the total inormation rate by a actor o will require an average transmit power o s 5 «. Equivalently, increasing the average ^ * power by a actor o gains us an additional bit rate o bits/symbol. Thus, the uncoded systems also ollow the exponential relationship between transmission rate and transmit power. B. Power Control In this paper, we restrict our attention to schedulers which guarantee a certain level o reception reliability. Our notion o reliability is motivated by the concept o outage proposed in [25], where a packet is reliably received i the instantaneous mutual inormation is higher than the required rate. To invoke mutual inormation based reliability, we require that 2 is suiciently long to transmit D packets close to channel capacity. I the scheduler chooses to transmit O = packets o length? bits in a time-slot o length 2 seconds, then the required transmission rate is given by ˆŠ C C Œ~ - Ž nats/second. Using the results in [3], we deine the capacity unction as = ( 29? ^ * = = [ Y * d *šc œž B ^ (5) where, or simplicity, we have assumed that the system bandwidth is one Hertz. For simplicity, we also assume 29. The quantity is the hannon capacity o a Gaussian channel with noise variance - and an average power o =, and represents the maximum reliably achievable rate. Using (5), the appropriate power control to ensure reliable reception o packets ollows immediately. To transmit O = packets, the scheduler selects the power = such that O =,=, leading to the ollowing power control, = O = ( = *Š Note that,= is a strictly convex unction o O =, a linear increase in the number o packets requires an exponential increase in required transmit power. The convex relation between required transmit power and rate is also evident in the non-asymptotic regime, either by considering error exponents [26] or inite length codes. The error exponent or Gaussian channels varies logarithmically in power and linearly in rate. Thus, to guarantee the same error perormance or all scheduler decisions O =, the required power control will be similar to (6). As an example o practical methods, consider a simple uncoded system using -ary QAM modulation. For large ( ƒ ) rectangular QAM constellations, the average power required to achieve a certain minimum distance between constellation points is only marginally greater than the average power required to achieve the same minimum distance using the best known QAM [27]. The constellation points are given ~M by two-tuples 3 M 3 _, or CB ^ *. The transmission rate is bits/symbol and (6) C. Zero Outage chedulers In its general orm, we deine a randomized scheduler as a memoryless probabilistic mapping rom the buer state N = and channel ading state = to the number o packets O = and transmission power =, i.e., ²± N = &³ O = = ª. Thus, a scheduler is characterized by probabilities, Prob O = = N=. Based on the discussion in ection II-B, the power,= is a unction o O = which reduces the mapping to a single dimensional variable, µ± N= ³ O =. For the special case o AWGN channel ( =V ), is urther reduced to ± N= ³ O =. For the sake o clarity, all results in this section are derived or the case o AWGN channels and can be easily extended to inite state ading channels. With some abuse o notation, we use ¹ªº» to denote the probability o transmitting ¼ packets given that the queue has packets, i.e.,,¹"º» Prob O = ¼ Nk= ½ scheduler can be represented by an I ¾ C À packets than available in the buer, thus,¹"º» or ¼jÁ with entries ¹ªº». ince,¹"º» are probabilities,,¹"º» ng ¹ÃÂ. Thus, every upper triangular matrix (since any scheduler cannot transmit more ), The set o valid schedulers will be denoted by Ä, where each scheduler Å Ä satisies the above properties. A scheduler is labeled deterministic i ¹ªº»uÅ F H, i.e., the scheduler takes the same action or each queue state at all times. It ollows immediately rom the deinition that the deterministic schedulers orm a subset o randomized schedulers. Later in this section, we will show that the deterministic schedulers are a basis o the set o randomized schedulers. ince the source is assumed to be i.i.d. and the schedulers are assumed to be memoryless, all schedulers lead to the stationarity