Utility Optimal Coding for Packet Transmission over Wireless Networks - Part I: Networks of Binary Symmetric Channels

Utility Optimal Coding for P ack et T ransmission o v er W ireless Netw orks – Part I : Netw orks of Binary Symmetric Channels K. Premkumar , Xiaomin Chen, a nd Douglas J. Leith Hamilton Institute, Nationa l Univ ersity of Ireland, Mayn ooth, Ireland E–mail: { Premkuma r .Karumbu, Xiaomin.Chen, Doug.Leith } @nuim.ie Abstract —W e consider multi–hop networks comprising Binary Symmetric Channels ( BSC s). The network carries unicast fl ows fo r multiple users. The utility of the network is th e sum of the utilities o f the flows, wher e the utility of each flo w is a conca v e function of its throughput. Giv en that the network capacity i s shared by the flows, ther e is a contention f or network reso urces like coding rate (at the physical layer), scheduling time (at th e MA C layer), etc., among the flows. W e propose a proportional fair transmission scheme that maximises the sum u tility of flo w throughputs subject to the ra te and the scheduli ng constraints. This is achieve d b y jointly optimising the pack et coding rates of all the flows through th e network . Index T erms —Bin ary symmetric channels, code rate selection, cross– layer optimisation, network utility maximisation, schedul- ing I . I N T RO D U C T I O N In a co mmunicatio n network, the network capacity is shared by a set of flo ws. There is a c ontention for resources among the flows, which lead s to many interesting p roblems. One such problem , is how to allocate th e resour ces optimally acr oss the (competing) flows, when the physical la yer is err o neous . Specifically , schedule/tran smit time for a flow is a resour ce that has to be optimally allocated among the competing flows. In this work, we p ose a n etwork utility ma ximisation problem sub ject to sche duling con straints that so lve a r esource allocation pr oblem. W e consider packet commun ication ov er multi–hop net- works comprising of B inary Symmetric Channels ( BSC s, [1]). The n etwork consists of a set of C ≥ 1 ce lls C = { 1 , 2 , · · · , C } which define the “interference domains” in the network. W e allow intra–cell inter ference ( i.e tr ansmissions by nodes w ithin the same cell interfere) but assume that there is no inter–cell interference. This captures, for example, common network arch itectures where nodes within a given cell use the same radio chann el while neig hbou ring cells u sing or thogon al radio channels. Within each cell, a ny two nodes are within the decodin g rang e of each other , and hence, can communicate with eac h other . The cells are interc onnected using m ulti– radio bridging no des to cre ate a multi–h op wir eless netw ork. A multi–radio bridging n ode i connecting the set of cells B ( i ) = { c 1 , .., c n } ⊂ C can be thought of as a set of n single This work is supported by Sci ence Foundat ion Ireland under Grant No. 07/IN.1/I901. a b c d 4 flow f 1 flow f 2 flow f 3 Fig. 1. An illustration of a wirele ss mesh network with 4 cells. Cells a , b , c , and d use orthogonal channe ls CH 1 , CH 2 , CH 3 , and CH 4 respect iv ely . Nodes 3, 5, and 6 are bridg e nodes . The bridge node 3 (resp. 5 and 6) is provi ded a time slice of ea ch of the channe ls CH 1 & CH 2 (resp. CH 2 & CH 4 for no de 5 a nd CH 2 & CH 3 & CH 4 for no de 6). Three flows f 1 , f 2 , and f 3 are considered. In this exa mple, C f 1 = { a, b } , C f 2 = { d, b, a } , and C f 3 = { c, d } . radio nodes, one in each cell, interco nnected b y a high–speed , loss–free w ired ba ckplane (see Figure 1). Data is tran smitted across this mu lti–hop network as a set F = { 1 , 2 , · · · , F } , F ≥ 1 of unicast flows. The rou te of each flow f ∈ F is given by C f = { c 1 ( f ) , c 2 ( f ) , · · · , c ℓ f ( f ) } , where the source no de s ( f ) ∈ c 1 ( f ) and the d estination node d ( f ) ∈ c ℓ f ( f ) . W e assume loo p–fre e flows (i.e. , no two cells in C f are same). Figur e 1 illustrates th is network setup. A scheduler assigns a time slice of duratio n T f ,c > 0 tim e units to each flow f that flows through cell c , subject to the constraint that P f : c ∈C f T f ,c ≤ T c where T c is the period of th e schedule in cell c . W e c onsider a perio dic schedu ling strategy in which, in each cell c , ser vice is giv en to th e flows in a round rob in