roceedings of the 46th IEEE Conference on Decision and Contro New Oreans, LA, USA, Dec. 12-14, 27 FrB2.5 ower Contro and Transmission Scheduing for Network Utiity Maximization in Wireess Networks Min Cao, Vivek Raghunathan, Stephen Hany, Vinod Sharma and. R. Kumar Abstract We consider a joint power contro and transmission scheduing probem in wireess networks with erage power constraints. Whie the capacity region of a wireess network is convex, a characterization of this region is a hard probem. We formuate a network utiity optimization probem invoving time-sharing across different transmission modes, where each mode corresponds to the set of power eves used in the network. The structure of the optima soution is a time-sharing across a sma set of such modes. We use this structure to deveop an efficient heuristic approach to finding a suboptima soution through coumn generation iterations. This heuristic approach converges quite fast in simuations, and provides a too for wireess network panning. Index Terms Network utiity maximization, power contro, transmission scheduing, coumn generation I. INTRODUCTION In wireess networks, it is we known that the traditiona ayers of the communication network cannot be considered in isoation. Many authors he proposed joint approaches to issues such as power contro, scheduing, and routing [2], [3], [5], [6]. There are many reasons for this. In the present paper, we focus on the foowing particuar characteristics of wireess channes, namey, that they tend to be hafdupex, and of a broadcast nature. The primary haf dupex constraint that a node cannot be simutaneousy transmitting and receiving impies that inks must be carefuy schedued. The broadcast characteristic of wireess communications brings with it the fundamenta issue of interference management, and scheduing is an important mechanism to aeviate interference between inks. Ceary scheduing decisions are strongy connected to the ower ayer functions of power contro, and ink ayer rate aocation, since scheduing and power contro together determine the quaity of the communication inks. But routing decisions (and, eventuay, end-to-end transport capacity) are aso determined by the quaity of inks, and hence transport, routing, scheduing and power contro need to be considered jointy: a the ayers are intimatey connected in a wireess network. In this paper we formuate and tacke the joint probems of bit rate aocation, power contro, scheduing, and routing, using an optimization framework where power consumption is taken into account. We begin with the simpest case of a set of parae inks, where routing is not an issue, and This materia is based upon work partiay supported by NSF under Contract Nos. CCR-325716 and CNS 5-19535, DARA/AFOSR under Contract No. F4962-2-1-325, DARA under Contact Nos. N14--1-1-576 and N661-6-C-221, Oakridge-Battee under Contract 239 DOE BATT 444522 and AFOSR under Contract No. F4962-2-1-217. consider the probem of maximizing the sum of the rates of a the inks, subject to erage power constraints on the transmit nodes. Even here, the haf dupex constraints, and the possibiity of scheduing the inks to mitigate interference, provide a combinatoria aspect to the probem, and it is in fact we known that the genera probem we tacke is N-compete [2]. Our main contribution is a inear programming formuation which expresses the optima schedues and power eves in terms of inear combinations of basic modes of the system. A characterization of the dimensionaity of the search space, eads to an iterative approach in which od modes are pivoted out, and new modes are pivoted in, using a coumn generation technique. This approach does not circumvent the fundamenta compexity of the probem, but we provide numerica evidence that it converges rapidy and obtains good (but suboptima) soutions for reasonaby arge-sized networks. Then, we extend the method to maximize end-to-end utiities across genera wireess mesh networks, using a muti-commodity fow to hande routing and fow contro constraints. There he recenty been many works on cross-ayer design of wireess networks, in which a resource aocation approach is used to formuate and sove a network-wide utiity maximimization (NUM) probem [2], [3], [5], [6]. In this framework, the soution to the optimization probem automaticay decomposes into a number of subprobems at each ayer, and the agorithms at each of these ayers interact with each other via dua variabes, which act to provide pricing information for cross-ayer coordination. However, the HY/MAC subprobem of wireess networks is typicay N-Compete. There he been efforts to find approximation schemes to sove this probem [4]. However, a simpified interference mode is assumed in [4], which does not appy for genera wireess networks. In [7], the authors consider the probem of finding the jointy optima end-to-end rates, routing, power aocation and transmission scheduing for wireess networks. Noninear coumn generation is used to sove the cross-ayer probem, and to prove convergence to the goba optima soution. To converge to the goba optimum, this procedure needs to find the optima soution of each coumn generation sub-probem. This generation subprobem is aso typicay N-Compete. The authors address the noninearity, but not the computationa compexity, of the cross-ayer soutions using coumn generation. In this paper, we further estabish a dimensionaity bound for the optima soution of the probem. This provides an approach to identify good HY/MAC modes to be used by the higher 1-4244-1498-9/7/$25. 27 IEEE. 5215
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 ayer operations. An aternative formuation woud be to find the optima aocation of power spectrum, and anaogous resuts can easiy be obtained. This approach has been taken in [8] where a simiar dimensionaity bound was obtained. The rest of this paper is organized as foows: in Section II, we describe the network mode. In Section III we consider the probem of maximizing the sum of ink utiities. In Section IV, we consider the probem of maximizing the sum of end-to-end utiities. We describe numerica resuts in Section V, and extensions in Section VI. Finay, we concude in Section VII. II. NETWORK MODEL AND ASSUMTIONS We make the foowing assumptions about the wireess transceiver, refecting characteristics of IEEE 82.11 wireess card: 1) it is haf-dupex, so a node cannot transmit and receive simutaneousy; 2) a singe-user radio is used, hence, a node can communicate with exacty one another node at any time. These are referred to as primary constraints. We refer to a transmitter-receiver pair as a ink. We consider a wireess network with N nodes and L = N(N 1) possibe inks. Denote the set of a the nodes and inks by N and L, respectivey. Let O(n) and I(n) denote the set of outgoing and incoming inks of node n, respectivey; and I{ } be the indicator function. A transmission mode m can be represented as a transmit power vector =(1 m,,l m ) that satisfies the primary constraints I(n) I{ m > } = 1. Due to primary constraints, not a inks can be simutaneousy active in a transmission mode, and thus many of the m s might need to be. Thus, different transmission modes need to be activated at different times to provide ong term end-to-end connectivity. The specific procedure by which transmission modes are seected and activated affects both throughput and deay performances of end-to-end fows. Denote the channe gain from the transmitter node of ink k to the receiver node of ink by G k. For any transmission mode m, the SINR achieved at the receiver of ink is γ m ( )= k G m G k k m, (1) + σ2 where σ 2 is the noise power. Thus, for each transmission mode m, there is an achievabe rate vector R m = (R1 m,,rl m) corresponding to. R m is assumed to be a non-decreasing function of γ m, depending on the moduation and coding schemes used. Theoreticay, it can be determined by the Shannon function as R m ( )=W og 2 (1 + γ m ( )), (2) where W is the bandwidth of the wireess channe. Note that when m =, R m =; thus inactive inks he zero data rates. We assume that the channe gains are fixed in what foows. In this paper, we are interested in finding power contro and transmission scheduing (C-TS) schemes that maximize the system utiity. With the transmission modes defined as above, the joint C-TS probem consists of seecting the transmission mode at each time instant so as to optimize the system performance objective. In finding the optima soution, we impose erage power constraints on the nodes. However, as discussed above, interference between nearby inks requires some scheduing of ink activations. A joint C-TS scheme can be represented as { (t),t (,T)}, where (t), t (,T), is the set of aowabe transmission modes, and T is the duration of the transmission. Note that scheduing is subsumed in such a power contro scheme since switching off a ink is accompished by setting its power eve to zero. Our goa is to find the optima C-TS scheme that maximizes the system utiity. The system utiity can be defined in various ways to meet different objectives. In Section III, we consider the probem of maximizing the information-transport capabiity which can be expressed as the sum of ink utiities. In Section IV, we consider the probem of maximizing the sum of end-to-end user utiities. III. OWER CONTROL AND TRANSMISSION SCHEDULING FOR UTILITY MAXIMIZATION A. arae Links Case We first formuate the