Decomposition Principles and Online Learning in Cross-Layer Optimization for Delay-Sensitive Applications

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1 Techncal Report Decomposton Prncples and Onlne Learnng n Cross-Layer Optmzaton for Delay-Senstve Applcatons Abstract In ths report, we propose a general cross-layer optmzaton framework n whch we explctly consder both the heterogeneous and dynamcally changng characterstcs of delay-senstve applcatons and the underlyng tme-varyng network condtons. We consder both the ndependently decodable data unts (DUs, e.g. packets) and the nterdependent DUs whose dependences are captured by a drected acyclc graph (DAG). We frst formulate the cross-layer desgn as a non-lnear constraned optmzaton problem by assumng complete knowledge of the applcaton characterstcs and the underlyng network condtons. The constraned cross-layer optmzaton s decomposed nto several cross-layer optmzaton subproblems for each DU and two master problems. These two master problems correspond to the resource prce update mplemented at the lower layer (e.g. physcal layer, AC layer) and the mpact factor update for neghborng DUs mplemented at the applcaton layer, respectvely. The proposed decomposton method determnes the necessary message exchanges between layers for achevng the optmal cross-layer soluton and t explctly consders how the cross-layer strateges selected for one DU wll mpact ts neghborng DUs as well as the DUs that depend on t. However, the attrbutes (e.g. dstorton mpact, delay deadlne etc) of future DUs as well as the network condtons are often unknown n the consdered real-tme applcatons. The mpact of current cross-layer actons on the future DUs can be characterzed by a state-value functon n the arkov decson process (DP) framework. Based on the dynamc programmng soluton to the DP, we develop a low-complexty cross-layer optmzaton algorthm usng onlne learnng for each DU transmsson. Ths onlne optmzaton utlzes nformaton only about the prevous transmtted DUs and past experenced network condtons. Ths onlne algorthm can be mplemented n real-tme n order to cope wth unknown source characterstcs, network dynamcs and resource constrants. Our numercal results demonstrate the effcency of the proposed onlne algorthm. Keywords- Cross-layer optmzaton, delay-senstve applcatons, wreless multmeda transmsson, decomposton prncples, onlne optmzaton.

2 Techncal Report I. INTRODUCTION To maxmze ts utlty, a wreless user needs to jontly optmze the varous protocol parameters and algorthms avalable at each layer of the OSI stack. Ths jont optmzaton of the transmsson strateges at the varous layers s referred to as cross-layer optmzaton [][2]. A. Related research Cross-layer optmzaton has been extensvely nvestgated n recent years n order to maxmze the applcaton s utlty gven the underlyng tme-varyng and error-prone network characterstcs. For nstance, cross-layer optmzaton solutons for sngle-lnk communcatons [3][4][6], ad-hoc networks [7][8], and cellular networks [9] have been proposed. The majorty of cross-layer optmzaton solutons can be dvded nto two man categores: Statc approaches, n whch the network condtons and applcaton characterstcs are descrbed usng statc models (.e. whch reman unchanged over tme), and the goal of the cross-layer optmzaton s to maxmze a certan utlty gven such a statc envronment. Such solutons, ncludng network utlty maxmzaton (NU) [0] (and the references theren), do not explctly consder and account for the tme-varyng source characterstcs and network condtons, thereby resultng n suboptmal performance for the delay senstve applcatons (e.g. wreless multmeda streamng) consdered n ths report. Sequental approaches, n whch the tme-varyng network condtons (e.g. channel condtons at the physcal layer, allocated tme/frequency bands at the AC layer etc.) and applcaton characterstcs (e.g. packet arrvals, delay deadlnes, dstorton mpact etc.) are explctly modelled as (controlled) stochastc processes, and the goal s to sequentally determne the cross-layer actons over tme to control ths stochastc process such that the long-term utlty s maxmzed [4][7]. The most mportant advantage of such sequental approaches s that they allow the wreless users to consder the experenced source and network dynamcs (whch are affected by both the uncertanty n the envronment and the actons chosen by the wreless user) and, based on the users knowledge about these dynamcs up to that moment, select ther cross-layer transmsson strateges to maxmze ther 2

3 Techncal Report utlty over tme. These solutons can sgnfcantly mprove the transmsson performance of delaysenstve applcatons n tme-varyng wreless networks, as compared to the statc approaches. However, current approaches consder smple models for both the tme-varyng applcaton characterstcs and dynamc network condtons whch cannot satsfy the requrements of the delaysenstve applcatons as explaned below. Based on the network dynamcs and decson granulartes n dfferent layers, most sequental approaches for wreless transmsson can be further classfed nto two categores: flow-based transmsson decsons and DU-based transmsson decsons. In the flow-based decson used n e.g. [3][4], the applcaton data s assumed to be homogeneous (.e. havng the same dstorton mpact and same delay deadlnes), and the network condtons are assumed to be tme-varyng (e.g. the network condtons are tme-slotted and changes across the slots). The goal of the flow-based approaches s to optmze the average or worst case qualty of servce (QoS), e.g. average/worst case packet delay, packet loss rate, bt rate etc., for the supported applcatons. However, snce the heterogeneous attrbutes of the packets n terms of delay deadlnes and dstorton mpacts etc. are gnored, the flow-based approaches often result n suboptmal utltes for the delay-senstve applcatons [24]. In DU-based transmsson scenaros [][5], each DU can contan one packet or multple packets. Each DU s characterzed by ts dstorton mpact (e.g. the decrease n the applcaton qualty when that DU s lost), ts packet length, the tme at whch the DU s ready for transmsson and ts delay deadlne. For example, n vdeo streamng applcatons, the DU can be one frame or one group of pctures, whch may comprse multple packets []. The decson s made for each DU to select the optmal transmsson strateges across multple layers such that the total qualty of the applcaton (e.g. the Peak Sgnal-to- Nose Rato (PSNR) for multmeda streamng) s maxmzed. In [6], the optmal packet schedulng algorthm (.e. DU-based) s developed for the transmsson of a group of packets to mnmze the consumed energy, whle satsfyng ther common delay deadlne. Ths optmal soluton s obtaned by assumng that the nter-arrval tme and delay deadlnes of the packets are known a pror. Ths soluton also assumes that the underlyng channel condtons are the same for all the packets. Ths packet 3

