II. PROBLEM FORMULATION. A. Network Model

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1 Cross-Layer Rate Optimization in Wired-cum-Wireless Networks Xin Wang, Koushik Kar, Steven H. Low Department of Electrical, Computer and System Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180, USA Department of Computer Science and Electrical Engineering, California Institute of Technology, Pasadena, California 91125, USA Abstract In this paper, we address the end-to-end rate optimization problem in a wired-cum-wireless network, where CSMA/CA based wireless LANs extend a wired backbone and provide access to mobile users. The objective is to achieve proportional fairness amongst the end-to-end sessions in the network. Since the network contains wireless links whose attainable throughput is elastic and is a non-convex and non-separable function of transmission rates of the neighboring links, the problem requires joint optimization at both the transport layer and the link layer. A dual-based algorithm is proposed in this paper to solve the rate optimization problem. It is implemented in the distributed manner, and works at the link layer to adjust transmission rates for the wireless links in the basic service sets, and at the transport layer to adjust session rates. We prove rigorously that the proposed algorithm converges to the globally optimum points. Simulation results are provided to support our conclusions. I. INTRODUCTION In recent years, CSMA/CA based wireless LANs are being successfully used as the last-mile technology in the present-day pervasive computing environments: they provide sufficient bandwidth for office applications with relatively limited mobility, and typically the users may roam inside a building or a campus. Although wireless LANs do not replace wired networks, they extend the wired networks where it is impractical or overly expensive to use cabling. In a typical wired-cum-wireless network, mobile hosts MHs), such as laptop computers, peripherals and storage devices can roam in the wireless networks, called basic service sets BSSs), which are attached at the periphery of a wired backbone infrastructure). The wired infrastructure can be an IEEE 802 style Ethernet LAN or some other IP based networks. A typical wired-cum-wireless network is shown in Fig. 1. Access Point Access Point Wired Backbone Access Point Access Point Fig. 1. A Wired-cum-Wireless Network. The wired and wireless networks are inter-connected via Access Points APs), which are actually fixed base stations that provide interfaces between the wired and wireless parts of the network and control each BSS. The

2 MHs can roam from one BSS to another. A MH within a BSS can only access the infrastructure through its AP, and it is assumed that, in each BSS, all the MHs are within the broadcast region of that particular AP. This study addresses the problem of rate optimization for the end-to-end sessions in wired-cum-wireless networks, in which end-to-end sessions will run across both wired and wireless links. The goal of the rate control is to provide proportional fairness amongst the end-to-end sessions in the network. The rate optimization problem in wired networks has been extensively studied, e.g., [3], [4], [5], and [6]. It has been proved that, since the feasible rate region can be represented by a set of simple, separable, convex constraints in wired networks, the globally fair rate is attainable via distributed approaches based on convex programming. The rate optimization for single-hop flows in wireless networks has also been widely considered. In [9] Tassiulas et al. have proposed a centralized algorithm to attain max-min fair rate in certain ad-hoc networks. On the other hand, Nadagopal et al. [7] and Ozugur et al. [8] have proposed decentralized algorithms that try to achieve some fair rate allocations. Kar et al. [11] have derived distributed strategies that achieve proportional fairness for single-hop flows in an Aloha network using only local topology information. In [13] Wang et al. provide distributed algorithms to achieve max-min fair rates in Aloha networks. In [14] the authors consider fair rate provisioning amongst the single-hop flows in CSMA/CA based ad-hoc networks, and propose distributed algorithms to provide fair rate allocations with local topology information. Even rate control for multi-hop sessions in wireless networks has been considered, e.g. [10] and [15]. However, none of these works considers the special feature when a session may run across both wired and wireless links, and their results are not readily applicable to a wired-cum-wireless network. In this paper, we address the problem of rate optimization in the wired-cum-wireless