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Jointly Optimal Source-Flow, Transmit-Power, and Sending-Rate Control for Maximum-Throughput Delivery of VBR Traffic over Faded Links Enzo Baccarelli, Nicola Cordeschi, and Tatiana Patriarca Abstract—Emerging media overlay networks for wireless applications aim at delivering Variable Bit Rate (VBR) encoded media contents to nomadic end users by exploiting the (fading-impaired and time-varying) access capacity offered by the “last-hop” wireless channel. In this application scenario, a still open question concerns the closed-form design of control policies that maximize the average throughput sent over the wireless last hop, under constraints on the maximum connection bandwidth available at the Application (APP) layer, the queue capacity available at the Data Link (DL) layer, and the average and peak energies sustained by the Physical (PHY) layer. The approach we follow relies on the maximization on a per-slot basis of the throughput averaged over the fading statistic and conditioned on the queue state, without resorting to cumbersome iterative algorithms. The resulting optimal controller operates in a cross-layer fashion that involves the APP, DL, and PHY layers of the underlying protocol stack. Finally, we develop the operating conditions allowing the proposed controller also to maximize the unconditional average throughput (i.e., the throughput averaged over both queue and channel-state statistics). The carried out numerical tests give insight into the connection bandwidthversus-queue delay trade-off achieved by the optimal controller. Index Terms—Multimedia wireless connections, cross-layer management, throughput energy saving, flow control.
Ç 1
INTRODUCTION AND GOALS
O
the last years, a strong proliferation in the use of multimedia technology has been experienced, triggered by several reasons. First, wireless networking architectures (such as media Content Delivery Networks (CDNs)) are becoming an integral part of the communication environment [1, Chapters 5, 6, and 7]. Second, the utilization of new handheld wireless devices (such as PDAs and laptops) is becoming common. Third, new multimedia applications (such as streaming stored/live audio/video) became popular first in Internet wired environments and now in Internet mobile ones [2, Chapters 14 and 15]. However, these wireless media opportunities bring with them also several technological challenges. In fact, wireless networks suffer from a large variation in connection conditions, mainly due to fading phenomena, mobility, and multiple-access interference. Since currently deployed network protocols (as, for example, the Internet one [1, Chapter 4]) offer best effort services to the higher layers of the protocol stack, it is necessary that the Application (APP) and/or Transport layers implement adaptive bandwidthcontrol (e.g., flow-control) mechanisms, in order to delaysensitive media connections be successful [2, Chapter 4]. VER
. The authors are with the DIET Department, “Sapienza” University of Rome, via Eudossiana 18, Rome 00184, Italy. E-mail: {enzobac, cordeschi, patriarca}@infocom.uniroma1.it. Manuscript received 5 June 2009; revised 8 Oct. 2010; accepted 21 Dec. 2010; published online 17 Mar. 2011. For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number TMC-2009-06-0210. Digital Object Identifier no. 10.1109/TMC.2011.68. 1536-1233/12/$31.00 ß 2012 IEEE
Thus, as the exploitation of wireless networks drifts from delay-tolerant data-transfer to delay-sensitive multimedia applications, an adaptive smart control of the connection bandwidth that jointly accounts for the states of the underlying Physical (PHY) and Data Link (DL) layers and the energy limitations dictated by the battery-powered media terminals [3] becomes mandatory for attaining the best end-to-end performance [4], [5]. Since media applications are delay and jitter sensitive and the slow-start phase of standard Transport Control Protocols (TCPs) tends to introduce sudden unpredictable large delays [6], [7], an emerging trend in multimedia over wireless networks is to directly resort to cross-layer design of application-driven transport protocols.
1.1 The Tackled Problem About the focus of this paper, after considering a clientserver network architecture, we model the transmit node (e.g., the server node) as a time-slotted fluid G/G/1 queuing system fed by a Variable Bit Rate (VBR) encoder whose output rate (e.g., the connection bandwidth) may be controlled. In this operating context, the problem we go to tackle deals with the closed-form (i.e., no iterative) cross-layer (i.e., queue and channel-state aware) design of the controller that jointly performs optimal adaptive management of the connection bandwidth, transmit energy, and available throughput. In our framework, the queue state (i.e., the queue length), channel state (i.e., the fading level), connection bandwidth, and throughput are all modeled as random variables (r.v.s) that may be of discrete, continuous, or even mixed type. Published by the IEEE CS, CASS, ComSoc, IES, & SPS
BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM...
Hence, the corresponding number of allowed values may be finite, continuous infinite, or even uncountable infinite, so that we cannot resort to approaches/algorithms based on the Dynamic Programming (DP) for attaining finite-complexity solutions. Therefore, according to [24, Chapters 13 and 17], we adopt the throughput averaged over the fading pdf and conditioned on the queue state at the current slot (shortly, the conditional average throughput) as performance metric of the considered queuing system and then we proceed to maximize it. In the considered application scenario, this maximization must be accomplished under four constraints dictated by the APP, DL, and PHY layers. The first one is on the available average energy per slot and it is imposed by energy-saving limitations typically arising at the PHY layer [3], [9]. The second constraint we consider is on the allowed peak energy per slot. It is still dictated by the PHY layer and may arise from spectral compatibility issues and/or limitations on the maximum throughput allowed by the considered wireless standard [1, Chapter 6]. The third constraint involves the DL layer and (upper) limits the available buffer capacity. It guarantees that the considered queuing system is stable and also provides an upper bound on the resulting average queue delay (see the last part of Section 4).1 The fourth constraint arises from the APP layer and limits the maximum instantaneous bandwidth allowed the connection. It is typically dictated by the finest granularity level of the adopted encoder (such as, for example, the minimum size allowed the quantization step) [2, Chapters 5 and 6], and, together with the limit on the buffer capacity, contributes to upper bound the resulting average queue delay (see Section 4).
1.2 Related Works Although the emerging trend for multimedia delivery over wireless channels demands for an application-aware crosslayer approach to optimize the protocols [2, Chapters 1 and 12], [4], [9], up to date two tasks of bandwidth management at the APP/Transport layers and delay-versus-energy trade-off at the DL/PHY layers have been tackled by pursuing more traditional layered approaches (see, for example, [2, Chapter 12], [4] for a media applicationoriented list of still open questions and challenges about these topics). Specifically, according to a traditional “layered” perspective, for a long time, bandwidth management has been mainly considered as an effective mean for implementing congestion and flow control on the basis of loss-rate and Round-Trip-Time (RTT) measurements performed only at the Transport layer [1, Chapter 3]. Classic analysis of the steady-state performance offered in wired environments by layered congestion-control closed-loop systems that exploit various TCP versions in tandem with several Active QueueManagement (AQM) policies is detailed, for example, in [10]. However, the basic bandwidth-management algorithm common to all TCP versions (namely, Tahoe, Reno, NewReno, Vegas, Sack, TFRC, and Veno) reacts to segment loss 1. We anticipate that, under the control policy we go to develop, no overflow phenomena may occur.