probem for maximizing the information-transport capabiity of L parae inks where there are no primary constraints. Assume that the reward received for transporting one bit over ink is r, then the reward we get per unit time with a fixed transmission mode vector is V ( )= L =1 r R ( ). In practice, there are discrete power eves aiabe to each ink, and we wi assume that each ink can seect a power eve from the foowing set of K power eves: {, ax K 1, 2 max K 1 max 2),...,(K,ax } K 1 where ax denotes the maximum possibe power eve. Let denote the set of a K L power vectors aiabe to the network, and et M = {1, 2,...,K L } index this set. We refer to the mth eement in the set,,asthemth transmission mode. We aow a schedue to determine the activation of the transmission modes. Thus, over any transmission interva (,T) we aocate a fraction of time, α m, to the mth transmission mode; the time fraction can of course be zero. The utiity obtained is then α m V ( ). We assume that the inks are constrained by ong-term erage power constraints. Thus, the optimization probem is the foowing 5216
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 inear program (L): max α m V ( ) (3) α m m, L, (4) α m =1. (5) This L has ony L +1 constraints, but it has K L variabes. Unfortunatey, there are in genera of the order of K L(L+1) basic feasibe soutions, and the standard Simpex agorithm is just too sow in genera, even for this simpe network of L parae inks. In fact, it is not reaistic to expect to find the optima soution to this L for more than a smaish number of inks. Instead, we wi attempt to find suboptima, but good soutions via somewhat heuristic methods. An approach that is widey used for soving inear programming probems with a arge number of variabes, but few constraints, is the method of coumn generation. The coumns correspond to the transmission modes in the present probem. Coumn generation of the L provides a decomposition of the probem into a master probem and a subprobem, and it identifies a good subset of modes by iterativey soving the master probem and subprobem. The master probem is the same as (3) except that we repace M by a sma subset of modes M M. Note that the dimensionaity of the probem is such that basic feasibe soutions he at most L +1 basic variabes, since there are ony L +1 constraints. Thus, the optima soution wi be a time-sharing amongst at most L +1 transmission modes. Thus, we choose to set M = L +2, and sove the reduced L as the master program. Initiay we can randomy pick L +2 transmission modes. When soving the master probem, we obtain the optima dua variabes {λ, L} and β, corresponding to the constraints (4) and (5), respectivey. These variabes can then be used to identify a new mode to enter the basis. The best mode woud be obtained by soving the foowing subprobem, which is a separation probem for the dua L. The coumn generation subprobem is min \M L L λ V ( )+β (6) λ V ( )+β<. (7) We cannot sove this probem exacty, because the size of the set M \ M is huge; an exhaustive search over a the modes is prohibitive. However, it provides motivation for heuristic approaches for seeking a new transmission mode. To improve the objective of the master program, it is enough to find just one transmission mode m M \ M such that L λ m V ( )+β<. The objective of the master probem can be improved by adding such a mode into the active set M. Typicay, the corresponding time-sharing variabe wi increase from zero, and another basic variabe wi go to zero (become non-basic) uness we he reached a degenerate basic feasibe soution. Repeating the procedure in this way, we can obtain a monotonicay improving sequence of soutions as the coumn generation iteration evoves. If at some iteration, L λ m V ( )+β for a the modes m M \ M, then we he achieved the optimum of the origina probem by the prima-dua reation. At each coumn generation iteration, we need to find a new mode m from the set M \ M that satisfies (7). Note that λ can be interpreted as the margina price for increasing V ( ) the power eve of ink, and is the margina V utiity gained by increasing. We define π = ( ) λ, and π =(π 1,,π L ). We use a simpe heuristic greedy agorithm as foows: Agorithm 1: [Mode generating agorithm (parae ink case)] 1) Initiay, choose =, and compute π. 2) If max L π >, et = arg max L π, and raise the power of ink to the next eve. Update π. 3) Continue unti max L π. Agorithm 1 greediy attempts to maximize the function V ( ) λ, but it is not guaranteed to reach the L goba maximum since the function is not conce, and the power eves are not continuous. Nevertheess, if it provides a mode that satisfies (7) then this provides a new coumn to improve the objective in