Utility Optimal Coding for Packet Transmission over Wireless Networks - Part II: Networks of Packet Erasure Channels

Utility Optimal Coding for P ack et T ransmission o v er W ireless Netw orks – P art II: Netw orks of Pa cke t Erasure Chann els 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 define a class of multi–hop erasur e networks that approximates a wireless multi–hop network. T he network carries unicast flows f or multiple users, and each information packet within a flow is required to be decoded at the flow destination within a specified delay d eadline. The allocation of coding rates amongst flows/users is constrained by network capacity . W e propose a proportional fair transmission scheme th at maximises the sum u tility of flow throughputs. This is achieved by jointly optimising the pa ck et coding rates and th e alloca tion of bits of coded pack ets across transmission slots. Index T erms —Code rate selection, cross laye r optimisation, network utili ty maximisation, packet erasure chann els, schedul- ing I . I N T RO D U C T I O N In a co mmunicatio n network, t he ne twork capacity is shared by a set of flo ws. There is a co ntention for resources am ong the flows, which leads to many interesting pro blems. One such problem , is how to alloca te the res ources optimally acr oss the (competin g) flows, when the physical layer is err oneous . Specifically , schedule/tr ansmit time f or a flow is a resource that has to be optimally allocated among the competing flows. In this work, we pose a network u tility m aximisation problem sub ject to sched uling constrain ts that solve a resou rce allocation problem. In anoth er w ork, we studied the prob lem of op timal resour ce allocation in networks [1]. W e define a class of multi–h op erasure networks, and consider pa cket commu nication over this class. The network consists of a set of C ≥ 1 cells C = { 1 , 2 , · · · , C } wh ich define the “interfer ence domain s” in the ne twork. W e allow intra–cell interferen ce ( i.e tran smissions by nodes within th e same cell interfere) but assume that there is no inter–cell interferen ce. T his cap tures, for example, commo n network architecture s where n odes within a g i ven cell use the same radio channel while neighb ouring cells u sing orthogo nal r adio channels. W ithin each cell, any two nodes are within the decodin g r ange of e ach o ther, and he nce, can commu nicate with ea ch other . The cells ar e intercon nected using mu lti– radio bridging no des to create a multi–hop wireless network. A multi–r adio br idging node i conn ecting th e set of cells B ( i ) = { c 1 , .., c n } ⊂ C can b e thoug ht o f as a set o f n sing le radio nodes, o ne in each cell, in terconn ected b y a h igh–speed , loss–free wir ed back plane (see Figur e 1). This work is supported by Science Foundatio n 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 wir eless mesh network w ith 4 cells. Cells a , b , c , and d use orthogona l channels CH 1 , CH 2 , CH 3 , and CH 4 respect iv ely . Nodes 3, 5, and 6 are bridge nodes . The bridge node 3 (resp . 5 and 6) is provi ded a time slic e of each of the cha nnels CH 1 & CH 2 (resp. CH 2 & CH 4 for node 5 and CH 2 & CH 3 & CH 4 for node 6). Three flows f 1 , f 2 , and f 3 are considered. In this example, C f 1 = { a, b } , C f 2 = { d, b, a } , and C f 3 = { c, d } . Data is tran smitted across this mu lti–hop network as a set F = { 1 , 2 , · · · , F } , F ≥ 1 of unica st flo ws. The r oute of each flow f ∈ F is given by C f = { c 1 ( f ) , c 2 ( f ) , · · · , c ℓ f ( f ) } , where the source node s ( f ) ∈ c 1 ( f ) an d the destination node d ( f ) ∈ c ℓ f ( f ) . W e assume loop –free flows (i.e., no two cells in C f are sam e). Figures 1 an d 2 illustrate this network setup. A scheduler assign s a time slice of dur ation T f ,c > 0 tim e units to each flow f that flows th rough cell c , subject to th e constraint that P f : c ∈C f T f ,c ≤ T c where T c is the period of the schedule in cell c . W e con sider a p eriodic sch eduling strategy (see Figure 2) in which , in each cell c , service is given to the flows in a r ound robin fashion, and that each flow f in cell c gets a time slice o f 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 en coded into cod ew ords of leng th n f = k f /r f symbols, with coding rate 0 < r f ≤ 1 . The co de employed for encod ing is d iscussed in Section II. W e requ ire sufficient t ransmit time s at each cell alon g route C f to allow n f coded sy mbols to be tr ansmitted in every schedule period. He nce there is n o qu eueing at the cells along the route of a flow . It is not ap parent at this p oint w hether it is optim al for flow f to tran smit a single code –word of n f symbols or transmit a block of n f symbols wher e each blo ck carries some portion s o f each of a set of cod ed packets. Channel Model: The channel in cell c for flow f is considered to be a pac ket erasu re chan nel with the prob ability of p acket erasure b eing β f ,c ∈ [0 , 1] . Thus, the end –to– end ch annel for flow f is a packet erasure c hannel with the probab ility of packet erasur e being β f = 1 − Y c ∈C f [1 − β f ,c ] Let th e Bern oulli rand om variable E f [ i ] in dicate the end– to– end erasure seen b y the i th block of flow f (independ ent of the erasure seen by other blocks) o f flow f . Note that E f [ i ] = 1 means that the i th blo ck is erased, an d E f [ i ] = 0 mean s that the i th block is r eceiv ed successfully . Note that P { E f [ i ] = 1 } = β f = 1 − P { E f [ i ] = 0 } . Each packet has a d eadline of D f slots, by wh ich time it mu st be d ecoded. Suc h a d elay con straint is n atural in applications such as video streaming. A packet is in error if the destination fails to decode the pac ket by the d eadline. Lettin g e f ( r f ) deno te the e rror p robability that a packet fails to b e d e- coded b efore its deadline, th e expe cted n umber o f inf ormation symbols successfully received is S f ( r f ) = k f (1 − e f ( r f )) . Other th ings being eq ual, we expect that dec reasing r f (i.e., increasing the nu mber of coded symbols n f = k f /r f sent) decreases error pro bability e f and so increases S f . Ho wever , since the network capacity is limited, and is shared by multiple flows, incre asing the coded pa cket size n f 1 of flow f 1 gen- erally requires d ecreasing th e packet size n f 2 for some other flow f 2 . That is, increasing S f 1 comes at the cost of decreasing S f 2 . W e are interested in understanding this trade– off, and in analysing the optimal fair allocation of co ding rates a mongst users/flows. Our main contribution is the analysis of fairness in the allocation of coding rates between users/flo ws competing for limited network capacity . In particular, we ado pt a utility– fair frame work, an d propose a scheme for obtain ing the propo rtional fair allocation of coding rates, i.e. the allocation of cod ing rates that maxim ises P f ∈F log S f ( r f ) subject to network capacity constraints. This pr oblem, which we show in Section II I, req uires solving a non–convex optimisation problem . Sp ecifically , at the ph ysical layer, the (channel) coding rate of a flow can be lo wered (to alleviate its c hannel errors) on ly at the expense of in creasing the cod ing rates of other flows. Also, at th e ne twork layer, the len gth of sch edules of each flow should be cho sen in such a way that it maxim ises the network utility . Interesting ly , we sh ow in our pro blem formu lation th at th e coding rate and the schedulin g are tightly coupled . Also, we s how that for a log (network) utility function (which typically g iv es proportional fair allocation o f resources) Fig. 2. An illustrati on of transmission scheme in cell a of the network in Figur e 1: Every transmission schedul e of T a time uni ts is ti me–shared by nodes 1 an d 3. Note that φ ∆ ( f ) N f R f symbols of the enc oded packet p are transmitt ed in transmission sche dule p + ∆ , where ∆ ∈ { 0 , 1 , 2 , · · · , n f − 1 } . The scheduling or capacit y constraint of cell a may not be tigh t (indica ted by empty time slice in the figure), as the