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as if they were generated by congestion. Thus, in wireless environments, where there may be losses due to fadinginduced channel errors and/or mobility-induced handovers, TCP tends to penalize throughput performance, without any positive impact on the experienced segment loss [6], [7]. Over the last years, two main approaches have been developed to limit as much as possible the fading/mobilityinduced occurrences of timeout and/or slow-start phases that penalize TCP performance in wireless environments [7]. The goal of the first approach is to recover from wireless-induced losses directly at the DL/PHY layers, so to “hide” these losses from TCP. For this purpose, TCP-aware DL protocols have been developed in [7], [11], and [12] that implement optimized ARQ/FEC policies at the DL/PHY layers. Being these ARQ/FEC mechanisms quite bandwidth wasting, the ultimate net bit rate (e.g., the goodput) available at the PHY layer may result strongly penalized [12]. Main task of the second approach is to develop TCPfriendly control mechanisms that guarantee the same average bandwidth of the TCP one, while sustaining smooth connection rates (see, for example, [13], [14], [15] and, more recently, [16]). Although TCP-friendly control mechanisms exploit, indeed, a minimum of cross-layer information for attempting to classify loss events; however, 1) they do not exploit the knowledge of the channel state for adjusting the connection bandwidth available at the APP/Transport layers; 2) they do not account for the energy-saving requirements dictated by the PHY layer; and 3) they do not utilize the full knowledge of the residual space available at the DL buffers for proactively coping with incoming overflow phenomena. Furthermore, in principle, the ultimate target of TCP and TCP-friendly control mechanisms is not to maximize the DL throughput averaged over the fading statistics [17], [10, Table I]. Optimized joint energy/queue control policies at the DL/PHY layers derived by exploiting the analytical tool of the Markov Decision Process (MDP) and implemented via Dynamic Programming are presented, for example, in [8], [18], and [19]. However, all these contributions do no address the bandwidth-management problem. Perhaps, works exploiting analytical approaches closest to that followed in this paper are those based on the Convex Optimization and Nonlinear Programming [20]. In [21], tools from convex optimization and deterministic (e.g., no stochastic) control theory are used to joint optimize congestion-control and queue-management policies for the wired Internet. In [22], the Lagrangian method is applied to develop flow-control algorithms for wired networks with multiple source-destination pairs, so to maximize the aggregate source utility. Both contributions [21], [22] refer to wired application scenarios, so that neither fading effects nor energy constraints are accounted for. The works in [28], [29], and [30] present a unified framework for joint rate control and scheduling in multihop wireless networks. Using a dual approach, they show that control problems can be decomposed and solved individually, while they are still coupled by the implicit cost associated with each queue. By fact, each user is associated with a concave utility function of the rate in order to model the principle of diminishing returns for elastic applications
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(e.g., weighted proportionally fair resource allocation) and adjust its sending rate according to the local congestion price, where a natural choice for a local source/congestion price is the queue length (alternative to standard “global” wired solutions). However, in [28], channel variation due to fading is not considered, while even if schedulers in [29] and [30] are allowed to know the current queue length and current channel condition (as in our work), they only consider a finite-state channel process. In this regard, the interesting work [30] presents a systematic approach to cross-layer design to make the interactions between different layers more easier, and they extend old (wired) utility maximization formulations to time-varying channel and adaptive multirate devices. Nevertheless, the authors consider a joint optimal design over time-varying channel networks only from a 2-3-4-layer perspective. Specifically, they do not take into account also for a joint physical layer optimization. They just consider the maximal capacity the physical layer can provide slot by slot to higher layers, and do not use any information related to fading or queue state in order to jointly optimize Transport layer and Physical layer performances (like, for example, analyzing as the knowledge of the congestion price can improve energy allocation and, conversely, the “energy safe” use of the current time-varying channel under its capacity can impact and improve congestion control).
1.3 Main Contributions and Outline of This Paper Thus, from the outset, main contributions of this work may be so summarized: First, we develop closed-form expressions for the optimal solution of the tackled cross-layer constrained optimization problem, and then, we characterize the corresponding average performance via closed-form upper and lower bounds. Afterward, by exploiting some basic results borrowed from the theory of the Ordered Statistics [23], we specify the operating conditions allowing the resulting controller to maximize the unconditional average throughput. . Second, we present an on-the-fly implementation of the optimal controller that does not require any a priori knowledge about the (possibly, time-varying) fading pdf. . Finally, we numerically test actual performance of the optimal controller on Rayleigh-faded channels, both in terms of average connection bandwidth and queue delay. Robustness of the controller performance against the degrading effects induced by imperfect channel-state measurements, time-correlated fading, and a priori unknown variations of the fading pdf is also numerically tested. The rest of this paper is organized as follows: after introducing in Section 2 the considered client-server networking architecture and the problem setup, Section 3 develops the optimal controller and points out its basic structural properties. In Section 4, we develop the operative conditions guaranteeing that the controller is optimal even in the sense of the unconditional average throughput maximization (shortly, the unconditional optimality of the controller), and, then, we provide upper and lower bounds .
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Fig. 1. The considered server-client system model.
on the resulting average throughput and queue delay. Section 5 focuses on testing actual performance and robustness properties of the proposed controller on several application scenarios of practical interest. Some conclusive remarks are drawn in Section 6, while analytical proofs of main results are deferred to the final Appendices, which can be found on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.68. About the adopted notation, underlined letters denote vectors; scalar random variables are denoted by bold characters, while their outcomes are indicated by the corresponding no bold symbols. Ef:g is the expectation operator, IRþ 0 is the set of the nonnegative real numbers, 4 IRþ is the set of the strictly positive real numbers, ¼ þ means “equal by definition,” while ½x indicates maxfx; 0g 4 ; sÞg ¼ Rand p ðÞ is the pdf of the r.v. . Finally, E f’ð ’ð; sÞp ðÞd denotes the expectation of the biargumental function ’ð; sÞ carried out only over the pdf of the r.v. , while ½fðxÞba indicates maxfa; minffðxÞ; bgg.