o N=. Due to the memoryless nature o the schedulers the queue state orms a irst order Markov chain, which we assume to be aperiodic. Thus we have a concise representation o the queue state process as ollows. Denote by Æ / Ç ¹"º» 4 the matrix o transition probabilities, where Ç ¹"º» is the probability o transition rom buer state N= ½ to buer state N = X ¼ Æ. Note that is o size I È*A È* and is not a unction o time index ; due to stationarity ½ o schedulers. The stationary probability o buer state is denoted by É» where, É» Prob /N = ½ 4. Let /AÉ"eÉ É ^ ÉË 4 denote the vector o stationary probabilities. From the deinition o and, it ollows that (7) ()

5 D Ì p Ë : :» Ë ½ I I å ã F i D ã Ü RAJAN ET.AL.: DELAY-BOUNDED PACKET CHEDULING OF BURTY TRAFFIC OVER WIRELE CHANNEL 5 Note that Æ depends on the choice o, which implies is a unction o. The average packet delay o any scheduler can be expressed as `sabbdc Í /Nk= 4 imilarly, the average transmission power abdc ͻΠ/ = 4 ¹Ã»  ɻ É»Ï ¹ªº» = abdc ¼ * is given by (9) () where = is the power control deined in (6). For a given arrival process, more than one scheduler can have the same average delay and power; the ollowing deinition o equivalence class ormalizes this. Deinition 2 (Delay-power equivalence class): Two schedulers are said to belong to the equivalence class ÐÒÑKÓ ºÔ Ó i they have the same average delay and average power. Formally, Ð Ñ Ó ºÔ Ó F ±Õ ÖÅ Ä `sabbdc E` abdc t H Two schedulers, Å Ð Ñ Ó ºÔ Ó are then said to be equivalent, and represented by V. The importance o the above equivalence class deinition will be become apparent in the next section. The ollowing example illustrates that the equivalence classes o the set o all schedulers can have more than one element. Example : The ollowing two schedulers are equivalent or. a Ø Ø Ø Ø Ø Ø In this case, the two schedulers only dier in buer state 3 that occurs with zero probability. In the rest o the paper, we will restrict our attention to a special class o schedulers which ensure a preset level o reliability, ormally deined as ollows. Deinition 3 (Zero Outage cheduler): A scheduler is said to be zero outage i packets are not dropped at the transmitter, there are no buer overlows and power = is chosen to ensure reliable reception o the packets. The set o all zero outage schedulers is denoted by p, i.e., F ±_ ÀÅ Ä is zero outage H () The ollowing proposition completely characterizes the set o zero-outage schedulers, p. Proposition 4: Consider a queue with inite buer size I and an input process with no more than D packet arrivals in one slot. Assume that the buer is initially empty. A stationary stochastic scheduler is zero outage i and only i or each state N = ½ ½ Å I are satisied. ^ DE I one o the ollowing two conditions ½ ) The stationary probability is zero or state, i.e., É». 2) I Ú Y\[ ] Fd¼ ±_ ¹ªº»#Û H denotes the minimum number o packets transmitted ½ ½ in state under the given scheduler, then Ú I. Proo: ee Appendix A Now, we note that the set o deterministic schedulers spans the set o all randomized schedulers, Ä, and hence a basis or Ä. The basis elements can be constructed column by column as ½ ollows. Consider the JML column o ½ \*. The J element to * J element ½ o the JML ½ column are zero. Hence vectors o dimension I * ½ orm the basis or the JML column. The basis vectors are CB, BCÝ CB,, BC ½ CB where in the last basis, the is in the JML position. Taking all possible combinations o the basis o all I n* columns, we get a basis or the set o all schedulers. An important property o all schedulers in Ä which allows us to do this basis expansion is (7). Finally, note that the total number o deterministic policies is I Z* Þ and all o them are not zero outage. The set o deterministic schedulers orm a redundant basis or the set o all randomized schedulers, Ä, and hence any scheduler can possibly have multiple basis expansions. Below, we give a constructive method or inding the coeicients o the basis expansion, in terms o deterministic schedulers. This constructive method is used in the proo o Theorem. Let ßÅ Ä be a given randomized scheduler o size á I n*à, and the ollowing steps recursively generate the basis expansion o. ½ ) et and â» H, where the ¼ JML member o o â» äã is a two-tuple» ¹ Ïå»^¹ such that the irst element o the tuple is the coeicient o the expansion and the second element o the tuple is the scheduler in the expansion. Thus, the expansion in the irst step is ã ª å ~ n. The ollowing construction updates the number and the elements in the set â» in each step, till all schedulers ½ in the expansion are deterministic. 