4 Techncal Report schedulng algorthm s further extended to the case n whch each packet has ts own delay constrants n [5]. In [6], the authors further consder tme-varyng (tme-slotted) channel condtons. However, the above papers do not consder the heterogenety of the packets n terms of dstorton mpact on the supported applcatons (e.g. vdeo streamng) etc. In [], the vdeo packets wth varous characterstcs are scheduled consderng a common delay deadlne and an optmal soluton (ncludng optmal packet orderng and retransmsson) s developed assumng that the underlyng wreless channel s statc. In [5], a DAG model s used to capture the meda packet dependences and, based on ths, an optmal packet schedulng method s developed usng dynamc programmng [3]. However, the proposed soluton dsregards the dynamcs and error protecton capabltes at the lower layers (e.g. AC and physcal layers). Summarzng, a general cross-layer optmzaton framework whch smultaneously consders both the heterogeneous and dynamcally changng DUs attrbutes of delay-senstve applcatons and the underlyng tme-varyng network condtons s stll mssng. In ths report, we am to develop a soluton that addresses both of these challenges for the delay-senstve applcatons such as multmeda transmsson. B. Contrbuton of ths report We consder a DU-based approach, and assume that the cross-layer decsons are performed for each DU. We consder both the ndependently decodable DUs (.e. they can be decoded ndependently wthout requrng the knowledge of other DUs) and the nterdependent DUs (.e. n order to be decoded, each DU requres those DUs t depends on to be decoded beforehand and these dependences are expressed as a DAG). We frst formulate a non-lnear constraned optmzaton problem by assumng complete knowledge of the attrbutes (ncludng the tme ready for transmsson, delay deadlnes, DU sze and dstorton mpact and DAG-based dependences) of the applcaton DUs and the underlyng network condtons. The formulatons n [5][6][][6] are specal cases of the framework proposed n ths report. Ths s the case, for nstance, when the multmeda data was pre-encoded and hntng fles were created before transmsson tme [24]. However, n the real-tme encodng case, these attrbutes are known just n tme when the packets are deposted n the streamng buffer, whch wll be consdered n Secton V. 4

5 Techncal Report The constraned cross-layer optmzaton can be decomposed nto several subproblems and two master problems as shown n Fgure. We refer to each subproblem as Per-DU Cross-Layer Optmzaton (DUCLO) snce t represents the cross-layer optmzaton for one DU. For the nterdependent DUs, the DUCLOs are solved teratvely n a round-robn style. One master problem s called the Prce Update (PU), whch corresponds to the Lagrange multpler (.e. prce of the resource) update assocated wth the consdered resource constrant mposed at the lower layer (e.g. energy constrant); and the other master problem s called Neghborng Impact Factor Update (NIFU), whch s mplemented at the applcaton layer. The NIFU corresponds to the update of the Lagrange multplers (called Neghborng Impact Factors, NIFs) assocated wth the DU schedulng constrants between neghborng DUs 2. It s clear that the decson granularty s one DU for DUCLO, two neghborng DUs for the NIFU, and all the DUs for the PU, as shown n Fgure. Fgure. The decomposton of the cross-layer optmzaton and correspondng nformaton update The DUCLO problem for each DU s further separated nto two optmzatons: an optmzaton to determne the optmal schedulng tme 3, whch ncludes the tme at whch the transmsson should start and t should be nterrupted; and an optmzaton to determne the correspondng optmal transmsson strateges at the lower layers (e.g. energy allocaton at the physcal layer, DU retransmsson or FEC at the AC layer). In ths report, we often refer to the applcaton layer as the upper layer, whle referrng to the physcal layer, AC layer, network layer (or a combnaton of these layers) as the lower layer(s). As we wll show n ths report, the proposed decomposton provdes necessary message exchanges between 2 These are consecutve packets generated by the source codec n the encodng/decodng order. 3 The schedulng tme s forwarded to the lower layer (e.g. the AC layer) such that ths layer can nterrupt the transmsson of the current packet and move to the next packet. A packet should be nterrupted ether because the DU s delay deadlne has expred or because the next DU has hgher precedence for transmsson than the current DU due to ts hgher dstorton mpact. 5

6 Techncal Report layers and llustrates how the cross-layer strateges for one DU mpact ts neghborng DUs and the DUs t connects wth n the DAG. In delay-senstve real-tme applcatons, the wreless user s often not allowed or cannot know the attrbutes of future DUs and correspondng network condtons. In other words, t only knows the attrbutes of prevous DUs, and past experenced network condtons and transmsson results. The message exchange mechansm developed based on the decomposton of the non-lnear optmzaton s nfeasble snce t requres exact nformaton about future DUs. However, when the dstrbuton of the attrbutes and network condtons of DUs fulfl the arkov property [23], the cross-layer optmzaton can be reformulated as a DP. Then the mpact of the cross-layer acton of the current DU on the future unknown DUs are characterzed by a state-value functon whch quantfes the mpact of the current DU s cross-layer acton on the future DUs dstorton. Usng the obtaned decomposton prncples developed for the onlne cross-layer optmzaton, we develop a low-complexty algorthm whch only utlzes the avalable (causal) nformaton to solve the onlne cross-layer optmzaton for each DU, update the resource prce and learn the state-value functon. Thus, the dfference between the methods proposed n ths report and those n [5][6][6][4] s that we explctly take nto account both the applcaton characterstcs and network dynamcs, and determne decomposton prncples for cross-layer optmzaton whch adheres to the exstng layered network archtecture and llustrates the necessary massage exchanges between layers over tme to acheve the optmal performance. The rest of the report s organzed as follows. Secton II formulates the cross-layer optmzaton problem for the ndependently decodable DUs as a non-lnear constraned optmzaton assumng the knowledge of the characterstcs of the supported applcaton and underlyng network condtons. Secton III decomposes the optmzaton problem and presents the necessary message exchanges between layers and between neghborng DUs. Secton IV further formulates the cross-layer optmzaton for nterdependent DUs as a non-lnear constraned optmzaton and presents the decomposed cross-layer optmzaton algorthm based on the decomposton prncples developed n Secton III. Secton V 6

7 Techncal Report presents an onlne cross-layer optmzaton for each DU transmsson. Secton VI shows some numercal results, followed by the conclusons n Secton VII. II. PROBLE FORULATION We assume that a wreless user streams delay-senstve data over a tme-varyng wreless network. We focus on the DU-based cross-layer optmzaton. Specfcally, the wreless user has DUs wth ndvdual delay constrants and dfferent dstorton mpacts. In ths secton, we consder that the DUs are ndependently decodable and wll dscuss the cross-layer optmzaton for the nterdependent DUs n Secton IV. The tme the DUs are ready for transmsson s denoted by t, =,,. The delay deadlne of each DU (.e. the tme before whch the DUs must be receved by the destnaton) s denoted by d, and thus, the followng constrant needs to be satsfed: d t. The DUs are transmtted n the Frst In Frst Out (FIFO) fashon (.e. the same as the encodng/decodng order). The sze of each DU s assumed to be l bts. Each DU also has the dstorton mpact q on the applcaton. Ths dstorton mpact represents the decrease on the qualty of the applcaton when the entre DU s dropped [][8]. Hence, each DU s assocated wth an attrbute tuple ψ = { q, l, t, d }. In ths secton and the subsequent two sectons, we assume that the attrbutes are known a pror for all DUs. In Secton V, we wll dscuss the case n whch the attrbutes of all the future DUs are unknown to the wreless user, as s the case n lve encodng and transmsson scenaros. Durng the transmsson, DU s delvered over the duraton from tme x to tme y ( y x ), where x represents the startng transmsson tme (STX) and y represents the endng transmsson tme (ETX). The choce of x and y represents the schedulng acton of DU, whch s determned n the applcaton layer. The schedulng acton s denoted by ( x, y ) satsfyng the condton of t x y d. At the lower layer (whch can be one of the physcal, AC and network layers or combnaton of them), the wreless user experences the average network condton c + durng the transmsson duraton. For smplcty, we assume that the average network condton s ndependent of the scheduled tme ( x, y ), whch can be the case when the network condton s slowly changng. The wreless user can deploy the 7