2
SYSTEM ARCHITECTURE AND PROBLEM SETUP
By referring to the server-client system architecture of Fig. 1, we consider a G/G/1 fluid queue system fed by a VBR encoder. Time is slotted, slot length is unit, and slot t spans the (semiopen) interval ½t; ðt þ 1ÞÞ; t 2 INþ 0 . The Information Units (IUs) to be sent over the wireless channel are delivered by the media encoder at the end of each slot, and they are buffered into a server’s queue of finite capacity Nmax . Thus, ðtÞ 2 IRþ 0 ðIU=slotÞ in Fig. 1 is the number of IUs arriving at the input of the queue at the end of slot t, and, according to a current taxonomy, we refer to ðtÞ as the connection bandwidth at slot t. The fading phenomena affecting the wireless link are assumed constant (i.e., time invariant) over each slot (i.e., the so-called “block-fading” model is assumed [1]) and are considered i.i.d. from slot to slot. Thus, the channel state ðtÞ 2 IRþ 0 over slot t is modeled as a real nonnegative r.v.2 with pdf p ðÞ. Furthermore, the channel-state value ðtÞ is assumed to be perfectly known at the transmitter at the beginning of 2. The meaning of the channel state ðtÞ is application depending. Without loss of generality, we may consider ðtÞ be the Signal-toDisturbance (e.g., the Signal to Noise-plus-Interference) ratio measured at the input of the client terminal of Fig. 1 during slot t.
BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM...
slot t, so that, slot by slot, Perfect-Link-State-Information (PLSI) is available at the transmit scheduler of Fig. 1. The overall system operates in the steady state (i.e., it works under stationary and ergodic operating conditions). Let sðtÞ 2 IRþ 0 be the number of IUs buffered at the queue of Fig. 1 at the beginning of slot t (e.g., the queue’s backlog at slot t). Thus, after denoting by rðtÞðIU=slotÞ the number of IUs to be sent over the physical channel during slot t (e.g., the sending rate at slot t), the following Lindley’s equation: sðt þ 1Þ ¼ ½sðtÞ þ ðtÞ rðtÞþ ; t 0
ð1Þ
dictates the evolution of the discrete-time queue-length process fsðtÞ 2 IRþ 0 ; t 0g (since data are assumed arriving at the end of each slot, this implies rðtÞ sðtÞ). According to the seminal results of Wolff [24] and Loyes [31], under the above assumed steady-state operating conditions, there exists a unique stationary solution to the recursion in (1). For easiness, we recast the result as a lemma, in order to make it compliant with our framework and the notation we adopt [32]. Lemma. For given service (scheduling) policy rðtÞ, channel process, and arrival process, under queue stability conditions there exists a unique r.v. sð1Þ (depending on both schedulers rðtÞ, ðtÞ, and on channel statistics), such that, for any choice of the initial conditions sð0Þ, the state process fsðtÞg in (1) converges (in finite time) to this stationary solution sð1Þ. Thus, unless otherwise stated, in the following, we assume that the queue operates under the stationary regime, so that fsðtÞg indicates the stationary and ergodic solution of the recursion in (1) and ps ðsÞ is the corresponding steady-state pdf. The cost to send rðtÞ IUs over slot t is the amount of energy EðtÞðJouleÞ required for their transmission. Thus, we assume that the corresponding number rðtÞ of IUs sent over the channel of Fig. 1 during slot t depends on both EðtÞ and the channel state ðtÞ via the rate function Rð:; :Þ adopted to measure the goodput performance of the considered system, so that we can write 4
rðtÞ ¼ RðEðtÞ; ðtÞÞ; t 1:
ð2Þ
The rate function Rð:; :Þ in (2) is a real nonnegative function of two nonnegative real arguments Eð:Þ; ð:Þ, and it is measured in ðIU=slotÞ. Roughly speaking, Rð:; :Þ summarizes the goodput performance of the DL/PHY of the considered system, so that its behavior and analytical properties may depend on several system parameters, such as the requested QoS, the FEC mechanisms implemented at the PHY layer, the fading statistics, the performance of the ARQ/fragmentation mechanisms utilized at the DL layer, and so on. Therefore, in the following, we limit to introduce few (quite mild) assumptions on Rð:; :Þ, that, by fact, are retained by the rate functions of practical interest [4]. First, the rate function RðE; Þ is continuous on IRþ 0 þ IR0 and admits up to second-order continuous derivatives on IRþ IRþ 0 . Second, it vanishes at E ¼ 0 and ¼ 0, i.e., RðE ¼ 0; Þ RðE; ¼ 0Þ 0. Third, it is nondecreasing both for E 0 and 0. Fourth, for any assigned 6¼ 0, the rate function is assumed to be strictly concave in 4 the E-variable, that is, R"" ðE; Þ ¼ @ 2 RðE; Þ=@E 2 < 0;
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for E > 0 and 6¼ 0. Finally, we assume that its first-order 4 derivative R" ðE; Þ ¼ @RðE; Þ=@E done with respect to the E-argument is nondecreasing in the -argument for 0.