networks in the framework of nonlinear optimization. For simplicity of exposition, we assume that all end-to-end sessions originate and terminate in MHs, and the source and destination MHs of any session belong to different BSSs. However, our results can be easily extended to the scenarios where end-to-end sessions within a BSS are allowed. It is worth noting that each session in the network runs across both wired links which have fixed link capacities and wireless links whose capacities depend on the transmission rates of MHs in that particular BSS. Just like its counterpart in the wireless networks, the problem of rate optimization in wired-cum-wireless networks has a complex and non-convex feasible region. In circumstances when wireless links become the bottlenecks, it is possible that by adjusting the transmission rates of the MHs, the capacities of the bottleneck links are increased and the overall system utility can be further increased. Therefore, cross layer cooperation must be introduced, which find not only the best rate for each session but also the transmission rates on the wireless links that support the session rates. We propose a dual-based distributed algorithm to iteratively solve the problem of rate optimization in the wiredcum-wireless network. Intuitively, the algorithm works at the transport layer in a smaller time scale, solving the optimal end-to-end session rates under the given transmission rates for all wireless links. At the link layer, the algorithm works at a much larger time scale, adjusting the transmission rates for the wireless links in a way so that the bottlenecks are alleviated and the aggregate utility can be further increased. We make this intuitive approach precise and rigorous in this paper. The distributed algorithm we propose in this paper essentially adjusts the wireless link transmission rates in the gradient direction. Although the formulated problem appears to be non-convex, we show that our approach actually converges to the global optimum. Simulations in various network scenarios support our results, and some representative cases are presented in this paper. The paper is organized as follows. In Section 2, we describe the system model, provide the link rate expressions in a CSMA/CA based BSS, and formulate the end-to-end rate control problem as an optimization problem in Section. The solution approach along with its convergence analysis is discussed in Section 3. In Section 4 the dual-based distributed algorithm is described in detail, and in Section 5 the simulation results are presented. The paper is concluded in Section 6. All the proofs are provided in the appendix. A. Network Model II. PROBLEM FORMULATION We consider a general wired-cum-wireless network modeled as M, W, N, L), where M denotes the set of all MHs, W denotes the set of CSMA/CA based BSSs, N denotes the set of fixed nodes in the wired backbone, and L denotes the set of unidirectional links that connect the fixed nodes in the wired backbone. Note that the set of fixed nodes in the wired backbone are denoted as N A R, where A denotes the set of APs that act as interface between the wired and wireless parts of the network and control all the BSSs, and R is the set of fixed intermediate

3 nodes in the wired backbone that work as routers and forward packets. For simplicity, we assume in this paper that each MH belongs to one and only one BSS, and each BSS has one and only one AP. BSS w, w W, can be modeled as an undirected graph G w N w, L w ), where N w and L w respectively denote the set of nodes and the set of undirected links in that particular BSS. Note that N w M w Aw, where M w and A w are the set of MHs and the AP for BSS w respectively. Denote As) as the AP associated with MH s M, i.e. As) A w if s M w. In BSS w, a link exists between two nodes if and only if they can receive each other s signals. A directed edge s, t) represents an active communication pair for s, t N w, and E w is the set of directed edges in BSS w. For any node s N w, the set of s s out-neighbors, O s {t : s, t) E w }, represents the set of neighbors to which s is sending traffic. Also, the set of in-neighbors of s, I s {t : t, s) E w }, represents the set of neighbors from which s is receiving traffic. We assume that each node has a single transceiver. A node can not transmit and receive simultaneously, and cannot receive more than one frame at a time. We further assume that the scheduling point process for each used link s, t) E w is Poisson and is independent of all other such processes in the network. In addition, the frame lengths are assumed to be exponentially distributed. The transmission rate for a wireless link s, t) is denoted as ρ s,t, which is the average Poisson transmission attempts made during an average frame transmission time. The propagation delay is assumed to be zero in the network model. The RTS and CTS are assumed to be very small, and their transmission time can be ignored. Acknowledgements are obtained instantaneously. For simplicity, we assume that none of the nodes adopts BEB Binary Exponential Backoff) algorithm. This is reasonable, as each node will choose its maximum contention window size according to the extent of contention in the wireless network and hence dynamic adaption of the contention window size is no longer necessary. Since end-to-end sessions within a BSS are not allowed according to the assumption, an immediate result is that all links in a BSS are between its MHs and the AP, and hence no two links in a BSS can be scheduled at the same time. Let ρ ρ s,t, s, t) E w, w W ) be the vector of transmission rates for all wireless links in the wired-cum-wireless network. Based on the analysis in [14], it can be easily shown that the attainable throughput on link s, t) in BSS w, in which either s or t must be the AP A w, can be expressed as ρ s,t c s,t ρ) 1 k O Aw ρ Aw,k s, t) E w. 1) k I Aw ρ k,aw Note that the terms k O A ρ Aw,k and w k I A ρ k,aw are the sums of transmission rates on all downlinks and w uplinks respectively in BSS w. The wired backbone of the network connects the set of APs A using the set L of unidirectional wired links whose capacity is c l, l L. A w connects to A v through a path Lw, v), where the path LA w, A v ) is the set of links that are used for the communication from A w to A v. For each link l L, let Sl) {A w, A v ) w, v W and l LA w, A v )} be the set of communication pairs consisting of APs that use link l. The wired-cum-wireless network is shared by the set S of end-to-end sessions. A session u S can be usually expressed as i, j), where w, v W, i M w, and j M v, meaning that the origin of the session is MH i in BSS w, and the sink is MH j in BSS v. As mentioned before, for simplicity of exposition, we assume that w and v denote different BSSs, and the results can be easily extended to the scenarios where end-to-end sessions within a BSS are allowed. Also note that APs generate no traffic. They only forward the traffic between the wireless and the wired parts of the network. B. Rate Control Problem In the wired-cum-wireless network, we need to provide proportional fairness between the end-to-end sessions. The problem can be formulated as follows. P : max i,j) S logy ij) s.t. y ij c i,ai) ρ) i, j) S y ij c Aj),j ρ) i, j) S Ai),Aj)) Sl) y ij c l l L ρ s,t 0 s, t) E w, w W y ij 0 i, j) S 2)

4 where y ij is the session rate for the session that originates from MH i and ends at MH j. The objective of session rate control and wireless link capacity allocation in the wired-cum-wireless network is to achieve proportional fairness amongst the end-to-end sessions, which is formulated as the maximization of the aggregate utility of the sessions. Since the sessions originate from one wireless network and end at another, they will travel two wireless links, one is from the origin to the access point, and the other is from the access point in the destined wireless network to the sink. The first and the second sets of constraints state that the rate of the end-to-end sessions cannot exceed the capacities of the two wireless links that are traveled, while c i,j ρ) is given in 1). Also, such sessions will travel through a set of links in the wired backbone, and the third set of constraints states that, the aggregate session rates on a wired link cannot exceed the capacity of that link. The fourth and the last sets of constraints ensure respectively that all the transmission rates and all session rates are non-negative. In contrast to the wired backbone which has fixed capacities for each link, the attainable throughput on a link of a CSMA/CA based BSS has a changeable value. Note that, under certain resource allocation scheme, the wireless link rates are computed and the session rates are adjusted so that the aggregate utility is maximized. However, since the wireless link capacities are not fixed, it is possible that, through resource reallocation, capacities of those bottleneck links in the wireless network can be increased and the aggregate utility can be further increased. Therefore, P involves not only the rate control for each end-to-end session, but also the best resource allocation schemes to support the session rates. Also note that the link rates in a CSMA/CA based BSS are not a concave function of the transmission rates, and hence the feasible rate region does not constitute a convex set. This makes P even more difficult