8 Techncal Report transmsson acton a A based on the experenced network condton. The set A represents the possble transmsson actons that the wreless user can choose. The transmsson acton at the lower layer can be, for example, the number of DU transmsson retry (e.g. ARQ) at the AC layer, and energy allocaton at the physcal layer. When the wreless user deploys the transmsson acton a under the network condton c, the expected dstorton of DU due to the mperfect transmsson n the network s represented by Q ( x, y, a ) = q p ( x, y, a ) 4, where p ( x, y, a ) can be the probablty that DU s lost as n [5] or the dstorton decayng functon 5 due to partal data of DU beng receved as n [8]. The resource cost ncurred by ts transmsson s represented by w( x, y, a ) +. In addton, we assume that the functons p ( x, y, a ) and w ( x, y, a ) satsfy the followng condtons: C (onotoncty): p ( x, y, a ) s a non-ncreasng functon of the dfference y x and the transmsson acton a. C2 (Convexty): p ( x, y, a ) and w ( x, y, a ) are convex functons of the dfference y x and the transmsson acton a. Condton C means that the expected dstorton wll be reduced by ncreasng the dfference y x, snce ths results n a longer transmsson tme whch ncreases the chance DU wll be successfully transmtted. In condton C2, the convextes of p and assumpton s satsfed n most scenaros, as wll be shown n Secton VI. w are assumed to smplfy the analyss. Ths Based on the descrpton above, the cross-layer optmzaton for the delay-senstve applcaton over the wreless network s to fnd the optmal schedulng acton (.e. determnng the STX x and ETX y for each DU) at the applcaton layer and, under the scheduled tme, the optmal transmsson acton a at the lower layer. The goal of the cross-layer optmzaton s to mnmze the expected average dstorton experenced by the delay-senstve applcaton. Ths cross-layer optmzaton may also be constraned on 4 We consder here that the dstorton of the ndependently decodable DUs s not affected by other DUs, as n [20]. 5 The dstorton decayng functon represents the fracton of the dstorton remaned after the (partal) data are successfully transmtted. For example, when the source s encoded n a scalable way, the dstorton functon s gven by D = Ke θr when R bts has been receved [8]. In θr( x, y, a ) ths case, the dstorton decayng functon s gven as p( x, y, a) = e and q = K. 8

9 Techncal Report the avalable resources at the lower layer (e.g. energy at the physcal layer). Then, the cross-layer optmzaton problem wth complete knowledge (referred to as CK-CLO) can be formulated as: mn Q( x, y, a ) x, y, a,, = = st.. x y, x t, y d, x+ y, a A, (CK-CLO) w( x, y, a) W. = where the constrant x+ y ndcates that DU + has to be transmtted after DU s transmtted (.e. FIFO), and the last lne n the CK-CLO problem ndcates the resource constrant n whch W s the average resource budget (e.g. the avalable energy for transmsson). III. DECOPOSITION FOR CROSS-LAYER OPTIIZATION In ths secton, we dscuss how the cross-layer optmzaton n the CK-CLO problem can be decomposed usng dualty theory [2], what nformaton has to be updated among DUs at each layer and what messages have to be exchanged across multple layers. Such decomposton prncples are mportant for developng optmal cross-layer solutons, because t adheres to the current layered network archtecture. A. Lagrange dual problem We frst relax the constrants n the CK-CLO problem by ntroducng the Lagrange multpler λ 0 T assocated wth the resource constrant and Lagrange multpler vector μ = [ μ,, ] 0, whose μ elements are assocated wth the constrant x+ y,. The correspondng Lagrange functon s gven as L( xya,,, λ, μ ) = Q( x, y, a) + λ w( x, y, a) W + μ( y x+ ), () where x = [ x,, x ], y = [ y,, y ] and a = [ a,, a ]. Then, the Lagrange dual functon s gven by = = = g( λ, μ ) = mn Q( x, y, a ) + λ w( x, y, a ) W + μ( y x+ ) x, y, a, = = = =,, st.. x y, x t, y d, a A, =,, The dual problem (referred to as CK-DCLO) s then gven by (2) 9

10 Techncal Report max g ( λ, μ ) λ 0, μ 0 (CK-DCLO) where μ 0 denotes the component-wse nequalty. The CK-DCLO dual problem can be solved usng the subgradent method as shown next. The subgradents of the dual functon are gven by hλ = w( x, y, a ) W wth respect to the varable λ and hμ = ( y x + ) wth respect to the varable μ [2]. The CK-DCLO problem can then be teratvely solved usng the subgradents to update the Lagrange multplers as follows. Prce-Updatng: and NIF Updatng: where z + = max { z,0} and α k and = (3) k+ k k λ = λ + α w( x, y, a) W = k β k+ k k μ = ( μ + β ( y x )), (4) + + are the update step sze and satsfy the followng condtons: k k α =, ( α ) 2 k k < and β, ( ) 2 = β < 6. The proof of convergence s gven n [2]. k= k= k= k= From the subgradent method, we note that the Lagrange multpler λ s updated based on the consumed resource and avalable budget, whch s nterpreted as the prce of the resource and t s determned at the lower layer, whle the Lagrange multpler vector μ s updated based on the schedulng tme of the neghborng DUs, whch s nterpreted as the neghborng mpact factors and s determned at the applcaton layer. The update s also llustrated n Fgure 2, and the detals of ths fgure are presented subsequently. Snce the CK-CLO problem s a convex optmzaton, the dualty gap between the CK-CLO and CK-DCLO problems s zero, whch s further demonstrated n Secton VI. Based on the multpler update gven n Eqs. (3) and (4), we can make the followng remark, whch s essental for mplementng practcal cross-layer desgns. Remark : The update of the Lagrange multplers λ and μ can be performed separately n the dfferent layers, thereby automatcally adherng to the layered network archtecture. + 6 These condtons are requred to enforce the convergence of the subgradent method. The choce of convergence and performance obtaned. One example s α k = β k = / k. k α and k β trades off the speed of 0