2.1 Setup of the Tackled Optimization Problem 4 2 Let us indicate by xðtÞ ¼ ½ðtÞ; sðtÞ 2 ðIRþ 0 Þ the bidimensional overall state of the system of Fig. 1. Thus, roughly speaking, the control problem we tackle with focuses on the optimal design of both the number rðtÞ of IUs to be sent over the channel of Fig. 1 at the beginning of slot t and the number ðtÞ of IUs to be output by the VBR encoder at the end of slot t (e.g., the optimal control of the connection bandwidth) when xðtÞ is the current system state. The ultimate target is the maximization of the conditional average throughput EfrðtÞjsðtÞg under constraints on the available average energy per slot E ave ðJouleÞ, allowed peak energy E p ðJouleÞ, buffer capacity Nmax ðIUÞ, and maximum connection bandwidth max ðIU=slotÞ. To formally state the mentioned control problem, let 4
EðtÞ "ððtÞ; rðtÞÞ ¼ R1 ððtÞ; rðtÞÞ; t 1
ð3Þ
be the energy requested to send rðtÞ IUs when the channel state (e.g., the fading level) is ðtÞ. Therefore, the tackled optimization problem may be formally stated as follows: max E frðtÞjsðtÞg
rðÞ;ðÞ
s:t: :
E f"ð ; rðtÞÞjsðtÞg E ave ;
ð4Þ ð4:1Þ
0 "ððtÞ; rðtÞÞ E p ;
ð4:2Þ
0 sðt þ 1Þ Nmax ;
ð4:3Þ
0 ðtÞ max ;
ð4:4Þ
0 rðtÞ sðtÞ;
ð4:5Þ
where the number rðtÞ and ðtÞ of IUs to be sent and required to the VBR encoder must be searched over the overall set of nonnegative real-valued functions depending on both current queue and channel states, that is, ½rðtÞ; ðtÞ ½rððtÞ; sðtÞÞ; ððtÞ; sðtÞÞ:
ð5Þ
Remark 1. Some considerations about the considered optimization problem Equation (4) points out that we maximize (on a per-slot basis) the expected throughput, given (i.e., conditioned on) the number sðtÞ of currently buffered IUs. Several contributions (see, for example, [24, Chapters 13 and 17] and references therein) support the utilization of the conditional average throughput in (4) as an effective metric for characterizing performance of queuing systems, specially when the traffic at the queue’s input exhibits heavy-tailed distributions. From an application perspective, this is, indeed, the case of traffic flows generated by VBR media encoders (see Fig. 1), that typically output Pareto-like distributed coded streams [2, Chapters 2 and 5]. We remark as the main problem of a conditional approach arises from
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the dynamic nature of each source controller, responsible with channel statistics and the scheduler’s structure for the time evolution of the buffer occupancy (that is, the half4 infinite buffer temporal sequence s ¼ ½s0 ; s1 ; s2 ; . . . ; ), that is not taken into account by the conditional optimization. From this perspective, we anticipate that a further optimization step in the design of the joined controller will be the optimal choice of the source scheduler among the degree of freedom provided by the conditional approach, in order to accomplish a throughput maximization on a wider scale, that is, on a temporal dimension, so to obtain an unconditional maximization of the average throughput provided to the final user. In fact, from an analytical point of view, the reason for considering the conditional expectation in (4) in place of the (perhaps, more Rconventional) unconditional one [8], [18], R 4 [19]: EfrðtÞg ¼ s rð; sÞps ð; sÞdds is that closed-form computation of this last requires that the (steady-state) pdf ps ðsÞ of the queue state is also available analytically. In turn, this requires to solve the corresponding Lindley’s equation [23], [24]: Z Z ps ðsÞ ¼ pa ða ¼ s s þ rð; s ÞÞp ðÞps ðs Þdds ; s
that, unfortunately, resists to closed-form solution even when the source statistic pa ðaÞ is known (i.e., nonoptimization depending; see, for example, [26] for some recent remarks on this question). For the same reason, in (4.1), we consider a constraint on the available average energy conditioned on the current queue state sðtÞ, in place of the more conventional unconditional average constraint [8], [18], [19]: Ef"ð ; sÞg E ave . So doing, we are able to extend a model usual in cross-layer (finite-state) design of channel schedulers also to the cross-layer design of continuous-state bandwidth adaptive sources, in order to overcome the problems arising when reductions in channel bandwidth are not matched by reductions in the source coding rate. In this context, it is of primary importance to tackle the problem of the maximization of the average arrival rate EfðtÞg (or some objective related to it, in order to, for example, obtain the additional feature of fairness in a network) that can be supported by the system, avoiding at the same time service interruptions by lowering the media quality in a “graceful” manner. However, as already detailed in Section 1.2, for this problem there does not exist an optimal scheduler able to also take into account the physical layer parameters and optimization involved. Nevertheless, we recall that, for the unconditional time averages, at the steady state, it must be EfðtÞg EfrðtÞg. Therefore, under feasibility conditions, being the conditional average throughput radiated over the channel strictly related to E fðtÞg (even if generally different), the objective (4) is still expected to represent a good optimization metric. Inspired by these considerations, in (4), we maximize the expected average throughput rðtÞ given the number sðtÞ of IUs buffered at the current slot t. One of the best features of our conditional approach is to have recast the scheduler design as a convex constrained optimization problem for which the strong duality holds (see Appendix A, which is available in the online supplemental material). As a
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TABLE 1 Main Taxonomy of the Paper
consequence, opposite to the main (computational expensive) approaches used in literature, as detailed in the following, we are able to obtain the optimal schedulers ½r; in closed form, easy to implement. In Section 4, we develop the operating conditions guaranteeing that the controller solving the optimization problem in (4) is also able to maximize the corresponding 4 4 unconditional average throughput r ¼ Efrð ðtÞ; sðtÞÞg ¼ R R s rð; sÞpðjsÞps ðsÞdds. Table 1 summarizes the main taxonomy used in this paper.
3
THE CONTROLLER MAXIMIZING THE CONDITIONAL AVERAGE THROUGHPUT 4
Let us indicate by "r ðr; Þ ¼ @"ðr; Þ=@r the first-order derivative of the energy function in (3) carried out with respect to the r-argument. Thus, after recognizing that the optimization problem in (4-4.5) is an instance of convex optimization problem, the resulting optimal throughput/ bandwidth solution ropt ð:; :Þ, opt ð:; :Þ may be evaluated in closed form, as detailed in the following Proposition 1 (see Appendix A for the proof, which is available in the online supplemental material). Proposition 1. Under the above reported assumptions, for the optimal solution of the constrained optimization problem in (44.5), we have that 1.
the delivered throughput is optimally scheduled according to ropt ððtÞ; sðtÞÞ 8 > < rp ððtÞ; sðtÞÞ rp ððtÞ;sðtÞÞ ¼ 1 1 > : "r ðtÞ; ðsðtÞÞ 0
for sðtÞ s1 for sðtÞ > s1 ; ð6Þ
where "1 r ð:; :Þ denotes the inverse function of "r ð:; :Þ 4 with respect to the E-variable, while rp ððtÞ; sðtÞÞ ¼ minfsðtÞ; RððtÞ; E p Þg is the peak value of the throughput allowed at slot-t. The threshold s1 in (6) dictates the boundary between the underloaded and overloaded operating regions of the server’s buffer of
BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM...