to solve. In spite of the apparent non-convexity, we derive a distributed algorithm that converges to the global optimum of this problem. A. Solution Approach III. SOLUTION APPROACH AND CONVERGENCE ANALYSIS We now present a dual-based approach to solve P iteratively. Instead of solving P directly, we consider the parameterized version of an end-to-end proportionally fair rate optimization problem when the wireless link rates c s,t ρ) are parameterized as x s,t for any s, t N w, s, t) E w, and w W, ˆP : max i,j) S logy ij) s.t. y ij x i,ai) i, j) S y ij x Aj),j Ai),Aj)) Sl) y ij c l y ij 0 i, j) S l L i, j) S Comparing P with ˆP, we see that the only difference is that, the wireless link capacities in P, whose values are not fixed and depend on the transmission rates of the wireless links, are parameterized in ˆP. Note that the optimum value in ˆP is a function on x, where x is the vector of capacities of all wireless links, i.e. x {x i,ai), x Aj),j : i, j) S}. We define Ûx) as the optimum value in ˆP when x is parameterized, i.e. { } Ûx) max logy ij ) y ij x i,ai), y ij x Aj),j, y ij c l, y s 0. 4) i,j) S Ai),Aj)) Sl) Since the vector of capacities of the wireless links in P in turn is a function on the link transmission rates, we can define function Ũρ) Ûcρ)), where cρ) c s,tρ) : s, t) E w, w W ). Therefore problem P can be rewritten as P : max Ũρ) 5) s.t. ρ s,t 0 s, t) E w, w W. In summary, to solve P, we update the transmission rate on link s, t) in BSS w using the following equation ρ n1) s,t ρ n) s,t δ c k,m ρ n) ), 6) v W k,m) E v λ n) k,m 3)

5 where [z] max{z, 0}, δ is the step size, c k,m is computed with the following formula ρ k,m 1 u O A ρ w Aw,u u I A ρ u,aw ) c 2 if v w and k, m) s, t); w k,m 1 u O Aw ρ Aw,u u I Aw ρ u,aw ρ s,t 1 u O A ρ w Aw,u u I A ρ u,aw ) 2 if k, m) s, t); w 0 otherwise, and λ n) s,t is the optimum solution to the dual problem of ˆP when x cρ n) ), i.e. λ n) arg min max 0, 0 y Ln) y, λ, γ) 8) where λ λ s,t : s, t) E w, w W ) are the Lagrange multipliers for the capacity constraints on the wireless links, γ γ l : l E) are the Lagrange multipliers for the capacity constraints on the wired links, y y ij : i, j) S) are the end-to-end session rates, and L n) y, λ, γ) is the Lagrange function of ˆP when x cρ n) ). Note that L n) y, λ, γ) is given by L n) y, λ, γ) i,j) S logy ij) ) l γ l Ai),Aj) Sl) y ij c l [ ) )] 9) i,j) S λi,ai) yij x i,ai) λaj),j yij x Aj),j. 7) We then solve y n) from ˆP when x cρ n) ), i.e. y n) arg max logy ij ) y ij c i,ai) ρ n) ), y ij c Aj),j ρ n) ) Ai),Aj)) Sl) y ij c l, y s 0 i,j) S. 10) B. Convergence Analysis The end-to-end proportionally fair rate optimization problem in P appears to be a non-convex problem. However, the following theorem proof in the appendix) states that following the procedures stated above, the solutions converge to the globally optimum points. Theorem 1: Let {ρ n) δ), y n) δ)} denote the sequence of vectors of transmission rates of the wireless links and the end-to-end session rates computed with the iterative procedures stated in 6)-10) when the step size is δ. Then there exists an R such that for δ <, the limit point of {ρ n) δ), y n) δ)} is the global optimal solution to the problem P. Intuitively, the procedures from 6) to 9) adjust the transmission rates of the wireless links in the gradient direction. Therefore the sequence of {ρ n) δ)} converge to a local optimal point or KKT point in P, where the Karush-Kuhn-TuckerKKT) conditions hold. Since P and P are equivalent, it can be shown that the KKT point of P actually gives the KKT point of P if y is solved by 10). Therefore the procedures from 6) to 10) converge to the KKT point of P. We further prove that, although P appears to be non-convex, its KKT points are actually globally optimum. IV. THE DUAL-BASED DISTRIBUTED ALGORITHM In this section, we describe in detail the distributed implementation of the dual-based algorithm to solve the proportionally fair rate control problem P. The algorithm works at both the transport layer and the link layer. Every time a CSMA/CA based BSS chooses the transmission rates for its links, the wireless link rates are updated accordingly. The algorithm then works at the transport layer, searching for optimal end-to-end session rates and the corresponding optimal link prices under the updated link rates. Once the optimal session rates and link prices for the wireless links are found, the algorithm works at the link layer again and adjusts the transmission rates using link prices and transmission rates of wireless links in its neighborhood. Therefore the proposed algorithm works at the link layer in a larger time scale and at the transport layer in a much smaller time scale.