2) In step, we consider each element o â» separately ½ * and generate elements or â» X. Thus â» ½ * X â». 3) For each å» ¹ ½ e*, write it as the convex combination o matrices ½ æ that dier only in the JML column. The ½ ¾ JML column o scheduler, å Ž» X ¹3ç where ¼> ¼ ½ *tn BC ¼ ½ *, has value in row ¼> ¼ ½ ¾*~ and zero in all other» ½ ¾ rows, i.e., the dierent columns orm a basis or X. The schedulers å Ž» X ¹3ç are ½ é identical to å» ¹ or all columns other than column. The coeicients are given by Ž» X ¹ ç ã» ¹»^¹ªº Ž ¹ ç «9Ž ¹ «~Ž» X ê ~Ž» X, where å»^¹ªº ëì is the element in the í JML row and îäjml column in å»^¹. ince scheduler satisies (7), the weighted sum o the schedulers å Ž» X ¹ ç give the scheduler å»^¹ and all schedulers å Ž» X ½&ïW½ ¹ ç satisies j (7). ½ ð 4) et and repeat the above steps 2-4 i I j. 5) The elements in â Ë X give the basis expansion o Ž Ë X ~ñ ¹ÃÂ Ë ¹ å Ë ¹.

6 ½ ø ý D õ õ ½ õ i Ü Ü i 6 IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 The basis expansion technique is best illustrated in the ollowing simple example. Example 2 : ä ò ó ä ó ô ô Ø M Ø Ø ô ó ô ô ô ò ò ä ô ó ô ó M Ø Ø We end this section with two propositions. In the next two propositions we consider a queue with inite buer size I and an input process with no more than D packet arrivals in one slot. Further deine the set o deterministic schedulers, õí F ±_ ÀÅ Ä ¹"º» Å F HAö ¼ BB I H and recall that the set o zero outage schedulers is denoted by p. The ollowing lemma shows that or every zero outage deterministic scheduler ÀÅ Ð Ñ Ó ºÔ Ó which satisies Condition o Proposition 4, there exists another deterministic scheduler ÖÅ ÐÒÑKÓ º Ô Ó which satisies Condition 2 o Proposition 4. Proposition 5 (Equivalent elements o ÐÒÑKÓ ºÔ Ó ): Let ùå ÐÒÑKÓ ºÔ ÓQú be a deterministic ½ scheduler such that there exists a buer state such ø that É». Then there exists a deterministic scheduler Å Ð Ñ Ó ºÔ Ó ú ½ such that Ú I where Ú ûy\[^] Fd¼ ± ¹ªº» ø Û H Proo: ee Appendix B. In the next proposition, we show that in the basis expansion o zero outage schedulers, only zero outage deterministic scheduler are needed and the coeicients corresponding to the non-zero outage schedulers are zero. Proposition 6 (Basis expansion o elements in p ): Let F"ü» HCý»Î represent the basis set o Ä, where ü»lå õ or all and þ õ is the cardinality o the set. For any stationary randomized scheduler ÿ Å p, let its expansion in terms o basis vectors be ÿ ý ã»»  ü» ã where Òi», and ã»»  ã. Then,» or all ü» Å õ p õ, where p is the set dierence. Proo: ee Appendix C. The ollowing example illustrates the results o propositions 5 and 6. Example 3 : Consider the ollowing our schedulers (scheduler a and are same as in Example, and reproduced here or the convenience o the reader): Ø Ø ae Ø Ø Ø Ø It is easy to see that aò. Now, i D, then is zero outage because it satisies condition 2 o proposition 4. It is also easy to veriy that a and are zero outage schedulers. Again when D ½ oó, state has zero probability o occurance under scheduler. However, scheduler a satisies condition 2 o proposition 4 and has the same delay and power as scheduler. The converse to proposition 6 is however not true, i.e., a convex combination o two zero outage deterministic schedulers is not always a zero outage randomized scheduler, as illustrated in Example 4. Example 4: Let and be as in Example 3. Let D and ô ô ince É» or buer state 3 under scheduler, it is zero outage. A convex combination o and does not guarantee a scheduler with zero outage. The probability o being in buer state 3 is non-zero under scheduler, and i we then switch to scheduler, we will remain in buer state N = µó indeinitely and continue to loose packets due to buer overlow. Instead a convex combination o schedulers and, will result in schedulers that have