fashion, an d that each flow f in cell c gets a time slice of T f ,c units in e very schedule. The scheduled tran smit times for flow f in source cell c 1 ( f ) define time slots for flow f . W e assum e that a new information packet arri ves in each time slot, which allows us to simplify the analysis by igno ring queueing. Information packets of each flow f at the source node S ( f ) consist of a block of k f symbols. Each packet of flow f is e ncoded in to c odewords of len gth n f = k f /r f symbols, with coding rate 0 < r f ≤ 1 . The code em ployed for encoding is discussed in Section II. W e requ ire sufficient transmit times at each c ell along rou te C f to allow n f coded sy mbols to b e tran smitted in every schedule period. Hence there is no q ueueing at th e c ells along the route of a flo w . Channel Model: T he channel in cell c for flow f is considered to be a binary symmetric channel ( BSC ) with th e cross–over probab ility (i.e., the probab ility of a bit error) being α f ,c ∈ [0 , 1] . The corresp onding transition probability ma trix is thu s given by H f ,c ( α f ,c ) =  1 − α f ,c α f ,c α f ,c 1 − α f ,c  . Thus, the end– to–end ch annel for flo w f is a cascaded channel (of ℓ f BSC s), which is a BSC , with the transition probab ility matrix H f ( α f ) = Q c ∈C f H f ,c ( α f ,c ) , th e cr oss– over p robab ility of which is given b y α f = X { x c ∈{ 0 , 1 } ,c ∈C f : P c ∈C f x c is odd } Y c ∈C f α x c f ,c (1 − α f ,c ) 1 − x c . Since, each tran smitted symb ol in a packet of a flow can, in general, take values from a 2 m = M –ary alphabet, there are m channe l uses of the BSC for every transm itted sym bol. Thus, the symbol error pro bability (for any m ≥ 1 ) is given by β f = 1 − (1 − α f ) m . Let th e Bernoulli ran dom variable E f [ i ] indicate the end –to–en d error of the i th cod ed symbol at the destinatio n in a code word of flow f . Note th at E f [ i ] s are indep endent an d identically d istributed (i.i.d .), an d that P { E f [ i ] = 1 } = β f = 1 − P { E f [ i ] = 0 } . In the channel model described, the channel p rocesses across time are independ ent copies o f the BSC s. This is realised in a wireless n etwork by mea ns o f an in terleaver of sufficient depth (af ter the channel e ncoder) , which interleaves the enco ded symbols. The interleaved symb ols see a fading channel (which is modelled as a chan nel with memo ry , e.g ., a Gilber t–Elliot chan nel [ 2]), but the de–interleaver ( before the channel decoder) brings back the original sequence of the en coded symbo ls, but interleaves the channel fades, the comb ined effect o f which can be mod elled as indepen dent channel processes across time. I n another work [3], we model the fading chan nel as a packet erasure channel (or a block fading channel), and obtain the optimal transmission stra tegy , which includes optima l interleaving of bits acr oss sch edules and the optimal coding rates. Letting e f ( r f ) denote the error probability that a packet fails to be decod ed, the expe cted numb er of in formation symb ols successfully receiv ed is S f ( r f ) = k f (1 − e f ( r f )) . Other things being equal, one expects th at decrea sing r f (i.e., in creasing the number of redund ant symb ols n f − k f ) decreases er ror probab ility e f , and so increases S f . Howe ver , since the network capacity is limited, and is shared by multiple flows, increasing the coded packet size n f 1 of flo w f 1 generally requires decr easing the packet size n f 2 for some other flo w f 2 . That is, incre asing S f 1 comes at the co st of decre asing S f 2 . W e are interes ted in und erstanding this trade–off, and in analysing the optimal fair alloca tion of coding rates amon gst users/flows . Contributions: Our main co ntribution is the an alysis of fairness in the allocation of c oding rates be tween users/flows competing fo r limited n etwork cap acity . I n pa rticular, we p ose a resource allocation problem in the utility–fair framew ork, and pro pose a scheme f or obtainin g th e pro portion al fair allocation of codin g rates, i.e. the allocatio n o f coding rates that m aximises P f ∈F log S f ( r f ) subject to n etwork capac ity constraints ( or schedulin g con straints). Specifically , at the physical layer, th e (chann el) coding rate of a flo w can be lowered ( to alleviate its channel erro rs) o nly at th e