6 A. Flow Rate Control Algorithm at the Transport Layer At the transport layer, the algorithm solves the rate control problem ˆP. When the link transmission rates have been updated, wireless link rates in every CSMA/CA based BSS are computed accordingly. The algorithm at the transport layer is then executed for the wired-cum-wireless networks. Although the network has CSMA/CA based BSSs, the wireless links therein have fixed capacities when the link transmission rates are given. The problem is essentially the same as the end-to-end proportionally fair rate control problem in a wired network. In fact, the algorithm at the transport layer follows the procedures stated in [5], i.e., each source adjusts its session rate and each link adjusts its link price in a response to the updated link transmission rates until the optimal solutions are achieved. Note that the algorithm not only gives the optimal rates, but also gives the corresponding Lagrange multipliers, which is also known as link prices. We now state the procedures to solve the dual problem of ˆP [2][5] when x cρ n) ). When the vector of transmission rates is λ n), the Lagrangian in 9) can be rewritten as L n) y, λ, γ) logyij ) y ij α ij) γ l c l ) λ i,ai) c i,ai) ρ n) ) λ Aj),j c Aj),j ρ n) ) 11) where i,j) S l i,j) S α ij λ i,ai) λ Aj),j l LAi),Aj)) γ l 12) is the sum of link prices for all the links the session i, j) S uses. Therefore the dual problem is min 0, 0 Dn) λ, γ) 13) where D n) λ, γ) max y Ln) y, λ, γ). 14) Since the logarithmic function is concave and the constraints for rate allocations are linear, ˆP is a convex program and has no duality gap. So at x cρ n) ), when the dual problem of ˆP achieves its optimum, denoted as λ n), γ n) ), the corresponding y, denoted as y n), is the optimum solution to the primal problem. Maximizing the dual function in 14) gives y ij λ) 1 α ij 15) where α ij is given in 12). The dual problem can then be solved using gradient projection method, where the Lagrange multipliers are adjusted in the opposite direction to the gradient D n) λ, γ): [ ] λ i,ai) k 1) λ i,ai) k) β Dn) [ λ λ i,ai) k)) λ i,ai) k) βy ij λk)) c i,ai) ρ ))] n) 16) i,ai) [ ] λ Aj),j k 1) λ Aj),j k) β Dn) [ λ λ Aj),j k)) λ Aj),j k)βy ij λk)) c Aj),j ρ ))] n) 17) Aj),j γ l k 1) [ γ l k) β Dn) γ l k)) γ l ] [ γ l k) βy l λk)) c l ] 18) where β > 0 is the step size, [z] max{z, 0}, λk) λ s,t k) : s, t) E w, w W ), and y l λ) Ai),Aj)) Sl) y ijλ) is the aggregate session rates at link l L. The rate control algorithm at the transport layer can be summarized as follows. When the transmission rates of the wireless links are given, the attainable throughput on the wireless links are updated. Then for each link in the network, wired or wireless, receives rates for all sessions that go through the link, computes the new price using 16) or 17) or 18), and communicates new link price to sources of the sessions that use the link. For each session i, j) S, its source MH i receives from the network the link prices in its path and calculate α ij using

7 12), chooses a new rate for the next period using 15), and communicates new rate to all the links in its path. The procedures at the link side and the session sources are repeated until the algorithm converges to the optimal session rates and optimal link prices. B. Transmission Rate Adjustment Algorithm at the Link Layer When the optimal link prices have been achieved at the given wireless link rates, the proposed algorithm will work at the link layer to update the transmission rates of the wireless links using 6) and 7). The main purpose of the transmission rate adjustment is to change the wireless link rates and ensures that the bottleneck wireless links are alleviated so that the aggregate utility can be further increased. It is worth noting that in 6) transmission rates, which are previously used at the link layer, and link prices, which are provided by the algorithm at the transport layer, are used to update the transmission rate of a link. Also note that the partial derivative ck,m is zero if wireless link k, m) and s, t) are not in the same