11 Techncal Report Fgure 2. essage exchange between layers and between neghborng DUs B. Decomposton for Lagrange dual functon Gven the Lagrange multplers λ and μ, the dual functon shown n Eq. (2) s separable and can be decomposed nto DUCLO problems: DUCLO problem {,, } : λ mn Q( x, y, a ) + w( x, y, a ) μ x + μy x, y, a st.. x y, x t, y d, a A (5) where μ 0 = 0 and μ = 0. Gven the Lagrange multplers λ and μ, each DUCLO problem s ndependently optmzed. From Eq. (5), we note that all the DUCLO problems share the same Lagrange multpler λ, snce the budget constrant at the lower layer s mposed on all the DUs (see Fgure 2). We also note that each DUCLO problem shares the same Lagrange multpler μ wth DUCLO problem and μ wth DUCLO problem + (see Fgure 2). Compared to the tradtonal myopc algorthm n whch each DU s transmtted greedly wthout consderng ts mpact on future DUs (e.g. flow-based approaches), the DUCLO problems presented here automatcally take nto account the mpact of the schedulng for the current DU on ts neghbours. Remark 2: The mpact between the ndependently decodable DUs takes place only through the Lagrange multplers λ and μ. Hence, we can separately fnd the cross-layer actons for each DU by estmatng the Lagrange multplers λ and μ, whch wll be used n the onlne mplementaton dscussed n Secton V.

12 Techncal Report C. Layered Soluton to the DUCLO problem In ths secton, we descrbe how the DUCLO problem can be separated nto two layered subproblems and what messages should be exchanged between layers. Gven the Lagrange multplers λ and μ, the DUCLO n Eq. (5) can be rewrtten as λ mn{ mn { Q( x, y, a ) + w( x, y, a )} μ x + μy} x, y a st.. x y, x t, y d, A (6) The nner optmzaton n Eq. (6) s performed at the lower layer and ams to fnd the optmal transmsson acton * a, gven STX x and ETX y. Ths optmzaton s referred to as LOWER_OPTIIZATION: λ f ( x, y ) = mn Q( x, y, a ) + w( x, y, a) a A The LOWER_OPTIIZATION requres the nformaton of the schedulng tme ( x, y ), dstorton mpact q and DU sze l whch are obtaned from the upper layer and the nformaton of transmsson actons a and prce of resource λ, whch are obtaned at the lower layer. The outer optmzaton n Eq. (6) s performed at the upper layer and ams to fnd the optmal STX x and ETX y, gven the soluton to the lower optmzaton n Eq. (7). Ths optmzaton s referred to as the UPPER_OPTIIZATION: mn f ( x, y) μ x + μy x, y st.. x y, x t, y d, The UPPER_OPTIIZATION requres the nformaton of f ( x, y ), whch can be nterpreted as the best response to ( x, y ) performed at the lower layer, and nformaton of μ and μ whch are obtaned at the upper layer. Hence, gven the message { q, l, x, y }, the LOWER_OPTIIZATION can optmally provde a * and the best response functon f ( x, y ). Gven the functon f ( x, y ), the UPPER_OPTIIZATION tres to * * fnd the optmal STX x and ETX y. Ths message exchange s llustrated n Fgure 2. Snce Q( x, y, a ) and w( x, y, a ) are convex functons of the dfference y x and a, the LOWER_OPTIIZATION and UPPER_OPTIIZATION are both convex optmzaton problems and (7) (8) 2

13 Techncal Report can be effcently solved usng well-known convex optmzaton algorthms such as the nteror-pont methods [2]. Remark 3: Ths layered soluton for one DU provdes the necessary message exchanges between the upper layer and lower layer, and llustrates the role of each layer n the cross-layer optmzaton. Specfcally, the applcaton layer works as a gude whch determnes the optmal STX and ETX by takng nto account the best response f ( x, y ) of the lower layer, whle the lower layer works as a follower, whch only needs to determne the best response f ( x, y ), gven the schedulng tme ( x, y ) determned by the upper layer. In summary, the algorthm for solvng the CK-CLO problem s llustrated n Algorthm. Algorthm : Algorthm for solvng the CK-CLO problem for the ndependently decodable DUs 0 0 Intalze λ, μ, λ, μ, ε, k = k k k k Whle ( λ λ + μ μ > ε or k = ) For =,, Layered soluton to DUCLO for DU End k k Compute λ +, μ + as n Eqs. (3) and (4). k k + End IV. CROSS-LAYER OPTIIZATION FOR INTERDEPENDENT DUS In ths secton, we consder the cross-layer optmzaton for nterdependent DUs. The nterdependences can be expressed usng a DAG. One example for vdeo frames s gven n Fgure 3. (ore examples can be found n [5]). Each node of the graph represents one DU and each edge of the graph drected from DU to DU represents the dependence of DU on DU. Ths dependency means that the dstorton mpact of DU depends on the amount of successfully receved data n DU. We can further defne the partal relatonshp between two DUs whch may not be drectly connected, for whch we wrte f DU s an ancestor of DU or equvalently DU s a descendant of DU n the DAG. The relatonshp means that the dstorton (or error) s propagated from DU to DU. 3

14 Techncal Report The error propagaton functon from DU to DU s represented by e ( x, y, a ) [ 0,] 7 whch s assumed to be a decreasng convex functon of the dfference y of DU can be computed as x and a. Then, the dstorton mpact Q( x, y, a ) = q q ( p( x, y, a )) ( ek ( xk, yk, ak )). (9) If DU cannot be decoded because one of ts ancestor s not successfully receved and p ( x, y, a ) represents the loss probablty of DU, then e ( x, y, a ) = p ( x, y, a ) as n [5]. k Fgure 3. DAG example wth IBPBP vdeo compressed frames The prmary problem of the cross-layer optmzaton for the nterdependent DUs s the same as n the CK-CLO problem by replacng Q ( x, y, a ) wth the formula n Eq. (9). The dfference from the CK- CLO problem s that Q ( x, y, a ) here depends on the cross-layer actons of ts ancestors and Q ( x, y, a ) may not be a convex functon of all the cross-layer actons ( x, y, a ) k, although k k k e ( x, y, a ) s a convex functon of ( x, y, a ). However, we note that, gven ( x, y, a ) k, k k k k k k k k k k Q ( x, y, a ) s a convex functon of ( x, y, a ). We wll use ths property to develop a dual soluton for the orgnal non-convex problem and we wll quantfy the dualty gap n the smulaton secton. The dervatve of the dual problem s the same as the one n Secton III. By replacng Q ( x, y, a ) wth the formula n Eq. (9), the Lagrange dual functon shown n Eq. (2) becomes g ( λ, μ ) = mn x, y, a, =,, ( ( ) ( ) ) ( ( )) ( ) ( ) { q q p x, y, a ek xk, yk, ak w x, y, a W y x + λ + μ + } = k = = st.. x y, x t, y d, a A, =,,.(0) 7 In general, the error propagaton functon e (,, ) x y a of DU also depends on whch DU t wll affect [20]. For smplcty, we assume the error propagaton functon only depends on the current DU and does not depend on the DU t wll affect. In ths report, to smplfy the analyss, we do not consder the mpact of error concealment strateges. Such strateges could be used n practce, and ths wll not affect the proposed methodology for cross-layer optmzaton. 4