expense of increasing the co ding rates of othe r flows. Also, at the n etwork layer, the length of sch edules of ea ch flow should be ch osen in such a way that it maximises the n etwork u tility . Interestingly , we show in our p roblem formu lation tha t the codin g rate and the scheduling are tightly coupled . Also, we show that for a log (network) utility function (which typically gi ves pr oportio nal fair alloca tion of resources) the optim um rate allocation (in general) gives unequal air–times which is q uite different from the previously k nown result of proportion al fair allocation being th e same a s th at of equal air –time allocation ([4]). This problem , which we show in Section III, r equires solving a non–c on vex o ptimisation pro blem. Our w ork differs fr om the previous work o n network utility maximisation (see [5] an d the referenc es th erein) in the following manner . T o the b est of our kn owledge, this is the fir st work that co mputes the optim al coding rate f or a given scheduling (or capacity) constraints in the utility– optimal fram ew ork. The rest of th e paper is organised as follo ws. I n Sectio n II, we obtain a measure for the end –to–en d packet decoding error, and describe the throu ghput o f the network. In Section III, we formu late a n etwork utility maximisation p roblem subject to constraints on th e transmission sche dule lengths. W e obtain the optimum coding r ates for each flow in th e network in Section IV. In Section V, we pr ovide some simple examples to illustrate our results. Th e proof s of v arious Lemmas are omitted d ue to lack of space. I I . P AC K E T E R RO R P RO B A BI L I T Y W e r ecall th at each transmitted symbol of flo w f reaches the destination node erroneously with probability β f . Hence, to recover the information packets, we employ a block code at the source nodes (a con v olution al code with zero–p adding is also a blo ck code). Since an ( n, k , d ) cod e can correct up to ⌊ d − 1 2 ⌋ er rors, we are interested in employing a code with a large distance d . Thus, a natu ral cho ice is the class of (linear) maximum –distance separable (MDS) codes. MDS cod es o f rate k /n have the prop erty that it achieves the Singleton bou nd ([6]), d 6 n − k + 1 , (1) i.e., the minimu m distanc e b etween any two co dew ords d , in an MDS cod e is n − k + 1 . Thus, the m aximum n umber o f errors that an MDS code can cor rect is  d − 1 2  =  n − k 2  . It is well known that in the case of b inary signalling, only tr i vial MDS cod es exist. He nce, in this pap er , we con sider M = 2 m – ary alphabet, whe re m > 1 . Examples for MDS codes in the case o f non–b inary alphabets include Reed–Solomo n codes ([6]), an d MDS–c onv olutional cod es ([7]). In [7], the authors show th e existence of MDS–conv olutional cod es for any code rate. W e note here th at Reed–Solomo n codes can also cor rect burst erro rs, and h ence, is more suitable for wireless network s (which does not emp loy an interlea ver). A. Network Constr aints on Coding Rate Based on the mo dulation and the bandw idth a vailable at each cell c , a flo w f , which passes through it, can obtain a maximum feasible phy sical (PHY) rate of transm ission in bits per second th at the cell c c an supp ort. Let w f ,c be the PHY rate of tr ansmission of flo w f in cell c . For ea ch transmitted packet of flow f , e ach cell c ∈ C f along its ro ute must alloca te at least n f w f,c units of time to transmit the packet (or encoded block) where we recall that n f is the length of the code word. Let F c := { f ∈ F : c ∈ C f } be the set of flows that are ro uted throug h cell c . W e recall that the transmissions in any cell c are scheduled in a TDMA fashion, and hence, the total tim e required fo r tran smitting packets for all flows in cell c is given by P f ∈F c n f w f,c . Since, fo r cell c , th e transmission sch edule interval is T c units of time, the coding rates r f must satisfy the schedulability c onstraint P f ∈F c k f r f w f,c 6 T c . B. Err or Pr obability – Upper bound The symbol errors E f [1] , E f [2] , · · · , E f [ n f ] are i.i.d. Bernoulli rand om variables, and h ence, the pro bability of a co dew ord (or en coded packet) bein g d ecoded inco rrectly is gi ven by P n P n f i =1 E f [ i ] > n f − k f 2 o . W e observe that P n f i =1 E f [ i ] is a bin omial rando m variable, and