no buer overlows., and have the same Note again that when D average delay and power. III. POWER EFFICIENT CHEDULER In this section, we provide a method to compute the minimal power scheduler in (4). Driven by the high computational complexity o solving or the optimal scheduler (using a dynamic program), we propose two suboptimal schedulers in ection III-B. In ection III-C, we derive a closed orm approximation or the average power as a unction o delay. Extension o the optimal schedulers to ading channel is briely discussed in ection III-D. Finally, we provide numerical results to illustrate the savings in power with increasing delay, and state some empirically observed properties o optimal schedulers in AWGN and ading channels.,

7 i ã ã õ 4 Ü ã ÿ ÿ ý ý ç ç ã i Ü Ü Ü RAJAN ET.AL.: DELAY-BOUNDED PACKET CHEDULING OF BURTY TRAFFIC OVER WIRELE CHANNEL 7 A. Optimal Packet chedulers We prove a lemma which is used in the proo o Theorem on convexity o set o randomized schedulers. Lemma 7 (Delay-power o 2 almost identical schedulers): Let and be two zero outage ½ schedulers which are identical in all columns except the JML column. The delay and power o scheduler which is given by a convex combination o and, equals a convex combination o the delays and powers o and, respectively. In other ` abdc words, i ã ã ã, Å / A then ã ç ` abdc \ ã ç 3`saBbdc and abdc ã ç abdc ã ç abdc ji ã ç. Further, ã ç É º» É º» ã É º», where where É º» and É º» are the stationary probabilities o being in state ½ under schedulers and respectively. Proo: ee Appendix D. An application o Lemma 7 is given in the ollowing example. Example 5 : Let schedulers a, and be as in Example 3. For D, and a uniorm arrival distribution, the delay and power under scheduler a ` abdc equals a j timeslot and abdc a s ` abdc. imilarly, sg ó*ó timeslots and abdc. The stationary probabilities or the dierent states under schedulers a and are / *4 and / 4. Note again that a. We ind that ` abbdc Í ã ç ` abdc a k ã ç Ï` abdc GMdó and where ã ç abdc time-slots ã ç abdc ay ã ç óa ó X. abdc By the repeated application o Lemma 7 it is easy to see that i we have more than 2 schedulers which dier in the same column, then the delay and power o the convex combination o such schedulers is given by the convex combination o the delays and powers o the individual schedulers. The ollowing theorem proves that the average power and delay or an arbitrary zero outage randomized scheduler can be obtained as the convex combination o delays and power o zero outage deterministic schedulers. Theorem (Characterization o delay and power): Consider a queue with inite buer size I and an input process with no more than D packet arrivals in one slot. For any randomized scheduler ÿ Å p, there exists ã» ç Å / ½ CB A 4 þ úup ý ç ã, with» ç, such»îâ that ` abbdc Í»  abbdc Í»  where ü» is a scheduler and ü»(å õ Proo: ee Appendix E. ã ç» ` abdc ü» ã ç» abdc ü» úup or all ½. (2) (3) Thus the average packet delay achieved by any zero outage randomized scheduler is given by a convex combination o the average packet delays achieved by all possible zero outage deterministic schedulers. In addition, the same convex combination o the average powers o the zero outage deterministic schedulers gives the average power o zero outage randomized policy. In proposition 6, it was shown that every zero outage scheduler in p admitted an expansion using only deterministic zero-outage schedulers. In the proo o Theorem, it is shown that among the several possible basis expansion o ÿ Å p, there is at least one which insures delay and power o ÿ is a convex combination o delay and power o some zero outage deterministic schedulers. The ollowing lemma shows a converse to this result, i.e., there exists schedulers which achieve delays and powers which are the convex combination o the delays and powers o any two arbitrary zero outage deterministic schedulers. Lemma 9 (Converse to Theorem ): Let and be two zero ` abdc outage deterministic schedulers with average delays `sabbdc and average powers abdc abdc respectively. Then, there exists a ãzero outage scheduler which achieves average delay ` abdc ã 3`saBbdc and average power abbdc ã abdc Proo : ee Appendix