15 Techncal Report Due to the nterdependency, ths dual functon cannot be smply decomposed nto the ndependent DUCLO problems as shown n Eq. (5). However, the dual functon can be computed DU by DU assumng the cross-layer actons of other DUs s gven, as shown n [5]. Specfcally, gven the Lagrange multplers λ, μ, the objectve functon n Eq. (0) s denoted as G( ( x, y, a ),, ( x, y, a ), λ, μ ). When the cross-layer actons of all DUs except DU are fxed, the DUCLO for DU s gven by where mn G( ( x, y, a),, ( x, y, a),, ( x, y, a ), λ, μ) x y, x t, y d, a A λ = mn ( Q ( x, y, a) + w( x, y, a ) μ x + μy ) + θ x y, x t, y d, a A Q ( x, y, a ) = qp( x, y, a ) ( ek ( xk, yk, ak )) k, (2) ( e( x, y, a )) q ( p ( x, y, a )) ( ek ( xk, yk, a k )) k k and θ represents the remanng part n Eq. (0), whch does not depend on the cross-layer acton ( x, y, a ). It s easy to show that the optmzaton over the cross-layer acton of DU n Eq. () s a convex optmzaton, whch can be solved n a layered fashon as shown n Secton III.C. As dscussed n [5], Q ( x, y, a ) can be nterpreted as the senstvty to (or mpact of) the mperfect transmsson of DU,.e. the amount by whch the expected dstorton wll ncrease f the data of DU s fully receved, gven the cross-layer actons of other DUs. It s clear that the DUCLO for DU s solved only by fxng the cross-layer actons of other DUs, unlke the solutons for the ndependently decodable DUs whch do not requre the knowledge of other DUs. Then, the optmzaton n Eq. (0) can be solved usng the block coordnate descent method [2], as n n n n n n descrbed next. Gven the current optmzer (( x, y, a ),, ( x, y, a )) at teraton n, the optmzer at n+ teraton n +, (( n+ n+ n+,, ),,( n+, n+ x y a x y, a )) s generated accordng to the teraton n+ n+ n+ ( x y a ),, = arg mn x y, x t, y d, a A n+ n+ n+ n+ n+ n+ n n n n n n ((,, ),, (,, ),(,, ),(,, ),, (,, ),, μ ) G x y a x y a x y a x y a x y a λ () (3) 5

16 Techncal Report At each teraton, the objectve functon s decreased compared to that of the prevous teraton and the objectve functon s lower bounded (greater than zero). Hence, ths block coordnate descent method converges to the locally optmal soluton to the optmzaton n Eq. (0), gven the Lagrange multplers λ and μ. In summary, the algorthm for solvng the CK-CLO problem for the nterdependent DUs s llustrated n Algorthm 2. Remark 4: From Eq. (), we note that, when we focus on the cross-layer optmzaton for DU, besdes the resource prce λ and NIF μ and μ as requested for the ndependently decodable DU, we further need some addtonal nformaton: the nterdependences wth other DUs (expressed by the DAG) and the values of p ( x, y, a ) and e ( x, y, a ) of all DUs k connected wth DU. For real-tme applcatons, k k k k k k k k the nformaton of future DUs s often unavalable when DU s transmtted. We show n Secton V how ths nformaton can be estmated onlne. Algorthm 2: Algorthm for solvng the CK-CLO problem for nterdependent DUs 0 0 Intalze λ, μ, λ, μ, ε, k = // for outer teraton k k k k Whle ( λ λ + μ μ > ε or k = ) Intalze : x, y, a, =,,,,δ, n =. // for nner teraton Whle ( Δ> δ or n = ) For =,, Layered soluton to DUCLO for DU as n Eq. (3). End n n n k k n n n k k Δ= G( ( x, y, a ), =,,, λ, μ ) G( ( x, y, a ), =,,, λ, μ ). n+ n+ n+ n n n ( x, y, a ) ( x, y, a ), =,,. n n + End k k Update λ +, μ + as n Eqs. (3) and (4). k k + End V. ONLINE CROSS-LAYER OPTIIZATION WITH INCOPLETE KNOWLEDGE The cross-layer optmzaton formulated n Sectons II and IV assumes complete a-pror knowledge of the DUs attrbutes and the network condtons. However, n real-tme applcatons, ths knowledge s only avalable just before the DUs are transmtted. Furthermore, the cross-layer optmzaton algorthms based on the decomposton prncples presented n Sectons III and IV requre multple teratons (as 6

17 Techncal Report shown n Sectons VI.B and VI.C) to converge, whch may be dffcult to mplement for real-tme applcatons. To deal wth the real-tme transmsson scenaro, we propose a low-complexty onlne crosslayer optmzaton algorthm motvated by the decomposton prncples developed n Sectons III and IV. A. Onlne optmzaton usng learnng for ndependent DUs In ths secton, we assume that the DUs can be ndependently decoded and that the attrbutes and network condtons dynamcally change over tme. The random versons of the tme the DU s ready for transmsson, delay deadlne, dstorton mpact and network condton are denoted by T, D, L, Q, C, respectvely. We assume that both the nter-arrval nterval (.e. T+ T) and the lfe tme (.e. D T ) of the DUs are..d. The other attrbutes of each DU and the experenced network condton are also..d. random varables ndependent of other DUs. We further assume that the user has an nfnte number of DUs to transmt. Then, the cross-layer optmzaton wth complete knowledge presented n the CK-CLO problem becomes a cross-layer optmzaton wth ncomplete knowledge (referred to as ICK-CLO) as shown below: N mn lm E Q( x, y, a ) x, y, a, N N T, D, L, Q, C = st..max( y, T ) x y D, a A, (ICK-CLO) N lm E w( x, y, a ) W N N T, D, L, Q, C = The optmzaton n the ICK-CLO problem s the same as the CK-CLO problem except that the ICK-CLO problem mnmzes the expected average dstorton for the nfnte number of DUs over the expected average resource constrant. However, the soluton to the ICK-CLO problem s qute dfferent from the soluton to the CK-CLO problem. In the followng, we wll frst present the optmal soluton to the ICK- CLO problem, and then we wll compare ths soluton wth that of the CK-CLO problem. Fnally, we wll develop an onlne cross-layer optmzaton for each DU. ) DP formulaton of the cross-layer optmzaton for nfnte DUs Smlar to the dual problem presented n Secton III, the dual problem (referred to as ICK-DCLO) correspondng to the ICK-CLO problem s gven by the followng optmzaton. 7