rates of flows f 1 and f 2 are gove rned by the whole netwo rk. the optim um rate a llocation (in general) gives un equal air– times which is quite d ifferent from the previously kn own result o f pr oportio nal fair allocation bein g the same as th at of equal air–time allocation ([2]). This problem, which we show in Section II I, requir es solving a no n–conve x optimisation problem . Our work differs f rom the previous work o n ne twork utility maximisation (see [3] and the r eferences th erein) in the following manner . T o the best o f our k nowledge, this is the first work th at com putes the optimal c oding rate for a giv en scheduling (or capacity) constraints in the utility– optimal f ramew ork. The rest o f the pa per is organised as follows. In Section II, we obtain a measur e for th e end– to–end packet erasure, and describe the throughpu t o f the network. W e then formulate a network utility maxim isation problem subject to constrain ts on the transmission sched ule lengths. In Section III, we ob tain the optimum tran smission strategy and the optimu m pa cket–lev el coding r ates for each flow in the network. In Section V, we provide some simple exam ples to illustrate ou r results. Du e to lack of space, the p roofs of v arious Lem mas are o mitted. I I . P R O B L E M F O R M U L A T I O N The encod ing has two stages. The fir st stage is the enco ding of each informa tion p acket using a standard g enerator ma trix such as a Reed–Solomon code or a fountain code [4]. Let P f [ t ] denote the inform ation packet that arrives at the sour ce of flow f in slot t . A packet P f [ t ] of flow f has k f symbols, the encoded packet C f [ t ] of which is of size n f = k f /r f with 0 < r f ≤ 1 , and we assume that the code is such that the packet P f [ t ] can be reconstructed from any of its k f encoded symbo ls (this is possible, for e xamp le, by Reed–So lomon codes). The second stage allocates the content o f the encoded pac ket C t of the first stage across the transmitted p ackets. Each encoded packet is segmented into D f portion s (whe re we recall that D f is the decoding deadline requiremen t for each packet of flow f ), the size of the ∆ th portio n b eing φ f (∆) n f , where ∆ ∈ { 0 , 1 , · · · , D f − 1 } and 0 6 φ f (∆) 6 1 . At transmission slot t , a transmitted packet is assembled from the φ f (0) po rtion o f C f [ t ] , the φ f (1) portion of C f [ t − 1] , and so on un til the φ f ( D f − 1) th po rtion of p acket C f [ t − D f + 1] . This proc edure is illustrated in Figure 3 for n f = 3 . Note that the transmitted packet is of size n f symbols. T o d ecode a packet P f [ t ] o f flow f , we use the tr ansmitted packets th at ar e received dur ing the tran smission slots t, t + 1 , · · · , t + D f − 1 . Note that the conventional strategy o f transm itting an encode d packet every transmission slot cor respond s to the special case: P f [1] P f [2] P f [3] C f [1] C f [2] C f [3] φ 0 C [1] φ 0 C [2] φ 1 C [1] φ 0 C [3] φ 1 C [2] φ 2 C [1] 1 2 3 T ime Fig. 3. T wo stage encoding (exampl e of D f = 3) : information packet P f [1] of size k f is encoded to C f [1] of size n f = k f /r f , the contents of which are alloc ated across s ubpack ets φ 0 C f [1] , φ 1 C f [1] , φ 2 C f [1] across 3 timeslots. φ f (0) = 1 and φ f (1) = φ f (2) = · · · = φ f ( D f − 1) = 0 . W e call the transmission scheme outlined above with ge neral φ · (∆) s a generalised block transmission scheme. A. Network Constr aints o n Codin g Rate Let w f ,c be the PHY rate o f transmission of flow f in cell c . For each transmitted packet o f flow f , e ach cell c ∈ C f along its route must allocate at least n f w f,c units of time to tran smit the packet (or enc oded block). Let F c := { f ∈ F : c ∈ C f } be the set o f flows that are r outed throug h cell c . W e reca ll that th e transmissions in any cell c are scheduled in a TDMA fashion, and h ence, the total time req uired fo r tr ansmitting packets for all flo ws in cell c is g iv en by P f ∈F c n f w f,c . Since, for cell c , the transmission s chedu le in terval