F. or all ji It should be emphasized that the convex span o õ úup is larger than p (some schedulers in convex span can be non-zero outage). Given that the randomized scheduler perormance can be obtained by appropriate linear combination o deterministic schedulers, the delay-power region can be characterized as ollows. Corollary (Characterization o delay-power region): The boundary o the achievable region in the delay-power plane is piecewise linear with the vertices achieved by deterministic schedulers. Proo : The total number o deterministic zero outage schedulers is inite, and the delay-power perormance o all randomized zero outage schedulers is given by a convex combination o delay-power o deterministic zero outage schedulers. Hence the corollary ollows. To ind the achievable delay-power region, we use a dynamic programming technique commonly known as Value Iteration Algorithm (VIA) [2]. The details o the VIA are given in Appendix G. ` r The delay bound is reduced monotonically and õ the minimal power deterministic policy is ound among ` r ú p or each. Thus, instead o solving (4), we are required to.

8 ƒ ó 7m 7 Ü IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 NUMBER OF PACKET TRANMITTED Fig. 2. CHEDULER CHEDULER BUFFER TATE chedulers or achieving dierent average delay bounds with minimal average power (Buer size "#, maximum arrival %&(, uniorm arrival distribution). solve the ollowing optimization problem, ) ` r ( Yu[^] ^[ Y -/. v+*, }32dw =G F>=,H (4) ` r ) The convex hull o the resulting ` r pairs gives the delay-power region, using Theorem and Corollary. Two examples o the transmission policy obtained using VIA are shown in Figure 2, which achieve two dierent points on the boundary o the achievable delay-power region. The igure illustrates the basic nature o the scheduler actions ( O = ) as a unction o queue state (N = ) as packet delay increases. Recall that the minimum possible average delay in our system is one, which is achieved by a scheduler which transmits all packets it receives as soon as possible. Thus, i schedulers start rom zero buer state, then schedulers achieving average delay one are such that O = N=, at least or N= i D. As the delay increases beyond one time-slot, the scheduler chooses to transmit ewer than N= packets or some states, and delays some o the packets; this can be seen in the behavior o chedulers and 2 in Figure 2, when D µ. cheduler achieves a smaller delay (4. time-slots) compared to cheduler 2 (4 7.5 time-slots) with higher power consumption (6.2 versus 5. units o power). Note that neither o the schedulers in Figure 2 transmit N = packets or N =, i.e., they do not lush the buer like a unit delay scheduler. Furthermore, as the delay increases, the schedulers tend to transmit close to average arrival rate : ó packets/time-slot (notice the behavior or buer states i N = i or cheduler 2). For each o the scheduler, buer overlows are avoided by transmitting close to D packets when Nk= is close to I (see N= ƒ or cheduler 2). The reduction in power requirements due to additional delay can be attributed to convexity o the relation between power and rate o transmission (6). Transmitting large number o packets requires exponentially increasing amount o power, thereby resulting in a large average transmission power or small delays. As the delay deadlines become less stringent, packets can be delayed so that the transmitter can minimize the transmission o large number o packets, to reduce large power spikes and hence reduce average power consumption. As the average delay increases unboundedly (assuming that the buer size also increases to ininity), the scheduler delays every packet such that (approximately) only : packets are always transmitted in every time-slot. In ection III-E, the scheduler behavior is discussed urther using spectral analysis. Alternately, the power reduction with delay can be understood using the ollowing source coding interpretation. Note that i we consider the inormation in the packet arrival times, in addition to the contents o the packets, then our scheduler action is analogous to that o a lossy source coder. The higher the delay allowed, the more is the compression o inormation. For