BSS. Therefore, the transmission rate of link s, t) can be updated using the information located in its BSS, although 6) states that transmission rates and link prices of all wireless links are necessary. C. Distributed Implementation of the Algorithm The dual-based algorithm for the end-to-end proportionally fair rate allocations in a wired-cum-wireless network can be summarized as follows: 1. Set n to 1. For any wireless link s, t) E w, w W, choose transmission rate that satisfies ρ n) s,t > Compute the link price and session rates in a distributed manner using the rate control algorithm at the transport layer with procedures 15) to 18). 3. Update the transmission rates for the wireless links at the link layer using the procedures 6) and 7). 4. Increment n. Repeat step 2 and 3 until the transmission rates, the link prices and the session rates converge. V. SIMULATION INVESTIGATION In this section, we investigate the performance of the distributed algorithm in providing proportional fairness amongst the end-to-end flows in the wired-cum-wireless networks. A median-sized network is considered, which is composed of 4 access points, 8 mobile nodes, 4 wired links and 8 wireless links. The network is configured as follows. D d Access Point 1 c C 1 e E B b A 3 a Access Point 2 0 Access Point 0 G g f 2 Access Point 3 h F H Fig. 2. The network configuration. The network has 4 access points, denoted as 0, 1, 2 and 3. The wired part of the network connects the APs through the wired links, denoted as 0, 1, 2 and 3. The capacities of the wired links are 0.5, 0.2, 0.6 and 0.8 respectively. In each basic service set, there are two MHs, who connect to the network through the AP. We denote the 8 MHs as A, B, C, D, E, F, G, H, and the wireless links are a, b, c, d, e, f, g and h respectively. There are four end-to-end sessions in this network, labeled as f 0, f 1, f 2 and f 3, and they are set up as shown in Table I. To investigate the performance of the proposed algorithm, we compare the globally optimum solutions solved by AMPL with the solutions given by the dual-based algorithm. The results are presented in Table II. The comparisons show that the proposed algorithm converge to the global optimum.

8 TABLE I THE SOURCE, SINK, AND PATH OF THE FLOWS. Session Source Node Sink Node Links on the Path f 0 E A e, 0, a f 1 B G b, 0, 2, g f 2 C F c, 3, 2, f f 3 H D h, 2, 1, d TABLE II THE OPTIMUM RESULTS AND THE SOLUTIONS GIVEN BY THE DISTRIBUTED ALGORITHM. Variables y 0 y 1 y 2 y 3 U Optimum Value Solutions by the algorithm How session rates and the aggregate utility converge at each link layer iteration is also plotted in Fig. 3. Note that the algorithm works at the link layer and at the transport layer work in different time scales. At the link layer, the transmission rates are adjusted every time the link prices and session rates have converged at the transport layer. Therefore the algorithm at the link layer works at a larger time scale, while the algorithm at the transport layer works at a much smaller time scale. Since we do not care how the link prices and session rates converge at the transport layer in this paper, we only present results of the optimal end-to-end session rates and aggregate utility when link transmission rates have been adjusted in the link layer at outer loop iteration Session Rates f 0 f 1 f 2 f Aggregate Utility Fig. 3. The session rates and the aggregate utility. VI. CONCLUSIONS In this paper, we consider the problem of providing proportional fairness amongst the end-to-end sessions in a general wired-cum-wireless network. The BSS is CSMA/CA based wireless networks, which are deployed to provide accesses for mobile hosts. Wired links with fixed capacities connect the access points, through which wireless and wired parts interact. The problem is considered in the framework of nonlinear optimization, and since the network contains wireless links whose attainable throughput is elastic and is a function of the transmission rates on wireless links in the same BSS, the problem requires joint optimization at both the transport layer and the link layer.