hence, the probab ility o f deco ding err or can b e com puted exactly . How- ev er , the exact prob ability of error is not trac table fo r fur ther optimisation as the probab ility of error, which is a fun ction o f the codin g rate, is n either concave nor con vex. Hence, we p ose the pr oblem b ased o n the uppe r boun d on th e erro r p robability So, we obtain an upper boun d an d a lower bound for th e error probab ility . W e show that the boun ds are tight, and h ence, the problem of network utility maximisation can b e posed based on th e lower bo und on the error p robability . Lemma 1. An upper bo und for th e en d–to–en d p r o bability o f a packet d ecoding err or for flow f is bou nded by the following. e e f = P ( n f X i =1 E f [ i ] > n f − k f 2 ) ≤ exp  − k f r f I E f [1]  1 − r f 2 ; θ f  (2) =: e f ( θ f , r f ) . wher e θ f > 0 is the Chernoff–boun d p arameter and the function I Z ( x ; θ ) := θ x − ln( E  e θ Z  ) is called the rate function in lar ge deviations theory . C. Err o r Pr obab ility – Lower bound Lemma 2. The end–to–end pr obability of a packet de coding err or for flo w f is at least a s la r ge as e e f ≥  β f 1 − β f exp  − k f 1 − 2 x f H ( B ( x f ))  · exp  − k f 1 − 2 x f D ( B ( x f ) kB ( β f ))  (3) wher e B ( x ) is the Bernoulli distribution with parameter x , H ( P ) is th e entr o py of pr obability mass function (pmf) P , and D ( P kQ ) is the informa tion divergence between the pmfs P and Q . From the lower an d the upper bou nds for the prob ability o f packet decoding error, and for the optimal θ ∗ f (see Eqn. (15) in Section IV), we see that the expone nt o f the lower bo und is the same as that of th e uppe r bo und (Eq n. ( 15)) with a pre– factor . Th is motiv ates us to work with the lo wer bound e f as a ca ndidate to compute the utility o f flow f , which is giv en by ln( k f (1 − e f )) . W e recall that E f [1] is a Bern oulli random variable wh ich takes 1 with probability β f , and 0 with pro bability 1 − β f . Thus I E f [1]  1 − r f 2 ; θ f  = θ f  1 − r f 2  − ln  1 − β f + β f e θ f  . Let x f := 1 − r f 2 . Note that 0 6 x f < 1 2 . Ther efore, from Eqn. (2), e f ( θ f , x f ) := exp  − k f 1 − 2 x f  θ f x f − ln  1 − β f + β f e θ f   (4) I I I . N E T W O R K U T I L I T Y M A X I M I S A T I O N W e are intere sted in maximising the u tility of the network which is defined as the sum utility of flow throug hputs. W e conside r the log of throughpu t as the candidate for the utility function being mo tiv ated by the d esirable prop erties like pro portion al fairness that it possesses. W e define the following notation s: Chernoff–bound parame- ters θ := [ θ f ] f ∈F , co de rates r := [ r f ] f ∈F , an d x par ameters x := [ x f ] f ∈F (where we recall that x f = (1 − r f ) / 2 ). W e define th e network utility as e U ( θ , x ) := X f ∈F ln ( k f (1 − e f ( θ f , x f ))) = X f ∈F ln ( k f ) + X f ∈F ln (1 − e f ( θ f , x f )) . (5) The problem is to ob tain the op timum coding r ate parameter x ∗ and the op timum Ch ernoff–boun d parameter θ ∗ , which maximises the network utility . Since, k f , the size of infor ma- tion packets of each flow f is giv en, maximising the network utility is equiv alent to maximising U ( θ , x ) := X f ∈F ln (1 − e f ( θ f , x f )) . (6) Thus, we define the following p roblem P1: max θ , x U ( θ , x ) = X f ∈F ln (1 − e f ( θ f , x f )) subject to X f : c ∈C f k f (1 − 2 x f ) w f ,c ≤ T c , ∀ c ∈ C (7) θ f > 0 , ∀ f ∈ F x f ≤ λ f ∀ f ∈ F x f ≥ λ f ∀ f ∈ F (8) W e note that the Eqn. (7) en forces the network cap acity (or the n etwork schedulability ) con straint. The objective function U ( θ , x ) is separable in ( θ f , x f ) pair for e ach flow f . Importantly , th e compone nt of utility function f or each flow f giv en by ln (1 − e f ( θ f , x f )) is not jointly con cav e in ( θ f , x f ) . Howe ver , ln (1 − e f ( θ f , x f )) is concave in θ f (for any x f ), and in x f (for any θ f ). Hence, the ne twork u tility maximisation pr oblem P1 is not in the stan dard conve x optimisation framework. Instead, we po se the f ollowing problem , P2: max θ max x X f ∈F ln (1 − e f ( θ f , x f )) (9) subject to X f : c ∈C f k f (1 − 2 x f ) w f ,c ≤ T c , ∀ c ∈ C θ f > 0 , ∀ f ∈ F x f ≤ λ f ∀ f ∈ F x f ≥ λ f ∀ f ∈ F (10) In ge neral, the s olution to P2 need n ot be the same as the solution to P1 . Ho