example, i delay equals one then the arrival times o the packets are exactly known at the receiver; equivalent to lossless encoding o timing inormation. As the average packet delay increases, the scheduler delays certain packets more than the others and the receiver will only have a noisy version o the packet arrival times; this is similar to lossy coding. Asymptotically, as the delay goes to ininity, the receiver will have no inormation about the actual arrival times o the packets; this corresponds to encoding timing at zero rate. The loss o inormation with increasing delays can be attributed to ³ O is not invertible the act that the scheduler mapping ± N or N= i D except when delay equals (assuming that buer is initially empty). Finally, we characterize extremal source statistics, resulting in the lowest power ` abdc ` abdc or any æ delay constraint and highest power source or. Proposition (Extremal power source): Let the arrival distribution be denoted by ˆ 6 where» ½ is the probability that packets arrive in any timeslot. Then the input distribution that results in the least transmit power in an AWGN channel, or any average delay is given by 6» : :9 :; ½ <9 :=; : 9 :=; ½ <9 :=; î É 9 where :; represents the greatest integer no larger than :. Also, the input distribution that `sr results n in the highest transmit power or a delay bound 3 o is given by m ½ 6» ½ ½ n DE ^ > BD Proo: ee Appendix H. 3 It is our conjecture that or any average delay >@?ACBEDGFIHJ, this is the input distribution requiring highest average transmit power. In other words, the On-O arrival process requires the highest transmit power at any delay in an AWGN channel among all arrival processes with same average and inite maximum arrival rate. CB

9 4 ; D M ` M = _ 4 ` ø a : _ RAJAN ET.AL.: DELAY-BOUNDED PACKET CHEDULING OF BURTY TRAFFIC OVER WIRELE CHANNEL 9 B. uboptimal chedulers For large buer sizes I, the number o possible states in the VIA increase exponentially and hence computing the optimal scheduler is computationally intensive. For the cases where the arrival distribution is measured in real-time and the optimal scheduler is adapted over time, the implementation o VIA can lead to an intractable design. Motivated by the high complexity o computing the optimal scheduler, we present two suboptimal schedulers in this section. The suboptimal schedulers we introduce in this section are deterministic. To obtain randomized suboptimal schedulers, we can again use convexity properties o randomized schedulers (Theorem and Lemma 9). A simple method or complexity reduction is to reduce the number o states in the VIA (see Appendix G). The ollowing state reduction is most useul or moderate to large delay scenarios. As delay increases beyond, the scheduler tends to take the same action in several consecutive queue states. Thus, to reduce the state diagram size, multiple states can be morphed into one state. For instance, or all states N= ¼ CB ¼ LK, the scheduler can be constrained to take the same action O =, thereby reducing the state space. It is clear that the constrained system will be suboptimal, but the loss due to additional constraints can be minimized by appropriate choice o the number and lengths o constraint intervals; some guidelines or the choice o the intervals can be derived rom the empirical observations about the optimal schedulers made in ection III-E. The second suboptimal scheduler is labeled the log-linear scheduler, described as ollows. For a queuing delay o one time-slot the optimal scheduler 4 lushes the buer at all timeslots, i.e., O = N = and the corresponding power required in each time-slot is proportional to Ž.. As the delay increases, we observe that the optimal scheduler tends to choose O = N = so that the power in each time-slot is linearly proportional to N=, thereby equalizing the power penalty in large buer states Nk=. For equalizing the power, the scheduler picks packets O = N= N=. Combined with the natural constraint that we cannot transmit more packets than available, we propose the ollowing log-linear scheduler, O = i N= O = µy\[^] Nk= 9ä ^ * NM N= _ (5) M The parameter o the log-linear scheduler is chosen to meet the delay