the master program. If (7) is not satisfied, then the whoe coumn generation procedure terminates with a basic feasibe soution. In this case it is possibe, but not guaranteed and certainy not easiy checked, that the fina point wi be the optima soution to the origina L. In practice, it is aso possibe that the coumn generation procedure wi need to be terminated when the computationa budget is reached, given the sheer size of the L when L is arge. B. Genera Case Now we extend to the genera case where each node n can transmit to any other node in the network. In this case, we pace the erage power constraints on the nodes, and the set of feasibe transmission modes now has to take account of the primary constraints. Reca that the primary constraints require that I{ m > } =1. (8) I(n) Thus we now define to be the set of a discrete power mode vectors that satisfy (8), and we et M index the eements of. Let =( 1,, N ) be the vector of erage power constraints on the nodes. As before, we define the utiity V ( ) = L =1 r R m ( ) associated with each mode. The 5217
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 resource aocation probem becomes: max α m V ( ) (9) α m n, n N, (1) α m =1. (11) which is a inear program with N +1 constraints. Again we empoy the coumn generation method to identify a good subset of modes M. The master probem repaces M with M in (9), where M M is a sma subset of modes with M = N +2. Assume that by soving the master probem, we obtain the optima dua variabes {λ n,n N}and β corresponding to the constraints (1) and (11), respectivey. Then the coumn generation probem is min λ n \M n N n N λ n V ( )+β (12) V ( )+β<. (13) Instead of soving (12) for the optima mode at each coumn generation step, we can empoy a heuristic agorithm to find a good mode guided by the dua price information, but we need to consider the primary constraints in the genera case. V As before, we define π = ( ) λ n:. We adopt the foowing simpe heuristic: Agorithm 2: [Mode generating agorithm (genera case)] 1) Initiay, et T = L, =, compute π. 2) If max T π >, et = arg max T π, and raise the power of ink to the next eve. Update π. 3) Let L( ) be the set of inks in primary confict with, and et T = T\L( ). 4) Continue unti T = or max T π. IV. ROVIDING END-TO-END UTILITIES Now we proceed to consider network utiity maximization for end-to-end fows. Assume that there are F sourcedestination fows in the network. Let s f denote the of fow f, and U(s f ) be the utiity that the user gets by achieving this rate. U(s f ) is assumed to be a conce function, defined according to the objective. Denote the source and destination nodes of fow f by b(f) and e(f), respectivey, and the set of a fows by F. The objective is to maximize the sum of the utiities of a the fows F f=1 U(s f ). We aow muti-path routes, and use a muti-commodity fow mode for the routing of data packets in the network. Such a mode is widey used in the iterature of network routing and optimization. Each source-destination fow f corresponds to a commodity in the network. Let denote the traffic fow that is assigned to ink by the routing scheme corresponding to fow f. The fow assignment given by the routing ayer must satisfy the fow conservation constraints at each node n: =,n N\{b(f),e(f)}, f F, I(n) x f = = s f,f F, O(S(f)) I(D(f)) This can be compacty written as Ax f = s (f), f F, (14) where A is the node-ink incidence matrix. The set of feasibe transmission modes is the set defined in Section III-B, which consists of M power vectors that satisfy the primary constraints. The probem we are interested in is: given the per node power budget, what is the optima joint power contro, MAC scheduing and routing scheme that maximizes the sum of the end-to-end utiities. This can be formuated as max U(s) = U(s f ) (15) Ax f = s (f), f F, (16) R ( ) L (17) α m n, n N (18) α m 1. (19) which is the maximization of a conce function over a convex region defined by a set of intersecting inear regions. Inequaity (17) states that the effective capacity of ink provided by time sharing the modes must be greater than or equa to the sum of the fow rate through the ink assigned by the routing ayer; (18) corresponds to the erage power constraint. Note that the optima vectors α and define a vector of ink capacities, c, and a vector of node erage power eves, e, respectivey, via: c = α M α m R( ) (2) e = α M α m B (21) where B is the transmitter-ink incidence matrix, whose (n, ) entry is given by { 1, if n is the transmitter of ink, B(n, ) = (22), otherwise. Given the optima ink capacities, the optima s and x are 5218