Fig. 1, and it may be computed by solving the following algebraic equation: Z "ð; rp ð; s1 ÞÞ p ðÞ d ¼ E ave : ð7Þ
Furthermore, ðsðtÞÞ in (6) is the optimal value of the dual variable of the tackled optimization problem, and it may be computed by solving the following (functional) equation for sðtÞ > s1 : rp ð;sðtÞÞ ! Z 1 1 p ðÞ d ¼ E ave ; " ; "r ; ðsðtÞÞ 0 ð8Þ 2.
the optimal bandwidth management is dictated by the following relationship: opt ððtÞ; sðtÞÞ minfðNmax sðtÞÞ þ ropt ððtÞ; sðtÞÞ; max g:
ð9Þ
Remark 2. On the behavior of the optimal controller. We note that the optimal scheduler (6) depends on sðtÞ both directly, via the threshold condition sðtÞ < > s1 and the peak throughput expression rp ððtÞ; sðtÞÞ, and indirectly, via the Lagrange multiplier ðsðtÞÞ. Furthermore, it presents the intuitive feature to pick the maximum possible rate subject to the peak-energy constraint as far as possible, while picks lower rates in good channel conditions for high queue length (i.e., the congestion price), in order to ensure the average energy budget and better distribute energy also to serve worse channels. This is done via (8) derived from the averaged conditional energy constraint, that allows to directly embed the knowledge of the fading statistics of the time-varying channel into the scheduler’s expression via the Lagrange multiplier ðsðtÞÞ. This accounts for the utility of a conditional energy constraint. In fact, we can say our approach is a trade-off between wireless-channel aware schedulers that do not account for the possibility of a direct physical layer/higher layer joint optimization ([28], [29], [30]), and intractable temporal unconditioned averages that cannot be handled as a simple Lagrange multiplier ðsðtÞÞ in order to design joint PHY-APP cross-layer schedulers. Furthermore, from the analysis of (7) and (8), we can recognize as the resulting optimal throughput/bandwidth solution ropt ð:; :Þ; opt ð:; :Þ does not need the full knowledge of p ðÞ, but only the knowledge of the cumulative (integral) measures expressed in (7) and (8). Therefore, under the assumed ergodic operation conditions, schedulers can work without any a priori knowledge of channel statistics and (7) and (8) can be computed via online time averages (see Section 3.2 for more details). Also, in Section 4, we will show how temporal considerations can be taken into account. Finally, we highlight that condition (9) presents the wellknown basic characteristic of a congestion controller [29], that is, the higher the congestion level (the buffer capacity), the lower the data generation rate.
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3.1
Structural Properties of the Optimal Scheduler in (6)-(9) The optimal scheduler in (6)-(9) retains some basic properties we go to detail. First, a direct examination of the righthand side (r.h.s) of (9) shows that the optimal controller shuts down the connection bandwidth (e.g., sets opt ð:; :Þ at zero) when the buffer is full (e.g., when sðtÞ in (9) equates Nmax and ropt ððtÞ; sðtÞÞ vanishes). This means that the optimal controller implements a flow-control policy that does not allow, indeed, overflow phenomena. As a consequence, under the optimal control policy, the queue system of Fig. 1 is loss free. Second, as a consequence of the monotonic behavior of the rate function Rð:; :Þ (see Section 2), it is quite direct to prove [25] that, for any assigned value s of the queue length, ropt ð:; :Þ in (6) is nondecreasing over 0, that is, ropt ð0 ; sÞ ropt ð00 ; sÞ;
for 0 > 00 :
ð10Þ
Third, after indicating by IU
opt
4 ðsÞ ¼ E ropt ð ; sÞjs ; s 0; ðIU=slotÞ
ð11Þ
the (optimal) number of sent IUs averaged over the channelstate pdf when the queue state is s, the following Proposition 2 holds (see Appendix B for the proof, which is available in the online supplemental material). opt
Proposition 2. IU ðsÞ in (11) is nondecreasing and concave for any s 0. Furthermore, it is strictly increasing for s s1 , and strictly concave for s > s1 .
3.2
On the Implementation Complexity and Adaptive Implementation of the Optimal Controller About the implementation complexity of the optimal controller, we note that ðsðtÞÞ in (8) and the threshold s1 in (7) can be computed offline on the basis of the system parameters and pdf p ðÞ of the channel state. Thus, the online implementation of ropt ððtÞ; sðtÞÞ in (6) can be accomplished via a simple three-way memoryless threshold detector, whose input is the value assumed (slot by slot) by the system state x ¼ ½; s. This means that the optimal controller we derived may be actually implemented without resorting to cumbersome DP-based iterative algorithms. Furthermore, a key property of the optimal schedulers in (6) and (9) is that they utilize the pdf p ðÞ of the link state only for the computation of the expectations at the left-hand side of (7) and (8). Therefore, since, under the assumed ergodic operating conditions, these expectations converge to the corresponding sample averages; approximations sb1 bðsðtÞÞ of s1 and ðsðtÞÞ can be computed by solving the and following window-based sample equations: t 1 X "ððjÞ; rp ððjÞ; sb1 ÞÞ ¼ E ave ; W j¼tW þ1
ð12Þ
and rp ððjÞ;sðtÞÞ ! t 1 X 1 1 ¼ E ave ; " ðjÞ; "r ðjÞ; bðsðtÞÞ 0 W j¼tW þ1 ð13Þ
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instead of (7) and (8), where due to ergodicity and i.i.d. assumption on channels states, we have limW !1 sb1 ¼ s1 and bðsðtÞÞ ¼ ðsðtÞÞ. We conclude that, by using the limW !1 sliding-window sample averages reported at the left-hand side of (12) and (13), the optimal schedulers can be implemented on-the-fly, i.e., without any a priori knowledge about p ðÞ. In fact, online implementation only requires that a measurement of the actual link state ðtÞ is available on the transmit node of Fig. 1. The numerical plots of Section 5.4 support the conclusion that (short) observation windows W of about 5 10 slot times suffice for the convergence of the above reported sample averages. Hence, the on-the-fly implementation of the proposed controller also allows it to quickly track a priori unknown time variations of the channel-state pdf p ðÞ.