bound. For buer states greater than I D, the log-linear scheduler transmits at least N = packets I to prevent buer overlows. The log-linear scheduler greatly simpliies M scheduler design since it requires only one parameter. The delay and power using the log-linear scheduler is calculated using (9) and (), respectively. The delay-power region o the M log-linear scheduler is obtained by monotonically increasing. In all our simulations, the log-linear perormance is close to the optimal scheduler perormance; urther results are given in ection M III-E. The value o to achieve a certain average delay depends on the arrival distribution o the source. We now propose a simple 4 Note that the optimal scheduler is not unique or %POQ and i the schedulers start rom the zero buer state. M adaptive algorithm that computes the value o dynamically to achieve any given average delay constraint based only on the knowledge o the mean arrival rate. In the adaptive scheduler, = is a unction o time and is updated in every time-slot as given below. M =y«rut = ` (6) ø where = is the sample average delay given by the ratio o the sample average buer length and arrival rate. The perormance o the adaptive scheduler is shown to numerically converge to the perormance o a log-linear scheduler that has ull knowledge o arrival statistics in ection III-E. C. Approximate Delay-Power Relation In this section we derive an approximate closed orm relationship between the average transmit power,, and ` abdc the average delay in an AWGN channel. The derivation is based on two simpliying assumptions: the packets are ininitely divisible (luid model) and the output distribution o the scheduler is Gaussian. The luid model allows us to divide packets arbitrarily, and thus avoid the discrete nature o scheduler decisions, which is a signiicant reason or intractability o closed-orm analysis o the optimal ` abdc scheduler. As discussed in ection III-A, or large, the optimal scheduler transmits at rates close to average arrival rate : i buer overlows can be avoided. For large delays, the queue length Nk= varies around the average queue length FdN=H. The scheduler action, O = Nk= F"Nk=,H by the ollowing linear relation, O =U4WVN= WẌ abdc, can thus be approximated around (7) For a stable system, the average arrival rate should equal the average departure rate and hence F O =,H :. Taking expectations on (7) we ind V X :. The slope /N = 4 o the scheduler, V, at FdN = H is approximated by the slope o log-linear scheduler, V FdN = H. The variance o the output process, O =, denoted by, is computed as ollows. From (2,7), /N = X 4 /N = 4 /O = 4 /Î< = 4 /N = 4 : /N = VN = YXA From stationarity o Nk=, /N = X Ö 4 /N = 4 and ater algebraic manipulations, we can ind as a V V /N = 4 () where a is the variance o the arrival process F*<G=,H. Using \ the Gaussian approximation or the distribution o O =[Z :, the average transmit power is F>=H ] = O = «^`_Ca 2cb - -Ed - «O = m X d - - m X e/2dwx3z { d -x a - dò~ rom 4 m X e/2dwx3z { d -x or large : ` abdc_

10 Ÿ -x -x : : a = : : m m i q IEEE TRANACTION ON INFORMATION THEORY, VOL. XX, NO. Y, MONTH 23 Thus, ^ * abdc ` abdc (9) The eect o random source arrivals is clearly highlighted in (9) by the actor Ÿ m. ources which exhibit large variations in their arrivals compared to their mean rate and are more bursty, have large Ÿ m. For more bursty sources, even small ` abdc increase in allows considerable power savings; practical examples o such sources include web and traic. Asymptotically, Ô ` abdc ǵ as, the high delay approximation x3z { - gives :, which is the same as the hannon limit. As ` abdc h, the random arrivals o the source are completely smoothened and hence well known inormation theoretic analysis applies. D. cheduling in Fading Channels In the preceding sections, we showed that time scheduling packets rom a source that produces variable number o packets every time-slot results in signiicant power reduction, even in AWGN channels. An analogous problem to scheduling due to source burstiness is the one where scheduling is perormed due to time-varying channel conditions. To urther emphasize the act that channel time-variations lead to a non-trivial scheduling problem under delay constraints, we consider transmission o a constant rate source over a ading channel. 