wever , in our pro blem, we sh ow that P2 achieves the solution of P1 . Lemma 3. . F or a function f : Y × Z → R that is concave in y and in z , but n ot jointly in ( y , z ) , the solution to the joint optimisation pr oblem for conve x sets Y and Z max y ∈Y ,z ∈Z f ( y , z ) (11) is the same a s max z ∈Z max y ∈Y f ( y , z ) , (12) if f ( y ∗ ( z ) , z ) is a concave function of z , wher e for each z ∈ Z , y ∗ ( z ) := arg max y ∈Y f ( y , z ) . W e n ote that f or each x f , the pro bability of er ror e f ( θ f , x f ) is conve x in θ f , and hen ce, ln(1 − e f ) is con cave in θ f . Thus, we first solve for th e o ptimum Chernoff bo und parameter θ ∗ which we d escribe in Section IV -A. After h aving solved fo r the o ptimum θ ∗ , we show in Section IV -B th at U ( θ ∗ ( x ) , x ) is a conc av e function of x . Hence, f rom Lemma 3 , the solution to problem ( P2 ) (the maximisation problem that separately obtains the optimum θ ∗ and optimum x ∗ ) is globally optimum. W e study the rate optimisation pro blem that obtains x ∗ in Section IV -C. I V . U T I L I T Y O P T I M U M R A T E A L L O C A T I O N A. Optimal θ ∗ Consider the f ollowing o ptimisation pro blem, for any giv en x ∈ [ λ f , λ f ] F . max θ X f ∈F ln (1 − e f ( θ f , x f )) (13) subject to θ f > 0 , ∀ f ∈ F W e no te that the objective f unction is separable in θ f s, and that e f is convex in θ f . He nce, th e prob lem defined in Eqn. (13), is a concave maxim isation proble m. W e recall that e f ( θ f , x f ) = exp  − k f 1 − 2 x f  θ f x f − ln  1 − β f + β f e θ f   . (14) The p artial derivati ve of e f with respec t to θ f is gi ven by ∂ e f ∂ θ f = e f · − k f 1 − 2 x f  x f − β f e θ f 1 − β f + β f e θ f  . Observe that β f e θ f 1 − β f + β f e θ f is an increasing function of θ f . Thus, if, for θ f = 0 , x f − β f 1 − β f + β f < 0 or x f < β f (equiv alently , r f > 1 − 2 β f ), the de riv ati ve is p ositiv e for all θ f > 0 , or e f is an increasing function of θ f . He nce, for x f < β f , the optimum θ ∗ f is ar bitrarily clo se to 0 which yields e f arbitrarily close to 1 . Thus, for error recov ery , fo r any end–to– end er ror probab ility β f , the co ding r ate shou ld be smaller th an 1 − 2 β f , in which case, we o btain the optim al θ ∗ f by equating the par tial deriv ativ e o f e f with respect to θ f to zero. i.e., β f e θ ∗ f 1 − β f + β f e θ ∗ f = x f or , e θ ∗ f = x f β f 1 − β f 1 − x f or , θ ∗ f = ln  x f β f  − ln  1 − x f 1 − β f  . The probability of error for a given x f and θ ∗ f ( x f ) is th en giv en by e f ( θ ∗ f , x f ) = exp  − k f 1 − 2 x f  x f ln  x f β f  + (1 − x f ) ln  1 − x f 1 − β f  = exp  − k f 1 − 2 x f D ( B ( x f ) ||B ( β f )  (15) B. A con vex optimisation fr amework to obtain optima l x ∗ f If ln(1 − e f ( θ ∗ f ( x f ) , x f )) is a con cave fun ction of x f , th en one can obtain the op timum x ∗ f using conve x optimisation framework. T o show the concavity of ln(1 − e f ( θ ∗ f ( x f ) , x f )) , it is sufficient to show that e f ( θ ∗ f ( x f ) , x f ) is conve x in x f . Define Λ f := ln  x f (1 − x f ) β f (1 − β f )  . No te that ∂ e f ∂ x f = − e f · k f Λ f (1 − 2 x f ) 2 ∂ 2 e f ∂ x 2 f =  e f · k f (1 − 2 x f ) 2  ·  k f (1 − 2 x f ) 2 Λ 2 f − 4Λ f 1 − 2 x f − 1 − 2 x f x f (1 − x f )  e f ( θ ∗ f ( x f ) , x f ) is conv ex if k f (1 − 2 x f ) 2 Λ 2 f ≥ 4Λ f 1 − 2 x f + 1 − 2 x f x f (1 − x f ) , or , 4(1 − 2 x f ) Λ f + (1 − 2 x f ) 3 x f (1 − x f )Λ 2 f ≤ k f Since, we consider x f > λ f , wh ere λ f = β f + ǫ f for some arbitrarily small ǫ f > 0 , we have 1 Λ 2 f 6 K 2 0 where 1 /K 0 := ln  λ f (1 − λ f ) β f (1 − β f )  , and henc e, a suf ficient co ndition for the con vexity of e f (and hence, the co ncavity o f ln(1 − e f ) ) is 4(1 − 2 x f ) Λ f + K 2 0 (1 − 2 x f ) 3 x f (1 − x f ) ≤ k f (16) The above condition is a con vex fun ction of x f , and we include this as a con straint in the p roblem form ulation. Thu s, e f ( θ ∗ f ( x f ) , x f ) is co n vex in x f , an d henc e, we obtain the optimal x ∗ f using con vex optimisation method . Also, fr om Lemma 3, the optima l coding rate r ∗ f = 1 − 2 x ∗ f is un ique and g lobally op timum. The minimum k f required to ensure con vexity of e f ( θ ∗ f ( x f ) , x f ) is com puted nu merically , and is tabulated below . T ABLE I M I N I M U M k f T H A T E N S U R E S C O N V E XI T Y O F e f ( θ ∗ f ( x f ) , x