is T c units of time, the coding rates r f must satisfy the sched ulability constraint P f ∈F c n f w f,c 6 T c . B. Err or Pr obability – Upper b ound Lemma 1. The end– to–end p r ob ability of a packet e rasur e for flo w f is bounded by e e f = P    D f − 1 X ∆=0 φ f (∆) k f r f E f [∆] > n f − k f    ≤ exp   −   θ f (1 − r f ) − D f − 1 X ∆=0 ln  1 − β f + β f e θ f · φ f (∆)      =: e f . wher e θ f > 0 is the Chern off–bound parameter . Let the random variable α f [ t ] indicate whether packet P f [ t ] is successfully decoded or not, i.e., α f [ t ] =  1 , if packet P f [ t ] is decoded succ essfully 0 , otherwise . W e note here that the d ecoding erro rs for th e successive packets are correlated, as each encoded packet ov erlaps with the transmission of p revious D f − 1 packets and the successive D f − 1 p ackets. Hence, the sequence of rand om variables α f [1] , α f [2] , α f [3] , · · · are co rrelated. But, the pro bability of any α f [ t ] = 0 is upper boun ded by Lemm a 1. 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 For flow f , th e total expected thro ughpu t as a resu lt of transmitting T ≥ 1 packets is given b y S f ( T ) = X ( x 1 ,x 2 , ··· ,x T ) ∈{ 0 , 1 } T T X t =1 k f x t ! P  α f [ t ] = x t , t = 1 , 2 , · · · , T  Note that the joint probability mass function P { α f [ t ] = x t , t = 1 , 2 , · · · , T } is not a produ ct–form distribution as the packet erasures α f [ t ] s ar e correlated. Howe ver , the above expectation can b e written as S f ( T ) = T X t =1 X x t ∈{ 0 , 1 } k f x t P { α f [ t ] = x t } = T · k f · (1 − e f ) Thus, the (a verage expected) flow throug hput is defined as S f = lim T →∞ S f ( T ) T = k f · (1 − e f ) . W e are interested in maximising the u tility of the n etwork which is defined as the sum utility of flo w throu ghputs. W e con sider the log of th rough put as the can didate for th e utility function being m otiv ated by th e desirable p roperties like pro portion al fairness that it possesses. W e define the follo wing n otations: th e Cherno ff–bound parameters θ := [ θ f ] f ∈F , coding rates r := [ r f ] f ∈F , and the a llocation o f co ded b its across transmission slots Φ := [ φ f ] f ∈F where φ f := [ φ f (0) , φ f (1) , · · · , φ f ( D f − 1)] . Thus, we d efine the n etwork utility a s e U ( Φ , θ , r ) := X f ∈F ln ( k f (1 − e f ( φ f , θ f , r f ))) =: X f ∈F ln ( k f ) + U ( Φ , θ , r ) (1) The pro blem is to obtain the optim um coded bit allo cation Φ ∗ , the optimu m Chernoff–boun d para meter θ ∗ , and th e op timum coding rate r ∗ that maximises the network utility . Since, k f , the size of inform ation packets of each flow f is given, maximising the network utility is eq uiv alent to maximising U ( Φ , θ , r ) := P f ∈F ln (1 − e f ) . Thus, we define th e follow- ing pr oblem P1: max Φ , θ , r U ( Φ , θ , r ) subject to X f : c ∈C f k f r f w f ,c ≤ T c , ∀ c ∈ C (2) D f − 1 X ∆=0 φ f (∆) = 1 , ∀ f ∈ F (3) φ f (∆) ≥ 0 , ∀ f ∈ F , 0 ≤ ∆ ≤ D f − 1 θ f > 0 , ∀ f ∈ F r f ≤ λ f ∀ f ∈ F r f ≥ λ f ∀ f ∈ F W e n ote that the Eqn. (2) enforces the network capacity (or the network sched ulability) constraint. The o bjectiv e f unction U ( Φ , θ , r ) is separab le in ( φ f , θ f , r f ) fo r each flo w f . Impor tantly , th e compo nent of utility fu nction for each flow f g iv en b y ln (1 − e f ( φ f , θ f , r f )) is no t jointly concave in ( φ f , θ f , r f ) . Howev er, ln (1 − e f ( φ f , θ f , r f )) is c oncave in each of φ f ( · ) , θ f , an d r f . Hen ce, the network utility maximi- sation problem P1 is not in the stand ard c on vex o ptimisation framework. In stead, we pose the following prob lem, P2: max Φ max θ max r X f ∈F ln (1 − e f ( φ f , θ f , r f )) (4) subject to X f : c ∈C f k f r f w f ,c ≤ T c , ∀ c ∈ C D f − 1 X ∆=0 φ f (∆) = 1 , ∀ f ∈ F φ f (∆) ≥ 0 , ∀ f ∈ F , 0 ≤ ∆ ≤ D f − 1 θ f > 0 , ∀ f ∈ F r f ≤ λ f ∀ f ∈ F r f ≥ λ f ∀ f ∈ F In general, the solution to P2 need not b e the solution to P1 . Howe ver , in our problem , we show that P2 achieves the solution o f P1 . Lemma 2. . F or a function f : Y × Z → R tha t is concave in y and in z , but not jointly in ( y , z ) , the solution to the joint optimisation pr oblem fo r conve x sets Y and Z max y ∈Y ,z ∈ Z f ( y , z ) (5) is the same as max z ∈Z max y ∈Y f ( y , z ) , (6) if f ( y ∗ ( z ) , z ) is a concave function o f z , wher e for each z ∈ Z , y ∗ ( z ) := arg ma x y ∈Y f ( y , z ) . W e no te that f or each r f and θ f , th e prob ability of er ror e f is con vex in φ f , and hence, ln(1 − e f ) is concave in φ f . Thus, we first solve for the optimu m code b it allo cation φ ∗ f in Section IV -A. Th en, usin g the o ptimum code bit allocatio n, we solve for th e optimum Cher noff bound parameter θ ∗ which we describe in sub section IV -B. After having solved for the optimum θ ∗ , we show in Sectio n IV -C that U ( Φ ∗ , θ ∗ ( r ) , r ) is a co ncave fu nction of r . Hence, fr om Lemm a 2, the solutio n to pro blem ( P2 ) (the maxim isation p roblem that separ ately obtains the optimum θ ∗ and optimum r ∗ ) is globally optimum. W e study the rate optimisation problem that o btains r ∗ in subsection I V -D. I V . U T I L I T Y O P T I M U M R AT E A L L O C A T I O N A. Optimal Code Bit Alloc ation Φ W e con sider the maximisation problem defined in Eqn. 4 for a given coding rate vector r and Chernoff–bou nd parameter vector θ , and obtain the optimum φ f for each flow f ∈ F . The sub– proble m is gi ven by max φ f X f ∈F ln (1 − e f ( φ f , θ f , r f )) subject to D f − 1 P ∆=0 φ f (∆) = 1 , ∀ f ∈ F φ f (∆) ≥ 0 , ∀ f ∈ F , ∀ ∆ ≤ D f − 1 . This is a sep arable conve x op timisation prob lem, and henc e can be solved by Lag rangian method . Let µ f be a Lag rangian multiplier for the constrain t D f − 1 P ∆=0 φ f (∆) = 1 , an d define µ = [ µ f ] f ∈F . Th e Lagra ngian function is gi ven by L ( Φ , µ ) = X f ∈F ln (1 − e f ) − X f ∈F µ f  1 − D f − 1 X ∆=0 φ f (∆)  Applying KKT condition, ∂ L ∂ φ f ( i ) | φ f ( i ) ∗ = 0 , we g et 0 = − e f 1 − e f · β f θ f e θ f φ ∗ f ( i ) 1 − β f + β f e θ f φ ∗ f ( i ) + µ f or , e θ f φ ∗ f ( i ) = 1 − β f β f (1 − e f ) µ f θ f e f − µ f (1 − e f ) (7) for i = 0 , 1 , 2 , · · · , n f − 1 . Since, the RHS of Eqn. 7 is the same for all i , we get φ ∗ f ( i ) = φ ∗ f ( j ) , and hence φ ∗ f (∆) = 1 D f , ∀ ∆ = 0 , 1 , · · · , D f − 1 . Thus, Φ ∗ allocates equ al po rtions of a n enco ded packet a cross transmission schedules with a delay of 0 , 1 , · · · , D f − 1 , un like the c on ventional tra nsmission scheme which tran smits all the coded bits of a p acket in one shot. Hen ce, e f ( φ ∗ f , θ f , r f ) is e f = exp  −  θ f (1 − r f ) − D f ln  1 − β f + β f e θ f D f  . (8) B. Optimal θ ∗ W e now consider the o ptimum Cher noff–bound parameter problem with th e optimum coded b its allocation Φ ∗ , and for any giv en cod ing rate vector r ∈ [ λ f , λ f ] F . max θ X f ∈F ln  1 − e f ( φ ∗ f , θ f , r f )  (9) subject to θ f > 0 , ∀ f ∈ F W e note th at the ob jecti ve function is separable in θ f s, and that e f is con vex in θ f . Hence, t he problem defined in Eqn. ( 9), is a concave ma ximisation problem. The partial der i vati ve of e f with respect to θ f is gi ven b y ∂ e f ∂ θ f = − e f ·  (1 − r f ) − β f e θ f /D f 1 − β f + β f e θ f /D f  . Observe th at β f e θ f /D f 1 − β f + β f e θ f /D f is an increasing function o f θ f . Thus, if, for θ f = 0 , 1 − r f − β f 1 − β f + β f < 0 o r r f > 1 − β f , the d eriv ati ve is positive f or all θ f > 0 , or e f is an increa sing function of θ f . He nce, for r f > 1 − β f , the optim um θ ∗ f is arbitrarily close to 0 wh ich yield s e f arbitrarily close to 1 . Thus, for erro r re covery , for any end–to–e nd error p robability β f , the cod ing rate should be smaller than 1 − β f , in which case, we obtain the o ptimum θ ∗ f by equating the partial deriv ativ e of e f with respect to θ f to ze ro. i.e., β f e θ ∗ f /D f 1 − β f + β f e θ ∗ f /D f = 1 − r f or , e θ ∗ f /D f = 1 − r f β f 1 − β f r f or , θ ∗ f = D f h ln  1 − r f β f  − ln  r f 1 − β f i . Thus, the pro bability of a packet deco ding erro r fo r a g i ven r f with the o ptimum a llocation of