18 Techncal Report where g ( λ ) s computed by the followng optmzaton. max g ( λ), (ICK-DCLO) λ 0 N g( λ) = mn lm E ( Q( x, y, a ) w( x, y, a )) W x max ( y, T), y D, a,, N N + λ λ, (4) A Ψ, C = where the Lagrange multpler λ s assocated wth the expected average resource constrant, whch s the same as the one n Eq. (). Once the optmzaton n Eq. (4) s solved, the Lagrange multpler s then updated as follows: N + k+ k k λ = λ α lm E w( x, y, a ) W + N N. (5) T, D, L, Q, C = Hence, n the followng, we focus on the optmzaton n Eq. (4). From the assumpton presented at the begnnng of Secton V.A, we note that T+ T, D T and other attrbute of DU are..d. random varables. Hence, for the ndependently decodable DUs, f we know the value of T, the attrbutes and network condtons of all the future DUs (ncludng DU ) are ndependent of the attrbutes and network condtons of prevous DUs. As shown n Fgure 4, DU wll mpact the cross-layer acton selecton of DU only through ETX y snce x = max ( y, t ). In other words, DU brngs forward or postpones the transmsson of DU by determnng ts ETX y. If we defne a state for DU as s = max ( y t,0). Then, the mpact from prevous DUs s fully characterzed by ths state. Knowng the states, the cross-layer optmzaton of DU s ndependent of the prevous DUs. Ths observaton motvates us to model the cross-layer optmzaton for the tmevaryng DUs as a DP [3] n whch the state transton from state s to state s + s determned only by the ETX y of DU and the tme t + DU + s ready for transmsson,.e. s = max ( y t,0). + + The acton n ths DP formulaton s the STX x, ETX y and the acton a. The STX s automatcally set x = max ( y, t ). The mmedate cost by performng the cross-layer acton s gven by Q ( x, y, a ) + λw ( x, y, a ). Gven the resource prce λ, the optmal polcy (.e. the optmal cross-layer acton at each state) for the optmzaton n Eq. (4) satsfes the dynamc programmng equaton [3], whch s gven by, C 8

19 Techncal Report V( s) = E max [ Q( x, y, a) λw( x, y, a) V( max ( y T, 0) )] + + β DL,, Q, CT, x=+ s t y< D a A V s V 0 where V( s ) represents state-value functon at state s and the dfference ( ) ( ) represents the (6) total mpact that the prevous DU mpose on all the future DUs by delayng the transmsson of the next DU by s seconds; t s the tme the current DU s ready for transmsson; and β s the optmal average cost. It s easy to show that V( s ) s a non-decreasng functon of s because the larger the state s, the larger the delay n transmsson of the future DUs, and therefore the larger the dstorton. There s a well-known relatve value teraton algorthm (RVIA) [3] for solvng the dynamc programmng equaton n Eq. (6), whch s gven by { } ( ) V ( ) n+ s = E max [ Q( x, y, a) + λw( x, y, a) + Vn( max ( y T, 0) )] Vn 0 D, Q, LC,, T x= s+ t, y< D, a A where V () n s the state-value functon obtaned at the teraton n. (7) Fgure 4. State of DU and state transton from DU to DU + 2) Comparson of the solutons to CK-CLO and ICK-CLO In ths secton, we dscuss the smlarty and dfference between the solutons to the CK-CLO and ICK-CLO problems. We note that both solutons are based on the dualty theory and solve dual problems nstead of the orgnal constraned problems. Hence, both solutons use the resource prce to control the amount of resource used for each DU. In the CK-CLO problem, the soluton s obtaned assumng complete knowledge about the DUs attrbutes and the experenced network condtons, whch s not avalable for the ICK-CLO problem. Hence, n the DUCLO for the CK-CLO problem, the mpact on the neghborng DUs s fully characterzed by scalar numbers μ and μ. The cross-layer acton selecton for each DU s based on the assumpton that the cross-layer actons for neghborng DUs (prevous and future DUs) are fxed. However, n the RVIA for the ICK-CLO problem, the cross-layer acton selecton for each DU s based 9

20 Techncal Report on the assumpton that the cross-layer actons for the prevous DUs are fxed (.e. the sate s s fxed) and the future DUs (and the cross-layer actons for them) are unknown. The mpact from the prevous DUs s characterzed by the state s and the mpact on the future DUs s characterzed by the state value functon V( s ). Hence, the soluton to the CK-CLO problem cannot be generalzed to the onlne DUCLO whch has no exact nformaton about the future DUs. However, the soluton to the ICK-CLO problem can be easly extended to the onlne cross-layer optmzaton for each DU, snce t takes nto account the stochastc nformaton about the future DUs once t has the state value functon V ( s ). In the next secton, we wll focus on developng the learnng algorthm for updatng the state-value functon V( s ). 3) Onlne cross-layer optmzaton usng learnng In ths secton, we develop an onlne learnng to update the state-value functon ( ) V s and the resource prce λ. Assume that, for DU, the estmated state-value functon and resource prce are denoted by V s and λ, then the cross-layer optmzaton for DU + s gven by ( ) mn Q ( x, y, a ) + w ( x, y, a ) + V ( max ( y t,0)) x, y, a λ + st.. x = s + t, y d, a A Ths optmzaton can be solved as n Secton III.C. The remanng queston s how we can choose the rght prce of resource λ and estmate the state-value functon V ( ) s. From the theory of stochastc approxmaton [22], we know that the expectaton n Eq. (7) can be removed and the state-value functon can be updated as follows: (8) V+ ( s ) = ( γ ) V( s ) + γ { max [ Q( x, y, a ) + λw( x, y, a ) + V ( max ( y t+, 0) ) ] V( 0 )}, x= s, y< d, a A and V s = V s, f s s ( ) ( ) + (9) j = j <. We should note that, n ths proposed learnng algorthm, the j= j= where γ satsfes γ, ( γ ) 2 cross-layer acton of each DU s optmzed based on the current estmated state-value functon and resource prce. Then the state-value functon s updated based on the current optmzed result. Hence, ths learnng algorthm does not explore the whole cross-layer acton space lke the Q-learnng algorthm [26] 20

21 Techncal Report and may only converge to the local soluton. However, n the smulaton secton, we wll show that t can acheve the smlar performance as the CK-CLO wth = 0, whch means that the proposed onlne learnng algorthm can forecast the mpact of current cross-layer acton on the future DUs by updatng the state-value functon. Snce V ( s ) s a functon of the contnuous state s, the formula n Eq. (9) cannot be used to update state-value functon for each state. To overcome ths obstacle, we use a functon approxmaton method smlar to the work n [9] to approxmate the state-value functon by a fnte number of parameters. Then, nstead of updatng the state-value functon at each state, we use the formula n Eq. (9) to update the fnte parameters of the state-value functon. Specfcally, the state-value functon V ( s ) s approxmated by a lnear combnaton of the followng set of feature functons: V s ( ) K k k rv ( s) fs 0 k= (20) 0 ow.. where K r = r,, r s the parameter vector; ( ) ( ) K v s = v s,, v ( s ) s a vector functon wth each element beng a scalar feature functon of s [9]; and K s the number of feature functons used to represent the mpact functon. The feature functons should be lnearly ndependent. In general, the statevalue functon V( s ) may not be n the space spanned by these feature functons. The larger the value K, the more accurate ths approxmaton. However, the large K requres more memory to store the parameter vector. Consderng that the state-value functon V ( s ) s non-decreasng, we choose = K! ( ) K s v s s,, as the feature functons. Usng these feature functons, the parameter vector K r = r,, r s then updated as follows: k k r+ = ( γ ) r + { max [ (,, ) (,, ) ( max (, 0) ) (2) k γ Q x y a + λw x y a + V y t+ ] V( 0 )}/( Kv ( s )) x= s, y< d, a A Smlar to the prce update n Secton III, the onlne update for λ s gven as follows: 2