f ) β f minimum k f require d 0.1 6 0.01 10 0.001 33 0.0001 164 From the ab ove table, we see that the minimum packet size required to ensure con vexity is very small, an d in practice, the packet size k f is much larger than the minimum size re- quired. Hence , fo r all practical pu rposes, the optimal co de r ate problem is a c on vex problem. More impo rtantly , the constraint giv en b y Eqn. (1 6) is not an active constrain t. Howe ver , for the sake of c ompleteness, we inclu de this constraint in the problem d efinition below . C. Optima l Coding R ate r In this subsection, we o btain th e optima l coding rate using the optimal Chernoff–boun d p arameter v ector θ ∗ , by solving the follo wing network u tility maximisatio n problem max x X f ∈F ln  1 − e f ( θ ∗ f , x f )  (17) subject to X f : c ∈C f k f (1 − 2 x f ) w f ,c ≤ T c , ∀ c ∈ C x f ≤ λ f ∀ f ∈ F x f ≥ λ f ∀ f ∈ F 4(1 − 2 x f ) Λ f + K 2 0 (1 − 2 x f ) 3 x f (1 − x f ) ≤ k f ∀ f ∈ F (18) The objective function is separ able and concave, and h ence, can be so lved u sing Lagrangian relaxation me thod. Also, the constraint represented by Eqn. (1 8) is n ot an activ e constraint, and hence, ther e is no Lagrang ian cost to this constraint. W e note here that the codin g rate should be such that k f / (1 − 2 x f ) is an integer , and hence, obtaining x ∗ f is a discrete optimisation problem . This is, in general, an NP hard problem. Hence, we relax this constraint, and allow x f to take any real value in [ λ f , λ f ] . The Lagran gian function for the optimal rate problem is thu s L ( x , p , u , v ) = X f ∈F ln  1 − e f ( θ ∗ f , x f )  − X c ∈C p c   X f ∈F c k f (1 − 2 x f ) w f ,c − T c   + X f ∈F u f  x f − λ f  − X f ∈F v f  x f − λ f  Applying KKT condition, ∂ L ∂ x f | x ∗ f = 0 , we have − 1 1 − e f ∂ e f ∂ x f | x ∗ f = X c ∈C f p c w f ,c 2 k f (1 − 2 x ∗ f ) 2 + v f − u f = 2 k f (1 − 2 x ∗ f ) 2   X c ∈C f p c w f ,c   + v f − u f e f 1 − e f · k f Λ ∗ f (1 − 2 x ∗ f ) 2 = 2 k f (1 − 2 x ∗ f ) 2   X c ∈C f p c w f ,c   + v f − u f e f 1 − e f Λ ∗ f = 2   X c ∈C f p c w f ,c   + ( v f − u f )(1 − 2 x ∗ f ) 2 k f = λ f + ( v f − u f )(1 − 2 x ∗ f ) 2 k f where λ f := 2  P c ∈C f p c w f,c  and Λ ∗ f := ln  x ∗ f (1 − x ∗ f ) β f (1 − β f )  . If the optimal x ∗ f is either λ f or λ f , then it is u nique. If x ∗ f ∈ ( λ f , λ f ) , then u f = v f = 0 , and in this case (which is the most interesting case, and we consider o nly this case for the rest o f the paper), we ha ve e f 1 − e f · Λ ∗ f = λ f e f = λ f λ f + Λ ∗ f (19) exp − k f 1 − 2 x ∗ f D ( B ( x ∗ f ) kB ( β f )) ! = λ f λ f + Λ ∗ f k f 1 − 2 x ∗ f D ( B ( x ∗ f ) kB ( β f )) = ln  λ f + Λ ∗ f λ f  (20) In the above eq uation, both the LHS and the RHS are increasing in x ∗ f . Also, LHS is a strictly co n vex (increasing) function a nd RHS is a strictly concave (increasing) fun ction of x ∗ f . Hen ce, they in tersect at exactly on e point in the region ( β f , 0 . 5 ] which is the optimal x ∗ f for a g iv en L agrang ian price vector p . D. Sub –gradient Ap pr oach to Compute op timal p ∗ c In this section, we discuss the p rocedur e to o btain the optimal shadow costs o r the Lagrange variables p ∗ . The dual problem for the primal pr oblem defined in Eqn. (17) is gi ven by min p ≥ 0 D ( p ) , where th e dual function D ( p ) is given b y D ( p ) = max x X f ∈F ln(1 − e f ( x f )) + X c ∈C p c   T c − X f ∈F c k f (1 − 2 x f ) w f ,c   (21) = X f ∈F ln(1 − e f ( x ∗ f ( p ))) + X c ∈C p c   T c − X f ∈F c k f (1 − 2 x ∗ f ( p )) w f ,c   . (22) In the ab ove equ ation, e f ( x f ) denotes e f ( θ ∗ f ( x f ) , x f ) . Since the d ual fun ction (of a primal p roblem) is conve x, D is conv ex in p . Hence, we use a sub –gradien t m ethod to obtain the optimum p ∗ . Fr om Eq n. (21), for any x , D ( p ) ≥ X f ∈F ln(1 − e f ( x f )) + X c ∈C p c   T c − X f ∈F c k f (1 − 2 x f ) w f ,c   , and in particular, the dual fun ction D ( p ) is greater than that for x = x ∗ f ( e p ) , i.e., D ( p ) ≥ X f ∈F ln(1 − e f ( x ∗ f ( e p ))) + X c ∈C p c   T c − X f ∈F c k f (1 − 2 x ∗ f ( e p )) w f ,c   = D ( e p ) + X c ∈C ( p c − e p c )   T c − X f ∈F c k f (1 − 2 x ∗ f ( e p )) w f ,c   (23) Thus, a sub–grad ient of D ( · ) at any e p is given by the vector   T c − X f ∈F c k f (1 − 2 x ∗ f ( e p )) w f ,c   c ∈C . (24) W e obtain an iterative algo rithm based on s ub–g radient