code d bits Φ ∗ , and the optimum Cher noff–bound p arameter θ ∗ f , is e f = exp  − D f  (1 − r f ) ln  1 − r f β f  + r f ln  r f 1 − β f  = exp ( − D f · KL ( B (1 − r f ) ||B ( β f )) where K L ( f 1 , f 2 ) is the Kullback– Leibler divergence between the p robab ility mass functions (pmfs) f 1 and f 2 . C. A con ve x optimisatio n framew ork to obtain optimal r ∗ f If ln(1 − e f ( φ ∗ f , θ ∗ f , r f )) is concave in r f , the n o ne can obtain the optimum r ∗ f using con vex optimisation f ramew ork. T o show the concavity of ln(1 − e f ( φ ∗ f , θ ∗ f , r f )) it is su fficient to show that e f ( φ ∗ f , θ ∗ f , r f ) is con vex in r f . No te that ∂ e f ∂ r f = e f · θ ∗ f ( r f ) ∂ 2 e f ∂ r 2 f = e f  θ ∗ 2 f − D f r f (1 − r f )  e f is conve x if  ln  1 − r f β f  − ln  r f 1 − β f  2 ≥ D f r f (1 − r f ) , or , ln  1 − r f r f 1 − β f β f  ≥ p D f p r f (1 − r f ) or , p D f p r f (1 − r f ) − ln  1 − r f r f 1 − β f β f  ≤ 0 The fu nction 1 √ r f (1 − r f ) is con vex in r f . Also, ln  1 − r f r f  is decreasing with r f , and he nce, − ln  1 − r f r f 1 − β f β f  ≤ − ln  1 − λ f λ f 1 − β f β f  . Th us, we ha ve a su fficient c ondition √ D f p r f (1 − r f ) − ln  1 − λ f λ f 1 − β f β f  ≤ 0 (10) The above co ndition r equires the delay dea dline D f to be smaller than some D f ( r f ) . W e conside r D f s to satisfy th is condition , and hence, the rate o ptimisation problem is a concave max imisation p roblem. For the sake of completen ess, we include this as a co nstraint in th e p roblem formu lation. Howe ver , this condition is no t an acti ve constraint. D. Optimal Coding Rate r From the previous subsection , we ob serve un der the delay constraint Eq n. (10) th at e f ( φ ∗ f , θ ∗ f ( r f ) , r f ) is co n vex in r f , and hence, we obtain the optimum coding rate r ∗ f using conve x optimisation method . Also, from Lemma 2, it is clear that r ∗ f is the un ique globally optimu m rate. Th us, we solve the following network utility maxim isation problem max r X f ∈F ln  1 − e f ( φ ∗ f , θ ∗ f ( r f ) , r f )  (11) subject to X f : c ∈C f k f r f w f ,c ≤ T c , ∀ c ∈ C r f ≤ λ f ∀ f ∈ F r f ≥ λ f ∀ f ∈ F p D f p r f (1 − r f ) − a ≤ 0 ∀ f ∈ F (1 2) where a = ln  1 − λ f λ f 1 − β f β f  . It is clear th at th e objective fun c- tion is separable and concave, a nd hen ce, can be solved using Lagrang ian relaxa tion metho d. Also, we note here that the constraint represented b y Eqn. (12) is not an active con straint, and hence, there is no Lag rangian co st to this constraint. W e note h ere that the codin g rate shou ld be such that k f /r f is an inte ger, and hence, ob taining r ∗ f is a d iscrete optimisation problem . This is, in ge neral, an NP hard pro blem. Hence, we relax this con straint, and allow r f to take any r eal value in [ λ f , λ f ] . The Lagrang ian function for the rate optimisation problem is thus L ( r , p , u , v ) = X f ∈F ln (1 − e f ) − X c ∈C p c   X f ∈F c k f r f w f ,c − T c   + X f ∈F u f  r f − λ f  − X f ∈F v f  r f − λ f  Applying KKT condition, ∂ L ∂ r f | r ∗ f = 0 , we have − 1 1 − e f ∂ e f ∂ r f | r ∗ f = X c ∈C f p c − k f r ∗ 2 f w f ,c + v f − u f = − k f r ∗ 2 f   X c ∈C f p c w f ,c   + v f − u f e f 1 − e f · θ ∗ f = k f r ∗ 2 f   X c ∈C f p c w f ,c   + v f − u f . If the optimum r ∗ f is eithe r λ f or λ f , then it is u nique. If r ∗ f ∈ ( λ f , λ f ) , then u f = v f = 0 , which is the most intere sting case, and we consider only this case for the rest of the paper . Let λ f := P c ∈C f p c w f,c . Th e above equation be comes e f 1 − e f · θ ∗ f = λ f k f r ∗ 2 f (13) e f = λ f k f λ f k f + θ ∗ f r ∗ 2 f (14) exp  − D f D ( B (1 − r ∗ f ) kB ( β f ))  = λ f k f λ f k f + θ ∗ f r ∗ 2 f D f D ( B (1 − r ∗ f ) kB ( β f )) = ln λ f k f + θ ∗ f r ∗ 2 f λ f k f ! (15) In the above eq uation, the LHS is a strictly conv ex decr easing function o f r ∗ f . Sinc e, th e u tility max imisation prob lem is a concave maximisation pro blem, the optimum rate r ∗ f ∈ (0 , 1 − β f ) exists an d is unique. E. Sub –gradient Ap pr oach to Compute optimum p ∗ c In this section , we discuss the pro cedure to obtain the Shadow co sts or the Lagr ange variables p ∗ . The dual pr oblem for the primal prob lem defined in Eqn. (1 1) is given by min p ≥ 0 D ( p ) , where the dual functio n D ( p ) is giv en by D ( p ) = max r X f ∈F ln(1 − e f ( r f )) + X c ∈C p c   T c − X f ∈F c k f r f w f ,c   (16) = X f ∈F ln(1 − e f ( r ∗ f ( p ))) + X c ∈C p c   T c − X f ∈F c k f r ∗ f ( p ) w f ,c   . (17) In the ab ove equ ation, e f ( r f ) d enotes e f ( φ ∗ f , θ ∗ f ( r f ) , r f ) . Since th e dual func tion (of a p rimal problem) is conv ex, D is conv ex in p . Hence, we use a sub–grad ient method to obtain the o ptimum p ∗ . Fro m Eqn . (16), it is clear that for any r , D ( p ) ≥ X f ∈F ln(1 − e f ( r f )) + X c ∈C p c   T c − X f ∈F c k f r f w f ,c   , and in particular, D ( p ) is gre ater than tha t fo r r = r ∗ f ( e p ) , i.e., D ( p ) ≥ X f ∈F ln(1 − e f ( r ∗ f ( e p ))) + X c ∈C p c   T c − X f ∈F c k f r ∗ 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 r ∗ f ( e p ) w f ,c   (18) Thus, a sub–grad ient of D ( · ) at any e p is given by the vector   T c − X f ∈F c k f r ∗ f ( e p ) w f ,c   c ∈C . (19) W e obtain an iterative algorithm based o n s ub–g radient method that yields p ∗ , with p ( i ) being the La grangian s at the i th iteration. p c ( i + 1) =   p c ( i ) − γ ·   T c − X f ∈F c k f r ∗ f ( p ( i )) w f ,c     + . where γ > 0 is a sufficiently small stepsize, and [ f ( x )] + := max { f ( x ) , 0 } ensures th at the Lagrang e multiplier n ev er g oes negativ e. Note that the Lag rangian u pdates can be locally done, as each cell c is requ ired to kn ow only the rates r ∗ f ( p ( i )) of flows f ∈ F c . T hus, at the beginning o f each iter ation i , the flows cho ose their cod ing rates to r ∗ f ( p ( i )) , an d each cell computes its cost based o n the rates of flo ws through it. The updated costs alon g th e route of each flo w are then fed back to the source nod e to co mpute the r ate fo r the n ext iteration. The Lagran ge multiplier p c can be viewed as the cost of transmitting traffic throug h cell c . The am ount of serv ice time that is available is given by δ = T c − P f ∈F c k f r ∗ f ( p ( i )) w f,c . When δ is p ositiv e an d large, then the Lagrangian co st p c decreases rapidly ( because D is conve x), an d when δ is negativ e, then the Lagrangian cost p c increases rapidly to make δ ≥ 0 . W e n ote that the incr ease o r d ecrease o f p c between successiv e iterations is prop ortional to δ , the amou nt of service time av ailable. Th us, th e sub–g radient proced ure pr ovides a dynamic co ntrol scheme to balance the network loads. W e explore th e prop erties o f the o ptimum rate par ameter r ∗ f in Sectio n IV -F. In Section V, we p rovide some examp les that illustrate the op timum utility–fair resour ce allocation. F . Pr operties of r ∗ f Lemma 3. r ∗ f ( D f ) is an increasing function of D f . Lemma 3 is quite intuitiv e. For any given chan nel er ror β f , as the d eadline becom e less stringent, it is optima l to go for a high rate cod e. In other words, it is o ptimal for a flow to use as much sche duling time as p ossible (f or a large D f , an d he nce, use a high rate code); howe ver , th e resources are shared among multiple flows, and he nce, we ask the following question: “ what is th e optimal share of the scheduling time ” that ea ch flow should have. Interestingly , in o ur prob lem formula tion, the code rate r f also solves this optimal schedu ling times fo r each flows. V . E X A M P L E S A. Examp le 1: T wo cells with equal traf fic loa d W e begin by conside ring th e example shown in Figur e 4 consisting of two cells a and b having three nod es 1, 2, and 3. Eac h cell ha s the same packet e rasure prob ability β and th e schedule len gth T . Ther e are three flows f 1 , f 2 , an d f 3 , with two of the flo ws f 1 and f 3 having one–hop rou tes C f 1 = { b } and C f 3 = { a } , and o ne flow f 2 having a two–h op route a b 3 flow f flow f 1 Fig. 4. Cells with equal traffic load C f 2 = { a, b } . Each flow