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 characterized as the soution of the probem: max U(s) = U(s f ) 1 9 8 7 s1 d2 Ax f = s (f), f F, c, L This provides a natura decouping of the network operations: the higher ayers decide the optima source rates s and fow assignments x that achieve the maximum utiity, whie the physica and MAC ayers find the optima operating point c through time sharing among a set of transmission modes. Note that the optima (c, e ) ies in R+ L+N ; moreover, it ies in the convex hu of the set {(R( ), B ),m M}. By Caratheodory s theorem [1], it can be expressed as a convex combination of at most L + N +1 vectors from this set. Thus, as a generaization of the inear programming resuts from the previous sections, at most L + N +1 of the α m variabes are nonzero in the optima soution. Further, we can restrict ourseves to such vectors in the search to find the optima soution. This eads us to consider a generaization of the coumn generation method that we outined in the previous sections. At each coumn generation iteration, we choose a sma set of transmission modes M M with M = L + N +1, and sove the foowing restricted probem: max U(s) = U(s f ) (23) Ax f = s (f), f F, (24) α m R ( ), L (25) α m n, n N (26) α m =1. (27) Since U(s f ) is a conce function, (23) is a conce program with inear constraints, which can be easiy soved when M is a sma set. Furthermore, there is no duaity gap, and we can find the optima dua variabes corresponding to the constraints (25), (26) and (27), denoted by {µ, L}, {λ n,n N}and β, respectivey. The corresponding coumn generation subprobem is min λ n m µ R m + β(28) L \M n N n N λ n L µ R m + β<. (29) Note that (28) is identica to (12) if we et V ( ) = L µ R m. Simiary, we define π = L µ R ( ) λ n:. We can empoy the same heuristic Agorithm 2 to generate new transmission modes. Fig. 1. 6 5 4 3 2 1 s2 1 2 3 4 5 6 7 8 9 1 Location of the nodes in the simuated network. V. NUMERICAL RESULTS Due to space imitations, we demonstrate the C-TS scheme ony through a simpe numerica exampe (refer to [9] for more resuts). Note that the utiity function can be defined in different ways according to the objective. Here we choose the utiity function to be the transport capacity of the network, U(s) = U(s f )= s f d f, where d f d1 is the distance between node b(f) and e(f). We generate the network topoogy by randomy pacing N nodes in a 1 1 m square. The path oss between, where d ij is the distance between node i and j. The bandwidth is assumed to be W =2MHz.We do not assume any specific moduation and coding scheme, but assume that the rate function on the SINR is given by the Shannon function as in (2) where node i and j is G ij =1 4 d 3.5 ij the SINR γ m ( ) is given by (1), and the noise power is σ 2 =1 12. Assume that each node has a maximum power ax = 1 mw, and erage power budget =4 mw. We assume that each node has 5 power eves, from 2 mw to 1 mw, in increments of 2 mw. We construct a simpe network with N =14nodes, with the ocations shown in Fig. 1. We consider two end-to-end fows with source and destination nodes denoted by s1, s2 and d1, d2, respectivey. There are atogether L = 182 possibe inks. Theoreticay the size of the transmission mode set is M L + N + 1 = 197. racticay the number of transmission modes needed is often much ess. In this exampe, we set M =3, and start by randomy choosing M transmission modes. At each iteration, we sove the probem (23), and based on the optima dua variabes, find a new transmission mode according to Agorithm 2. Simutaneousy, a transmission mode with α m =is pivoted out. The evoution of the achieved utiity with the coumn generation iterations is shown in Fig. 2. Soving probem (23) aso gives the the corresponding muti-path routes. The muti-path routes for fow s1 d1 and s2 d2 are shown in Fig. 3 and Fig. 4, respectivey, where the thickness of the ines represents the fow rates. It can be sen that due to the power budget and interference, 5219
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 14 x 18 achieved utiity with coumn generation 1 mutipath fows for s2 >d2 9 12 8 d2 1 7 6 U(s) 8 5 4 6 3 2 s2 4 1 Fig. 2. 2 1 2 3 4 5 6 7 iteration Evoution of the achieved utiity with coumn generation iterations. 1 1 2 3 4 5 6 7 8 9 1 Fig. 4. Muti-path fows for s1 d1. 1 mutipath fows for s1 >d1 9 9 8 7 6 5 4 3 s1 8 7 6 5 4 3 2 2 d1 1 1 1 2 3 4 5 6 7 8 9 1 Fig. 3. Muti-path fows for s1 d1. 1 2 3 4 5 6 7 8 9 1 (a) m =1, α m =.438 Fig. 5. Transmission modes and corresponding activation time α m. the muti-path fows tend to use short rather than ong inks. When the coumn generation iterations converge, we see that there are ony 2 transmission modes with α m >. Due to space, we ony show a sampe transmission mode in Figure. 