method that yields p ∗ , with p ( i ) being the Lagrangians at the i th iteration. p c ( i + 1) =   p c ( i ) − γ ·   T c − X f ∈F c k f (1 − 2 x ∗ f ( p ( i ))) w f ,c     + where γ > 0 is a sufficiently small stepsize, and [ f ( x )] + := max { f ( x ) , 0 } ensures that the Lagra nge mu ltiplier n ev er g oes negativ e. No te that the Lagrangian u pdates can be locally done, as each cell c is required to know o nly the rates x ∗ f ( p ( i )) of flows f ∈ F c . Thus, at the beginning of each iteration i , the flows choose th eir cod ing rates to 1 − 2 x ∗ f ( p ( i )) , a nd each ce ll computes its co st ba sed on th e r ates of flows th rough it. The updated costs alo ng the r oute of each flow are then fed back to the source no de to compute the r ate f or the next iteration. The Lag range multiplier p c can be vie wed as the cost of transmitting traffic th rough cell c . The amo unt of serv ice time that is available is given by ∆ = T c − P f ∈F c k f (1 − 2 x ∗ f ( p ( i ))) w f,c . When ∆ is positive and large, then the Lagrangian co st p c decreases rap idly (becau se D is conve x), and when ∆ is negativ e, th en the Lagran gian cost p c increases rapidly to make ∆ ≥ 0 . W e note that th e increase or decrease of p c between successive iteration s is prop ortional to ∆ , th e amo unt of service time av ailable. Thus, the sub–grad ient procedure provides a dynamic control scheme to balan ce the network load. W e explore the properties of the optimum rate parameter x ∗ f in Section IV -E. In Section V, we provide some examples that illustrate the o ptimum utility–fair r esource allocatio n. E. Pr operties o f x ∗ f W e are in terested in studying the b ehaviour of th e optimum coding rate r ∗ f = 1 − 2 x ∗ f , when the PHY r ate w f ,c and the packet size k f increases such that k f /w f ,c is alw ays a constant. Lemma 4 . r ∗ f = 1 − 2 x ∗ f ( k f ) is an inc r easing fu nction of k f (with th e PHY r ate w f ,c being pr oportion al to k f ). Lemma 4 is quite intuitive. For any giv en chann el err or β f , as the block (or packet) len gth incr eases, it is optimum to g o for a h igh rate cod e. In other word s, it is optimu m f or a flow to use as m uch schedu ling time as po ssible (i.e., use a large block length k f , a nd h ence, u se a high rate co de); h owe ver , the resources are shared among multiple flows, an d hen ce, we ask th e following question: “ what is the op timum share of the scheduling time ” that each flow should ha ve. Interesting ly , in our pr oblem fo rmulation , the optim um cod e rate parame ter x ∗ f also so lves this o ptimum sched uling times for each flo ws. It is interesting to ask the question of how lar ge the p ack et sizes k f be for optimum r esource allocatio n, and Lemma 4 provides a hin t to the solution. From Lemma 4, we und erstand the fo llowing: if there are two flo ws f 1 , f 2 , th rough a cell c (seeing the same chan nel conditio ns, i.e., β f 1 = β f 2 ) with w f 1 ,c > w f 2 ,c then it is optimu m for flo w f 1 to use a large packet size k f 1 and flow f 2 to use a small packet size k f 2 . The o ptimum schedule len gth will be to allocate less sched ule time to flo w f 1 and m ore schedu le time to flow f 2 . In th e asymptotic case when w f ,c and k f gr ows to ∞ (and k f gr ows lin early with w f ,c , we see f rom Eqn. (20) th at the er ror exponent also goes to ∞ (as 1 − 2 x f > 0 ) , and hence, e f → 0 . In this case, we see that th e o ptimum rate can approach arbitrarily close to 1 − 2 β ∗ f . Thus, for any k f and w f ,c , the optimum co ding rate r ∗ f < 1 − 2 β ∗ f Previous studies on o ptimum resource allocation establish that the propo rtional fair allocation is the sam e as equal air–time allocation ([4]). But, in this pr oblem, we see an interesting ph enomen on that is unu sual of a propor tional–fair resource alloc ation. Lemma 5. The op timum rate allocation x ∗ (or equ ivalently r ∗ ) is not eq uivalent to eq ual air–time allocation which is a b 3 flow f flow f 1 Fig. 2. Cells with equal traf fic load typically the solution of a pr oportional– fair (o r ln utility) allocation . In par ticular, we see that th e flows that see a better c hannel get less air–times than the flows that see a worse chann el. This ph enomen on is evident in the case o f infin itely lon g co de words; with other p arameters being same, the air–times of flows in a