4
CONDITIONAL-VERSUS-UNCONDITIONAL AVERAGE THROUGHPUT MAXIMIZATION AND PERFORMANCE BOUNDS
At a first glance, it may be somewhat surprising that (9) dictates only an upper bound on the optimal bandwidth. However, an examination of the (last part of the) proof reported in Appendix A, which is available in the online supplemental material, unveils that this behavior arises from the fact that, by assumption (see Section 2), arrivals dictated by opt ððtÞ; sðtÞÞ occur at the input of the queue of Fig. 1 at the end of slot t, while the corresponding IUs sent over the channel are drained from the output of the queue at the beginning of slot t. Therefore, the value assumed by opt ððtÞ; sðtÞÞ cannot influence the current delivered throughput ropt ððtÞ; sðtÞÞ (or its conditional average value (4)), but only future values of the buffer state (sðkÞ; k > t) and, consequently, future values of instantaneous and average throughput. As a consequence, the degree of freedom provided by inequality (9) can be exploited in order to accomplish further optimizations. To this end, Proposition 2 points out that the conditional average optimal throughput in (11) does not decrease for increasing queue-state values. Hence, it could be expected that the bandwidth should assume the maximum value allowed by the bound in (9), in order to fill the queue as much as possible and, then, maximize the resulting throughput averaged over both channel and queue states. By exploiting some basic results of the Theory of Ordered Statistics [23], the following Proposition 3 formally confirms this expectation:
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se sopt ;
ð15Þ
where the above inequality is in the Pareto sense (e.g., opt component wise). Therefore, since IU ðsÞ in (11) is nondecreasing (see Proposition 2), the statistical dominance in (15) allows the following developments: opt ðaÞ E;es fropt ð ; sÞg Es~ IU ðsÞ opt Esopt IU ðsÞ E;sopt ropt ð ; sÞ ;
ð16Þ
where (a) in (16) stems from (15). Overall, (16) proves that the controller in (6) and (14) maximizes the unconditional average throughput over the set of control policies compliant with the relationships in (6) and (9). u t Unless otherwise stated, in the rest of this paper, we refer as optimal controller to the scheduler that jointly allocates throughput and bandwidth according to (6) and (14).
4.1
On the Average Throughput of the Optimal Controller 4 Let r ¼ Efropt ð ; sÞg ðIU=slotÞ be the unconditional average throughput generated by the optimal controller (that is, the optimal throughput averaged over both channel and queuestate statistics). The following Proposition 4 points out its limit values and (statistical) behavior. Proposition 4. Let the client-server system of Fig. 1 work under the optimal control policy. Thus, 1.
For max > Nmax , the queue state identically equates Nmax , e.g., sðt þ 1Þ Nmax ; for any t;
ð17Þ
so that we have (see (11)) ropt IU
opt ððtÞ; sðtÞÞ minfðNmax sðtÞÞ þ ropt ððtÞ; sðtÞÞ; max g ð14Þ
Proof. Let opt ððtÞ; sðtÞÞ be the controller policy dictated by e (14), and let ððtÞ; sðtÞÞ be any other controller policy working according to (9). Furthermore, for the fixed 4 channel controller (6), let sopt ¼ ½sopt ð0Þ; sopt ð1Þ; . . . and 4 e s ¼ ½e sð0Þ; seð1Þ be the resulting queue-state sequences
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triggered by the above bandwidth policies. By fact, different queue-dependent bandwidth-management policies determine (together with delivered throughput and channel statistics) different unknown steady-state es ðsÞ, so that it is generally queue-length pdf popt s ðsÞ and p (analytically) hard to compare unconditional performances. However, in Appendix C, which is available in the online supplemental material, we prove that sopt statistically dominates se (see the Strassen Theorem, [23]). This means that the following relationship holds:
Proposition 3. The following bandwidth-management policy
maximizes the unconditional average throughput Efrð ; sÞg over the set of schedulers provided by Proposition 1, meeting the bound in (9).
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opt
ðNmax Þ:
ð18Þ
Furthermore, in this case, the optimal controller in (6) and (14) also maximizes the unconditional average throughput over the full set of schedulers meeting the constraints in (4.1-4.5). For max Nmax , the steady-state pdf of the queue state identically vanishes out of the closed interval ½max ; Nmax , e.g., ps ðsÞ ¼ 0 for any s 62 ½max ; Nmax ;
ð19Þ
so that we have IU
opt
ðmax Þ ropt IU
opt
ðNmax Þ:
ð20Þ
BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM... opt
Proof. 1.
Tq ¼ When max exceeds Nmax , (14) becomes opt ððtÞ; sðtÞÞ ¼ Nmax sðtÞ þ ropt ððtÞ; sðtÞÞ, as a consequence of sðtÞ ropt ððtÞ; sðtÞÞ (see (6)). Thus, (17) directly arises from the Lindley’s equation in (1). Furthermore, for the unconditional average throughput, we have 4
ropt ¼ E;s fropt ð ðtÞ; sðtÞÞg Es E fropt ð ðtÞ; sðtÞÞg ¼ Es fIU
opt
ðaÞ
ðsÞg IU
opt
ðNmax Þ;
where ðaÞ stems from (17). The above relationship proves the validity of (18). e Finally, let ½e rððtÞ; sðtÞÞ; ððtÞ; sðtÞÞ be a generic scheduler that allocates throughput and bandwidth according to all constraints in (4.14.5). For the corresponding unconditional average throughput, the following chain of inequalities holds: ðaÞ f rð ; sÞg ¼ Ees fE fe rð ; sÞgg ¼ Ees IUðsÞ Ees; fe ðbÞ opt opt ð21Þ Ees IU ðsÞ Ees IU ðNmax Þ IU
2.
opt
ðcÞ
ðNmax Þ ropt ;
where ðaÞ arises by observing that, by design, the scheduling policy in (6) maximizes the conditional throughput, ðbÞ is justified with the nondecreasing property of the conditional throughput (see Proposition 2), and finally, ðcÞ follows from (18). Thus, the inequality (21) proves that the optimal controller is throughput maximizing over the full set of schedulers compliant with the constraints in (4.1-4.5). After introducing (14) into (1), this last may be recast in the following form: sðt þ 1Þ ¼ minfNmax ; sðtÞ ropt ðsðtÞ; ðtÞÞ þ max g: ð22Þ Thus, being sðtÞ ropt ððtÞ; sðtÞÞ and max Nmax , (22) leads to max sðt þ 1Þ Nmax ;
ð23Þ
sopt opt
ðaÞ
¼
sopt ðbÞ Nmax ¼ ; opt ropt IU ðNmax Þ
397
ð24Þ
for max > Nmax , and ðcÞ
max IU
opt
ðNmax Þ
opt
Tq ¼
sopt
opt
ðaÞ
¼
sopt ðdÞ Nmax ; opt ropt IU ðmax Þ
ð25Þ
for max Nmax . Step ðaÞ in (24) and (25) arises by observing that, under steady-state operating conditions, ropt must equate opt . Step ðbÞ in (24) stems from (17) and (18), while steps ðcÞ and ðdÞ in (25) follow from the combined exploitation of (19) and (20). In a dual way, a joint utilization of the lower bound in (23) and the upper bound in (20) leads to step ðcÞ in (25). The above relationships require some comments. First, opt for max > Nmax , Tq only depends on Nmax (see (24)), while, for max Nmax , both Nmax and max play a role (see opt (25)). Second, the rate of increment of IU ðsÞ for growing values of the s-argument also depends on the actual pdf adopted for modeling the channel state (see (11)), and it resists closed-form analytical evaluation. As a matter of fact, (24) and (25) do not allow, indeed, to arrive at firm (e.g., general) conclusions about the ultimate effect of Nmax opt and max on Tq . The numerical plots of Section 5 confirm, indeed, these conclusions.