22 Techncal Report κj j j j j= j= γj where κ satsfes κ ( κ ) 2 λ = λ + κ W, (22) + wj j = =, <, lm = 0. + In Eqs. (2) and (22), teratng on the state-value functon V ( y ) and the resource prce λ at dfferent tmescales ensures that the update rates of the state-value functon and resource prce are dfferent. The resource prce s updated on a slower tmescale (lower update rate) than the state-value functon. Ths means that, from the perspectve of the resource prce, the state-value functon V ( y ) appears to converge to the optmal value correspondng to the current resource prce. On the other hand, from the perspectve of the state-value functon, the resource prce appears to be almost constant. The algorthm for the proposed onlne optmzaton usng learnng s llustrated n Algorthm 3. Algorthm 3: Proposed onlne optmzaton usng learnng Intalze λ, r = 0, s = 0, = For each DU Observe the attrbutes and network condton of DU and the tme t + at whch DU + s ready for transmsson; Layered soluton to the DUCLO gven n Eq. (8); Update s+ = max ( y t+, 0), λ + as n Eq. (22) and r + as n Eq. (2); + End B. Onlne optmzaton for nterdependent DUs In ths secton, we consder the onlne cross-layer optmzaton for the nterdependent DUs as dscussed n Secton IV. In order to take nto account the dependences between DUs, we assume that the DAG of all DUs s known a pror. Ths assumpton s reasonable snce, for nstance, the GOP structure n vdeo streamng s often fxed. When optmzng the cross-layer acton ( x, y, a ) of DU, the * * * * * * transmsson results pk( xk, yk, a k ) and k( k, k, k ) e x y a of DUs wth ndex k < are known. Then, the senstvty Q ( x, y, a ) of DU s computed, based on the current knowledge, as follows: * * * Q ( x, y, a) = qp( x, y, a) ( ek( xk, yk, ak )) ( e( x, y, a) ) q ( p ) ( ej ( xj, yj, a j )), (23) k j j 22

23 Techncal Report * * * where q ( p ) s the estmated dstorton mpact of DU. The term ek( xk, yk, a k ) s the error propagaton functon of DU k <, whch s already known. If j * * * <, ej ( xj, yj, aj ) = ej( xj, yj, aj ), otherwse e ( x, y, a ) = 0 by assumng that DU j can be successfully receved. In other words, f DU j j j j * * * * * * k s transmtted, the transmtted results pk( xk, yk, a k ) and k( k, k, k ) assumed to be successfully receved n the future. e x y a are used, otherwse DU k s Smlar to the onlne cross-layer optmzaton for ndependent DUs gven n Secton V.A, the onlne optmzaton for the nterdependent DUs s gven as follows: mn Q ( x, y, a ) + w ( x, y, a ) + V ( max ( y t,0)) x, y, a λ + st.. x = s + t, y d, a A The update of the parameter vector r and the resource prce λ s the same as n Eqs. (2) and (22). (24) VI. NUERICAL RESULTS In ths secton, we present our numercal results to evaluate the proposed decomposton method and the onlne algorthm. We consder an example n whch the user streams the delay senstve DUs over a tme-varyng channel wth energy constrants. A. odels for dstorton mpact and energy cost functons In ths example, we consder the proposed cross-layer optmzaton soluton to determne the optmal schedulng and energy allocaton for DUs wth varous attrbutes at the applcaton layer transmtted over a tme-varyng channel at the physcal layer. The transmsson acton s the number of bts, a, to be transmtted. The consumed energy (cost) s gven, as n [5], by a N 0 (,, ) y 2 x w x y a = ( y x) c, (25) where N 0 denotes thermal nose. It s easy to show that w( x, y, a ) s a convex functon of the dfference y x and a. We assume that the applcaton data s compressed n a scalable way [] such that, gven the amount of transmtted bts, a, the expected dstorton of the ndependent DU wth ndex s gven, as n [8], by 23

24 Techncal Report θ mn (, ) (,, ) 2 a l Q x y a = q, (26) θ mn (, ) where θ > 0. That s, (,, ) 2 a l p x y a =. It s easy to show that p( x, y, a ) s a convex functon of a. For nterdependent DUs, the expected dstorton of DU s then gven by θ mn (, ) (,, ) ( 2 k a k l Q x y a q k ) = k θ mn (, ) That s, (,, ) 2 a l ek xk yk ak = 8. The dstorton reducton for each DU s gven by q Q. (27) In ths example, the dstorton mpact q s the realzaton of a unformly dstrbuted random varable n the range of [ 50, 50 ]. The DU sze l s assumed to be constant and equals 0000bts. The varyng DU sze s consdered n Secton VI.F for vdeo streamng. The arrval nterval t t s the realzaton of an exponentally dstrbuted random varable wth the mean of 50 ms. The DU lfetme d t s 50 ms. The parameter θ equals 0.5. We wll verfy the effcency of the proposed methods usng the model developed n ths secton n Sectons B~ E. We wll further consder a more realstc scenaro wth vdeo streamng over wreless networks n Secton F. B. Dual and prmal solutons and dualty gap for ndependent DUs Fgure 5 (a) shows the dualty gap between the dual solutons and prmal solutons over 0 teratons n a settng wth = 0 ndependent DUs. It s shown that the dualty gap goes to zero after around 00 teratons, whch demonstrates that the subgradent algorthm developed n Secton III converges to the optmal total expected dstorton gven by the prmal solutons. Fgure 5 (b) further shows that the prmal and dual solutons are equvalent. However, the subgradent method requres around 00 teratons to converge to the optmal solutons, whch may be hard to mplement n the real-tme applcatons (e.g. vdeo streamng) snce t requres a lot of computaton. Hence, n Secton V, we have developed an onlne algorthm whch can sgnfcantly reduce the complexty of the cross-layer optmzaton (.e. one teraton) and only use the current avalable nformaton. The smulaton results for the onlne algorthms are presented n Secton VI.D. 8 Here the error propagaton functon represents the fact that ncreasng the facton of DU reduces the amount of error propagated to other DUs. 24