cell c are propor tional to 1 1 − 2 β f,c , an d hen ce, flows with s mall β get less air–times. V . E X A M P L E S In this Section, we analyse some simple networks based on the utility optimu m solution that we obtained. In particular, we a nalyse the so–called parking –lot topology often used to explore fairness issues. I t is to be n oted that th e park ing–lot topolog y is a simple case of a line network, and the results of this sectio n extends in a simple way to a linear n etwork. A. Examp le 1: T wo cells with eq ual traffic load W e begin by considering the example shown in Figu re 2 consisting of two cells a and b having three nodes 1, 2, a nd 3. Each cell has the same symbol error pr obability β and the schedule length T . Th ere are three flo ws f 1 , f 2 , a nd f 3 , with two of the flows f 1 and f 3 having one–ho p r outes C f 1 = { b } and C f 3 = { a } , and one flow f 2 having a tw o–ho p rou te C f 2 = { a, b } . Each flow has the same inform ation packet size k and PHY transmit r ate, i.e. w f ,c = w . The en d–to– end packet err or pro bability exper ienced by the two–hop flow f 2 is greater th an that experie nced by the one hop flows f 1 and f 3 , since each ho p has the sam e fixed error probab ility . Hence, we need to assign a lower coding rate r f 2 to flow f 2 than to flows f 1 and f 3 in order to obtain the same err or pro bability (a fter deco ding) ac ross flows. However , when operatin g at the bou ndary of the network cap acity region (thereby maximising through put), decreasing the coding rate r f 2 of the two–h op flow f 2 requires that the coding rate o f both one– hop flows f 1 and f 3 be incre ased in order to r emain within the available n etwork capacity . In this sense, allo cating coding rate to the two–hop flow f 2 imposes a g reater margin al cost on the n etwork (in terms of the su m–utility) than the o ne– hop flo ws, and we expe ct th at a fair allocation will therefore assign h igher co ding rate to the two–hop flow f 2 . Th e solutio n optimising this trade–o ff in a pro portiona l fair man ner can be understoo d using the analysis in the previous section. In this example, both the cells are equally lo aded and, b y symmetry , the L agrange multiplier s p a = p b . Hence, λ f 1 = a b 3 flow f flow f 1 Fig. 3. Cells with unequal traf fic load λ f 2 2 = λ f 3 . Note that x ∗ f 2 < x ∗ f 1 and Λ ∗ f 2 < Λ ∗ f 1 . Hence, we find fr om Eqn. (19) that e f 1 e f 2 = λ f 1 λ f 2 λ f 2 + Λ ∗ f 2 λ f 1 + Λ ∗ f 1 < 1 . B. Examp le 2: T wo cells with u nequa l traffic load W e consider the same network as in the pr evious example, but now with only the flows f 1 and f 2 (i.e., th e flow f 3 is not p resent, see Figu re 3) in the network. In this example, c ell b carries two flo ws while cell a carries o nly on e flo w . The encodin g r ate co nstraints are gi ven by 1 r f 2 ≤ wT k , (from cell a ) , 1 r f 1 + 1 r f 2 ≤ wT k , (from cell b ) . Since, both r f 1 and r f 2 are at most 1, it is clear th at at the optimum point, the rate co nstraint o f cell a is not tig ht while the constraint of cell b is tight. Th us, the shadow prices (Lagran ge mu ltipliers) p a = 0 and p b > 0 . That is, at the first hop the cell is not operating at cap acity , and so the “price” f or using th is cell is zero. In this example, λ f 1 = λ f 2 , and hence , f rom Eqn. (19), we d educe that for low channe l errors, e f 1 ≈ e f 2 . This allo cation make sense intuitively since although flow f 2 crosses two hop s, it is only constrained at th e second h op and so it is natu ral to share th e available capacity of this second h op appro ximately equally b etween the flows. V I . C O N C L U S I O N S In this paper, we p osed a utility fair pr oblem th at yields the op timum codin g across flows in a capacity constra ined network. W e showed th at the problem is highly non–conve x. Howe ver , we p rovided some simp le cond itions under wh ich the glo bal n etwork u tility optim isation p roblem ca n be solved. W e obtained the optimum coding rate, and analysed some of its proper ties. W e also an alysed som e simple networks based on the utility optimum framework we propo sed. T o the b est of our knowledge, this is the first work o n cro ss–layer optimisation that studies op timum coding across flows which are competing for n etwork resource s. 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