5
PERFORMANCE TESTS AND COMPARISONS
In this section, we test actual performance and robustness properties of the optimal controller of (6) and (14) in terms of unconditional average bandwidth opt , average queue length opt sopt , and average queue delay Tq . In the simulated environments, the channel state ð:Þ plays the role of instantaneous fading-affected Signal-to-Noise Ratio (SNR) measured (on a per-slot basis) at the client side of Fig. 1. In the carried out numerical tests, we model ð:Þ as a central -squared r.v. with four degrees of freedom, e.g., p ðÞ ¼ expðÞ, for 0 [27]. Furthermore, the rate function Rð; EÞ adopted to numerically evaluate the throughput performance of the simulated systems is the logarithmic one, e.g., Rð; EÞ logð1 þ EÞ; ðIU=slotÞ:
ð26Þ
that, in turn, proves the validity of (19). Finally, by exploiting the nondecreasing behavior of opt IU ðsÞ (see Proposition 2), from (23) we directly arrive at (20). u t
The latter is commonly employed to measure the so-called Shannon Capacity of transmission links impaired by Gaussian-distributed additive noise-plus-interference disturbances [27].
On the Average Queue Delay of the Optimal Controller opt opt Let Tq ðslotÞ, ðIU=slotÞ, and sopt ðIUÞ be the unconditional average queue delay, connection bandwidth, and queue length under the optimal control policy, respectively. Hence, an application of the Little’s formula together with a suitable exploitation of the relationships in (18) and (20) leads to the following expressions for the resulting average queue delay:
5.1 Behavior of the Optimal Scheduling Policy Figs. 2 and 3 report the behavior (in log scale) of the optimal schedulers in (6) and (14) for 0 and some values of the queue length s and the available average energy E ave (see (4.1)). An examination of these plots leads to four main conclusions. First, it is not guaranteed that the optimal scheduler is work conserving, thus meaning that it may refrain to transmit when the channel state is low (e.g., the fading is deep) and the queue length s is not too large.
4.2
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Fig. 2. Behavior (in log scale) of ropt ðs; Þ at E p ¼ 20 ðJouleÞ; Nmax ¼ 11 ðIUÞ; max ¼ 9 ðIU=slotÞ. Fig. 5. Average queue length-versus-average energy behavior for the application scenario of Section 5.2 at E p ¼ 21 ðJouleÞ and max ¼ 2:5 ðIU=slotÞ.
Fig. 3. Behavior (in log scale) of opt ðs; Þ at E p ¼ 20 ðJouleÞ; Nmax ¼ 11 ðIUÞ; max ¼ 9 ðIU=slotÞ.
Fig. 6. Average queue delay-versus-average energy behavior of the optimal controller for the application scenario of Section 5.2 at E p ¼ 21 ðJouleÞ and max ¼ 2:5 ðIU=slotÞ.
5.2
Fig. 4. Average bandwidth-versus-average energy behavior of the optimal controller for the application scenario of Section 5.2 at E p ¼ 21 ðJouleÞ and max ¼ 2:5 ðIU=slotÞ.
Second, the scheduler’s behavior tends to become (more) work conserving (e.g., the lower envelopes of the plots tend to shift on the left-hand side) when the queue length s grows and/or the available energy increases. Third, at fixed s, the rate of increment of ropt ð; sÞ and opt ð; sÞ with grows for increasing values of E ave . In turn, this implies that higher values of E ave allow ropt ð; sÞ and opt ð; sÞ to faster approach their limit values s. Fourth, the optimal scheduler tends to (fully) empty the queue only when the channel state is high (e.g., the fading is light).
Connection Bandwidth-versus-Queue Length Trade-Off As already remarked in Section 1, at the best of the authors’ knowledge, neither cross-layer nor layered adaptive control architectures are currently available in the open literature for the optimal (or even suboptimal) allocation of connection bandwidth, throughput, and transmit energy under all constraints we considered in (4.1-4.5). So, since planning meaningful and fair performance comparisons seems to be problematic, in this section, we focus on the unconditional average bandwidth and queue-length performance of the optimal controller and give insight on the resulting tradeoff. The plots of Figs. 4, 5, and 6 report the behaviors of opt opt , sopt , and Tq versus the available average energy E ave , respectively. These plots give also insight about the effects of the buffer capacity Nmax on the average performance of the optimal controller. Specifically, at max ¼ 2:5 (IU=slot) and for values of Nmax ranging from 3 to 100 (IUs), we experience a (quasi) flat behavior of ropt for E ave > 10 ðJouleÞ (see Fig. 4). On the contrary, when E ave < 10 ðJouleÞ, the impact of Nmax becomes noticeable. Specifically, low values of Nmax
BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM...
Fig. 7. Average bandwidth-versus-maximum allowed bandwidth for some values of the channel-state estimation error parameter for the application scenario of Section 5.3. E p ¼ 21 ðJouleÞ, E ave ¼ 10 ðJouleÞ, and Nmax ¼ 100 ðIUÞ are considered.
(e.g., Nmax around 3 7 ðIUsÞ) penalize the attained average throughput with respect to the case of high Nmax values (e.g., Nmax ranging from 10 to 100). Furthermore, larger Nmax values tend to induce larger average queue delays (see Fig. 6), so that a suitable trade-off among average queue delay and throughput should be, indeed, pursued.