25 Techncal Report (a) (b) Fgure 5. (a) Dualty gap between the dual and prmal solutons for ndependent DUs; (b) Dual and prmal optmal schedulng tme for ndependent DUs C. Dual and prmal solutons and dualty gap for the nterdependent DUs Fgure 6 (a) shows the dualty gap between the dual solutons and prmal solutons for the nterdependent DUs wth = 0. Although the cross-layer optmzaton problem for the nterdependent DUs s not a convex optmzaton, t s shown here that the dualty gap n ths example goes to zero after around 230 teratons, whch demonstrates that the subgradent algorthm developed n Secton III also converges n the cross-layer optmzaton for nterdependent DUs. The subgradent algorthm for the nterdependent DUs requres two types of teratons: one s the outer teraton whch updates the prce of the resource λ and NIFs μ and the other one s the nner teraton whch s to fnd the optmal cross-layer acton for each DU gven λ and μ as shown n Eq. (3). Fgure 6 (b) shows the requred number of nner teratons per outer teraton usng the cross-layer actons obtaned n the prevous outer teraton as the startng pont n the current outer teraton. It s clear that 2~6 nner teratons are requred for each outer teraton to converge to the optmal cross-layer actons gven λ and μ. Hence, the subgradent method requres a total of 65 nner teratons, whch s unacceptable for the real-tme applcatons (e.g. vdeo streamng). As dscussed n Secton VI.B, ths motvates us to develop an onlne algorthm whch was presented n Secton V. The smulaton results for the onlne algorthm are presented n Secton VI.E. 25

26 Techncal Report (a) (b) Fgure 6. (a) Dualty gap between the dual and prmal solutons for nterdependent DUs, (b) Number of nner teratons per outer teratons for the cross-layer optmzaton of nterdependent DUs D. Onlne cross-layer optmzaton for ndependent DUs In ths smulaton, we consder three onlne algorthms for the scenaro wth ndependent DUs. The frst s the onlne cross-layer optmzaton for each DU proposed n Secton V. The second performs the cross-layer optmzaton every = 0 DUs by assumng complete knowledge of these DUs attrbutes and underlyng network condtons (we call ths -DU cross-layer optmzaton). The thrd one performs the cross-layer optmzaton for each DU (.e. =, called myopc onlne optmzaton). We wll refer to the transmsson of 0 DUs as one cycle. Fgure 7 depcts the dstorton reducton of each cycle under varous resource constrants for these three algorthms. From ths fgure, we note that, on the one hand, the onlne cross-layer optmzaton proposed n Secton V outperforms the myopc onlne optmzaton by around 6% for varous energy constrants because the proposed onlne optmzaton can predct the mpact on the future DUs through the state-value functon and allocate the energy for each cycle based on the mportance of DUs. On the other hand, the -DU cross-layer optmzaton outperforms the proposed onlne cross-layer optmzaton by around 4% snce -DU cross-layer optmzaton explctly consders the exact nformaton of future DUs whch s not avalable n the onlne cross-layer optmzaton. However, the proposed onlne crosslayer optmzaton has the followng advantages, compared to the DU cross-layer optmzaton: () t performs the cross-layer optmzaton for each DU and updates λ and state-value functon V( s ) for each 26

27 Techncal Report DU wthout requrng multple teratons, whch sgnfcantly reduces the computatonal complexty; () t does not requre exact nformaton about the future DUs attrbutes and network condtons. Fgure 7. The dstorton reducton under varous energy constrants for ndependent DUs E. Onlne cross-layer optmzaton for nterdependent DUs In ths smulaton, we also consder three onlne algorthms as descrbed n Secton VI.D for the scenaro wth nterdependent DUs. The nterdependences (represented by a DAG) are generated randomly every 0 DUs. The nterdependency between DUs happens only wthn one cycle (for nstance, a cycle could represent one group of pctures (GOP) of the vdeo sequences). Fgure 8 shows the dstorton reducton of each cycle under varous energy constrants. From ths fgure, we note that, for nterdependent DUs, our proposed onlne cross-layer optmzaton can sgnfcantly mprove the performance (more than 28% ncreased) compared to the myopc onlne optmzaton, and has smlar performance as the -DU optmzaton. We further show the dstorton reducton and energy allocaton for each cycle when the average energy constrant s 0 (.e. W = 0 ) n Fgure 9. From ths fgure, we observe that, after the ntal learnng stage (about 30 cycles), our proposed onlne soluton acheves the smlar performance as the -DU soluton. We wll also verfy ths observaton n a more realstc scenaro whch s presented n the next secton. The reason that our proposed soluton can have smlar performance as the -DU soluton s as follows: for the nterdependent DUs, the amount of the dstorton reducton s manly determned by the mportant DUs (on whch many other DUs depend on) 27

28 Techncal Report and our soluton can ensure that more mportant DUs are successfully transmtted by allocatng more energy to them. Fgure 8. Dstorton reducton under varous energy constrant for nterdependent DUs Fgure 9. (a) Dstorton reducton and (b) average energy consumpton for each cycle. F. Onlne cross-layer optmzaton for vdeo streamng In ths smulaton, we consder a more realstc stuaton n whch the wreless user streams the vdeo sequence Coastguard (CIF resoluton, 30 Hz) over the tme-varyng wreless channel. For the compresson of the vdeo sequence, we used a scalable vdeo codng schemes based on oton 28

29 Techncal Report Compensated Temporal Flterng (CTF) usng wavelets [25]. Such 3D wavelet vdeo compresson s attractve for wreless streamng applcatons because t provdes on-the-fly adaptaton to channel condtons, support for a varety of wreless recevers wth dfferent resource capabltes and power constrants, and easy prortzaton of varous codng layers and vdeo packets. We consder every 8 frames as one GOP and each DU corresponds to one frame at a certan temporal level, as shown n []. The dependency between DUs s llustrated n Fgure 0 (a). We compare three onlne optmzaton methods as n Secton VI.E. Fgure 0 (b) depcts the receved Peak Sgnal-to-Nose Rato (PSNR) n db under these methods. From ths fgure, we note that the myopc onlne optmzaton acheves the PSNR of 27.dB on average whch s generally consdered very poor vdeo qualty. However, our proposed onlne cross-layer optmzaton can mprove the vdeo qualty over tme through the learnng procedure and acheve the PSNR of 29.9 db (2.8dB better than the myopc soluton 9 ). oreover, the acheved vdeo qualty n our soluton s much smoother (.e. the PSNRs of all the frames do not vary dramatcally lke n the myopc case). We also demonstrated that the proposed soluton acheves the smlar performance (only 0.5dB less on average) as the -DU method, as ndcated n Secton VI.E. (a) (b) Fgure 0. (a) DAG for the nterdependency between DUs wth one GOP; (b) PSNR for the vdeo sequence coastguard under three cross-layer optmzaton methods 9 Note that t s well known that 0.5 db performance mprovement s vsble for a traned observer, db performance mprovement s vsble for any observer and 2dB of more results n sgnfcantly vsble performance mprovements. 29

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