5, where the active inks are marked by ines with arrows pointed from the transmitter to the receiver, and the number of red circes represents the transmit power eve. As can be seen from the resuts, the utiity maximization tends to choose high transmit power eves but we spaced active inks. the node power constraints: 1 T B(t)dt T where is the vector of node erage power constraints, and B is the transmitter-ink incidence matrix defined in (22). The genera continuous version of the end-to-end utiity maximization probem can then be written as VI. CONTINUOUS OWER LEVELS The discretization of the space of power vectors provides us with a combinatoria approach, which suffers from a corresponding exposion in compexity as the size of the network grows. It might be wondered if a more tractabe approach woud be to treat the power vectors as continuous variabes, in a continuous reaxation. Certainy, many of the observations made in this paper hod in the continuous case. For exampe, et be any measurabe, compact subset of R L + that satisfies the primary constraints. We can consider any time interva (,T), and any measurabe power aocation to the inks, (t),t (,T) for which (t) for a t, and which satisfies max U(s) = U(s f ) (3) Ax f = s (f), f F, (31) 1 T R ( (t))dt, L (32) T 1 T B(t)dt T, (33) which is an optimization over a feasibe, measurabe power aocation poicies, as we as ink end-to-end rates. This 522
46th IEEE CDC, New Oreans, USA, Dec. 12-14, 27 FrB2.5 optimization can be written more compacty as max U(s) = U(s f ) (34) Ax f = s (f), f F, (35) R ( ) dρ( ), L (36) B dρ( ), (37) where, for any power aocation (t), ρ is the corresponding probabiity measure on, defined for measurabe sets G by: ρ(g) = 1 T T I[ (t) G]dt. (38) Note that (3)-(33) optimizes over probabiity measures ρ on. Again, Caratheodory s convexity theorem can be used to simpify the form of the optima soution to (3)-(33). Let ρ now denote the unique measure that achieves the optimum, and write the ink capacities, and erage node power eves, respectivey, as c = R ( ) dρ( ) (39) e = B dρ( ). (4) Since (c, e ) ies in the convex hu of the set {(R( ), B) : }, it foows that there are M feasibe transmission modes (1), (2),..., (M) and nonzero time-sharing variabes α 1,α 2,...,α M, where M L + N +1, such that M c = α m R( (m) ) (41) e = M α m B (m). (42) If a genie were to provide the M transmission mode vectors, we coud repace (3)-(33) with the much easier: max U(s) = U(s f ) (43) which is the maximization of a conce function over a convex region defined by inear constraints, with ony a sma number of variabes. Unfortunatey, we he no insight into how to seect the optima transmission modes (1), (2),..., (M). Nevertheess, further work in this direction may be fruitfu if some additiona structure to the probem can be discovered to identify the optima transmission modes. VII. CONCLUSIONS We considered a joint power contro and transmission scheduing probem in wireess networks with erage power constraints in the network utiity maximization framework. This probem is known to be computationay intractabe. We formuate it as an optimization probem invoving timesharing across different transmission modes, where each transmission mode corresponds to the set of noda power eves used in the network. We estabish the structure of the optima soution as time-sharing across a sma fixed set of such modes. We use this structure to deveop a heuristic approach to find a suboptima soution through coumn generation iteration. It provides a too for wireess network panning and sow time scae operations. REFERENCES [1] H. G. Eggeston, Convexity. Cambridge University ress, 1969. [2] X. Lin and N. Shroff, Joint rate contro and scheduing in mutihop wireess networks, IEEE CDC, 24. [3] A. Eryimaz and R. Srikant, Joint congestion contro, routing and MAC for stabiity and fairness in wireess networks, IEEE JSAC, 24(8), 26. [4] A. Gupta, X. Lin, and R. Srikant, Low-compexity distributed scheduing agorithms for wireess networks, 26, preprint. [5] M. Neey and E. Modiano, Fairness and optima stochastic contro for heterogeneous networks, IEEE INFOCOM, 25. [6] A. Stoyar, Maximizing queueing network utiity subject to stabiity: Greedy prima-dua agorithm, Queueing Systems, 25. [7] M. Johansson and L. Xiao, Cross-ayer optimization of wireess networks using noninear coumn generation, IEEE Trans. on Wireess Commun., 5(2), 26. [8] R. Etkin, A.. arekh and D.Tse, Spectrum Sharing in Unicensed Bands, IEEE JSAC, 25(3), 27. [9] M. Cao, Anaysis and cross-ayer design of medium access and scheduing in wireess mesh networks, h.d. dissertation, University of Iinois at Urbana-Champaign, 27. Ax f = s (f), f F, (44) M α m R ( (m) ), L (45) M α m (m) n, n N (46) M α m =1. (47) 5221