5 Note that scheduling is equivalent to power control due to our emphasis on reliable communication, and their relation is deined by (6). Power control in ading channels has been an area o active interest (see [29] or a review). We illustrate, by an example, that the proposed method o dynamically changing the transmission rate and power, can lead to power savings in inite state ading channels with increasing delay. ` abbdc For a delay bound, time-slot, packet scheduling reduces to conventional power control. As the delay bound goes to ininity, the delay-bounded scheduling leads to maximal mutual inormation power control []. For simplicity, we consider a 2-state ading channel in the remainder o this subsection although our results are directly applicable to any channel with inite number o states. Consider a source that produces constant number o packets every time-slots, being transmitted over a two-state channel, with = Å F the two channel states given by, H and the transition probabilities 2 between i j ä ó k (2) where 2»Ïº ¹ is the probability o transitioning rom channel state = ½ to = X ¼. The corresponding stationary probabilities o being in the two channel states are.25 ( = n ) and.75 ( = ). At a delay o time-slot, the channel transmits 25% o the packets when = and the remaining packets when n. As the delay increases to 4.7 time-slots, only 2% o n, a savings o 9dB in the packets are transmitted when = power over case when delay equals time-slot. As the delay tends to ininity, the average number o packets transmitted in each state is given by the time water-illing solution, and is characterized in the ollowing Proposition. Proposition 2 (Time water-illing): For a two-state ading channel, power minimizing transmission rates, F" ž_h, or the two states, with no delay constraint, are given ^ ^ * The stationary probability o being in channel state ½ is denoted and / l» 4 q mon l» ½ p ½ p ½ p l» Á mqn ji l» l» ð q m mon Proo: Follows rom (6) and standard Lagrangian techniques. 6 Ü. Applying Proposition 2 to the channel given above suggests that at ininite delay r. We see that and these values o» are relected in the small raction o packets that are transmitted in state = at a delay o 4.7 time-slots. In practical systems both the source arrival rate and channel state are time varying; hence scheduling becomes doubly important. Our ormulation is applicable, in straightorward manner, to the case where both source and channel are time varying. The power savings consists o two components - one due to source burstiness and the other due to channel variations. The total gain depends on the ollowing actors: the extent o source burstiness a : `, the delay bound, the amount o channel variations F = H, and the stationary probabilities o dierent channel states. The larger the a :, more is the savings in power with increasing delay because the departure process, O =, is well smoothened out. Analogously, the larger the F = H, large dierence between the NRs o the two states leads to decreased average power with increasing delays. E. Numerical Results In this section, we present numerical results to study the perormance o the proposed schedulers. We irst compare the perormance o the optimal scheduler with the proposed suboptimal log-linear scheduler, and the closed orm approximation derived in ection III-C. pectral analysis is also used to analyze the behavior o the optimal and log-linear schedulers. An example or 4-state ading channel is also presented. Finally, we make some empirical observations about the optimal schedulers based on our numerical simulations. The plot o the minimal transmission power versus average delay bound is shown in Figure 3. For all delay-power pairs 5 This parallels our previous discussion on transmission o a bursty source over an AWGN channel. Preliminary results on bursty source-ading channel duality can be ound in [22]. 6 These values o s t are derived assuming that ractionally sized packets can be transmitted; packet integrity constraint must be imposed on top o these rate values to ind integral transmission rates.