5.3
Performance Robustness against Channel-State Estimation Errors To test the degrading effects induced on the average performance of the optimal controller by estimation errors possibly impairing the measured channel state, we have perturbed the actual channel-state sequence fðtÞg via a stationary zero-mean unit-variance Gaussian noise sequence fnðtÞg. Thus, the perturbed channel-state sequence fe ðtÞg is (slot by slot) generated according to the following relationship: pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ð27Þ eðtÞ ¼ ðtÞ þ 1 Ef2 ðtÞgnðtÞ; t 1; where the parameter 2 ½0; 1 is set so to control the desired average squared estimation error affecting eðtÞ. Specifically, ð=ð1 ÞÞ plays the role of (normalized) SNR affecting the channel estimate eðtÞ in (27), so that ¼ 1 is the error-free
Fig. 8. Average queue delay-versus-maximum allowed bandwidth for some values of parameter for the application scenario of Section 5.3. E p ¼ 21 ðJouleÞ, E ave ¼ 10 ðJouleÞ, and Nmax ¼ 100 ðIUÞ are considered.
399
Fig. 9. Average bandwidth-versus-average energy for the application scenario of Section 5.4. E p ¼ 21 ðJouleÞ, max ¼ 2:5 ðIU=slotÞ, and Nmax ¼ 100 ðIUÞ have been considered.
case, while lower values correspond to noisier channel estimates. Thus, after implementing the optimal controller in (6) and (14) for the same application scenario of Section 5.2, we have replaced fðtÞg by the perturbed sequence fe ðtÞg at the input of the optimal controller. The resulting numerically evaluated values for the unconditional average bandwidth and queue delay are reported in Figs. 7 and 8 for values ranging from 0.8 to 1. An examination of Fig. 7 shows that the decrement in the average bandwidth (e.g., the average throughput loss) induced by link-estimation errors is negligible at low/medium values of max (e.g., for values of max below 2:7 ðIU=slotÞ). However, when the maximum allowed bandwidth exceeds 2:7 3 ðIU=slotÞ, performance-loss induced by link-estimation errors becomes more noticeable, both in terms of average bandwidth and queue delay (see Figs. 7 and 8). In any case, an examination of these figures confirms that, even for values of the (%=ð1 %Þ) ratio as low as 5 6, the corresponding performance loss remains, indeed, limited up to about 15 percent of the corresponding error-free case.
5.4
Convergence Behavior of the On-the-Fly Implementation of the Optimal Controller and Performance Robustness against Time-Correlated Fading In order to test the convergence behavior of the on-the-fly implementation of the optimal controller, we have once a opt time evaluated the average performance sopt and Tq
Fig. 10. Average queue length-versus-average energy for the application scenario of Section 5.4. E p ¼ 21 ðJouleÞ, max ¼ 2:5 ðIU=slotÞ, and Nmax ¼ 100 ðIUÞ have been considered.
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[7]
experienced by the optimal scheduler in the application scenario of Section 5.2. Figs. 9 and 10 report the resulting average performance plots of the optimal controller (implemented on the basis of the actual p ðÞ pdf) and those of the corresponding on-the-fly versions. Both timeuncorrelated and exponentially time-correlated fading with correlation coefficient equal to 0.85 have been considered. As anticipated in Section 3.2, these plots support the performance robustness of on-the-fly implementation of the proposed controller, even for observation windows W as small as seven slot times. Furthermore, a comparison of the curves drawn in these figures also leads to the conclusion that performance of the proposed controller is (very) robust against the impairing effects induced by fading correlation, being the resulting performance loss limited up to 8 percent even for values of the average energy as small as 6 ðJouleÞ.
[14]
6
[15]
CONCLUSION
Most current transport protocols and standards have been designed pursuing layered application-agnostic approaches, so to guarantee durability and generality [1]. However, recent research gives evidence that cross-layer optimization could catalyze the development of new end-to-end transport protocols, with enhanced support for wireless multimedia applications. The optimal controller we developed complies with this new research line. In a nutshell, main features of the proposed controller are that: it is in closed form; it jointly allocates connection bandwidth, throughput, and transmit energy in an adaptive cross-layer fashion that forces information passing among APP, DL, and PHY layers; 3. it may be implemented on-the-fly; and 4. it also implements a flow-control policy that guarantees overflow-free (e.g., loss-free) working conditions. The carried out numerical tests support the performance robustness of the proposed controller against several system’s impairments, such as channel-estimation errors, time-correlated fading, and a priori unknown time variations of the channel-state pdf. 1. 2.
[8] [9] [10] [11] [12]
[13]
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
[26] [27]
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BACCARELLI ET AL.: JOINTLY OPTIMAL SOURCE-FLOW, TRANSMIT-POWER, AND SENDING-RATE CONTROL FOR MAXIMUM...
Enzo Baccarelli received the Laurea degree (summa cum laude) in electronic engineering, the PhD degree in communication theory and systems, and the PhD degree in information theory and applications from the Universita` degli Studi di Roma “La Sapienza,” Rome, Italy, in 1989, 1992, and 1995, respectively. He is currently with the INFOCOM Department, Universita` degli Studi di Roma “La Sapienza,” where he has been a full professor of data communication and coding since 2003. He has been the coordinator of the National Project Wireless 802.16 Multiantenna mEsh Networks (WOMEN), and the National Project Wireless multiplatform mimO active access netwoRks for QoS-demanding muLtimedia Delivery (WORLD). He is the author of more than 100 international IEEE publications. He has been an associate editor for the IEEE Communication Letters.
401
Nicola Cordeschi received the “Laurea degree” in communication engineering from the University of Rome “Sapienza” in 2004 (Summa cum Laude), and the PhD degree in information and communication engineering in 2008 from the INFOCOM Department of the same university. He is involved as a teaching assistant for several courses as information theory and coding, signals and systems, statistical signal processing, and broadband systems. Currently, he is a research assistant in the INFOCOM Department, where his activity is focused on dynamic radio resource allocation for multiantenna active access networks oriented to multimedia applications. The main research fields deal with the design and the optimization of high performing systems for wireless applications in centralized and decentralized environments based on multiantenna platforms, bandwidth adaptation mechanisms, and quality of service management. He is the author or coauthor of more than 47 publications. Tatiana Patriarca received the Laurea degree in communication engineering from the University of Rome “La Sapienza” in 2008 by defending a dissertation about joined buffer and sourcerate control over wireless channels. She is currently working toward the PhD degree in information and communication engineering at the INFOCOM Department of the “La Sapienza” University of Rome.
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