has the same information p acket size k , decodin g dead line D and PHY tr ansmit rate, i.e. w f ,c = w . This is analo gous to th e so–called park ing–lot to pology often used to explore fairness issues. The end–to–en d erasure probab ility experienc ed by the two– hop flow f 2 is gr eater than that experienced by the on e ho p flows f 1 and f 3 , since each hop h as the same fixed erasur e probab ility . Hence, we need to assign a lesser 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 (af ter decod ing) acr oss flows. Howev er, when operatin g at the bou ndary of the network capa city region (thereby maxim ising thro ughp ut), decreasing the coding rate r f 2 of the two–ho p flow f 2 requires that the coding rate of both one– hop flows f 1 and f 3 be increased in ord er to remain within the av ailable network capacity . I n this sense, alloca ting coding rate to the two–h op flow f 2 imposes a g reater marginal cost on the n etwork (in terms of the sum– utility) than the o ne– hop flo ws, and we expect that a fair allocation will therefore assign h igher co ding rate to the two–ho p flow f 2 . Th e solutio n optimising this trade–off in a p ropor tional fair m anner c an be understoo d u sing the analysis in the previous sectio n. In this example, both the cells are equally loaded and , by symmetry , the Lagran ge multipliers p a = p b . Hen ce, λ f 1 = λ f 2 2 = λ f 3 . F or th e Cher noff–bound parameter θ = [ θ , θ ] , we find fro m Eqn . (13), e f 2 1 − e f 2 · 1 − e f 1 e f 1 = λ f 2 λ f 1 · r ∗ 2 f 1 r ∗ 2 f 2 = 2 · r ∗ 2 f 1 r ∗ 2 f 2 . For sufficiently small era sure probab ilities, we have e f 2 e f 1 ≈ 2 · r ∗ 2 f 1 r ∗ 2 f 2 ≈ 2 Thus the pr oportio nal fair allocation is e f 1 = e f 3 ≈ 1 / 2 · e f 2 . That is, the coding rates are allo cated such that the one–ho p flows ha ve approximately half the error probability of the two– hop flow . B. Examp le 2: T wo cells with unequal traffic loa d W e consider the same n etwork as in the previous example, but n ow with o nly the flows f 1 and f 2 (i.e., the flo w f 3 is not present) in the network. In this exam ple, cell b carries two a b 3 flow f flow f 1 Fig. 5. Cells with unequal traffic load flows while cell a car ries only on e flow . T he enco ding rate constraints ar e given by 1 r f 2 ≤ wT k , (f rom cell 1) , 1 r f 1 + 1 r f 2 ≤ wT k , (f rom cell 2) . Since, both r f 1 and r f 2 are at m ost 1, it is clear that at the optimal poin t, the rate constrain t of cell a is no t tight while the co nstraint of cell b is tight. Th us, the shad ow p rices (Lagran ge multipliers) p a = 0 and p b > 0 . That is, at the first hop the cell is n ot ope rating at cap acity , and so the “p rice” for u sing this cell is zero. In this exam ple, λ f 1 = λ f 2 , an d hence, f rom Eq n. (13), we ded uce that for sufficiently low cell erasure prob ability β , e f 1 ≈ e f 2 . Alternatively , as the delay deadline D → ∞ , f rom Eqn. (13) we h av e e f 1 = e f 2 . These proportion al fair allocations make s ense intuitively since although flow f 2 crosses two hop s, it is o nly constrained at the second h op a nd so it is natural to share th e available capacity of this second hop approximately eq ually between the flows. When the era sure p robab ility is sufficiently small, this yields approx imately the same error pr obabilities for b oth flows. For larger erasure probabilities, it leads to the two–hop flo w ha ving higher error probability , in proportio n to the per–hop er asure probab ility β . R E F E R E N C E S [1] K. Premkumar , X. Chen, and D. J. Leith, “Utility optimal coding for pack et transmission ov er wireless networks – Part I: Networks of binary synchronous channels, ” in s ubmitte d , 2011. [2] A. Checco and D. J. Leith, “Proporti onal fairne ss in 802.11 wireless lans, ” to appear in IEEE Comm. Lette rs , 2011. [3] S. Shakkotta i and R. Srikant, Network Optimization and Contr ol . No w Publishers Inc., Boston - Delft, 2008. [4] A. Shokrollahi , “Raptor codes, ” Information Theory , IEEE T ran sactions on , vol. 52, no. 6, pp. 2551 –2567, Jun. 2006.