9 A dual-based algorithm is proposed in this paper to solve the rate optimization problem. It is implemented in the distributed manner, and works at the link layer to adjust transmission rates for the wireless links in the basic service sets, and at the transport layer to adjust session rates. Heuristically, the algorithm finds the optimal session rates when wireless links are given certain initial capacities. Then the link capacities are adjusted in a way so that the bottleneck wireless links are alleviated and the aggregate utility can be further increased. Although the formulated problem appears to be non-convex, we prove rigorously that the proposed algorithm converges to the globally optimal solutions. Simulation results are provided to support our conclusions. REFERENCES [1] D. Bertsekas and R. Gallagher, Data Networks, Prentice Hall, [2] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, [3] F. P. Kelly, Charging and Rate Control for Elastic Traffic, European Transactions on Telecommunications, vol. 8, no. 1, 1997, pp [4] F. Kelly, A. Maulloo, D. Tan, Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability, Journal of Operations Research Society, vol. 49, no. 3, 1998, pp [5] S. Low, D. E. Lapsley, Optimization Flow Control, I: Basic Algorithm and Convergence, IEEE/ACM Transactions on Networking, vol. 7, no. 6, December [6] K. Kar, S. Sarkar, L. Tassiulas, A Simple Rate Control Algorithm for Maximizing Total User Utility, in Proceedings of Infocom 2001, Anchorage, USA, April [7] T. Nandagopal, T.-E. Kim, X. Gao, V. Bharghavan, Achieving MAC Layer Fairness in Wireless Packet Networks, in Proc. ACM/IEEE MobiCom 00, pp. 8798, Boston, USA, August [8] T. Ozugur, M. Naghshineh, P. Kermani, C. M. Olsen, B. Rezvani, J. A. Copeland, Balanced Media Access Methods for Wireless Networks, in Proc. ACM/IEEE MobiCom 98, Dallas, USA, October [9] L. Tassiulas, S. Sarkar, Maxmin Fair Scheduling in Wireless Networks, in Proceedings of Infocom 2002, pp , New York, USA, June [10] M. Chiang, To Layer or Not To Layer: Balancing Transport and Physical Layers in Wireless Multihop Networks, in Proceedings of Infocom 2004, Hong Kong, China, March [11] K. Kar, S. Sarkar, L. Tassiulas, Achieving Proportionally Fair Rates using Local Information in Multi-hop Aloha Networks, Extended Abstract. in Proceedings of Annual Allerton Conference, Urbana-Champaign, USA, October [12] K. Kar, S. Sarkar, L. Tassiulas, Distributed Algorithms for Optimal Rate Allocation of Multipath Traffic Flows, Submitted for publication, December [13] Xin Wang, Koushik Kar, Distributed Algorithms for Max-min Fair Rate Allocation in Aloha Networks, Proceedings of Annual Allerton Conference, Urbana-Champaign, USA, October [14] Xin Wang, Koushik Kar, Throughput Modelling and Fairness Issues in CSMA/CA Based Ad-Hoc Networks, To appear in Proceedings of Infocom 2005, Miami, March [15] Xin Wang, Koushik Kar, Cross-Layer Rate Optimization in Multi-Hop Aloha Networks, To appear in Proceedings of ICC 2005, Seural, Korea, May Proof Outline of Theorem 1 APPENDIX To simplify the proof, we assume that the capacity constraints in ˆP are not degenerate for any x. Denote the optimal Lagrange multipliers for the link capacity constraints on the wireless links as λ when ˆP is parameterized at x. According to the Sensitivity Theorem[2], we obtain that x Û x) λ 19) and there exists an open sphere B centered at x such that for every x B there is an λx) which are the associated Lagrange multipliers. Furthermore, λ ) is continuous differential in B and λ x) λ. Therefore Û is total differentiable for x, and the differential is dû Û dx s,t λ s,t x)dx s,t x s,t s,t) E w,w W s,t) E w,w W Since Ũρ) Ûcρ)), and since cρ) is total differentiable for ρ, it follows that Ũρ) is total differentiable for ρ. Therefore we have the following property. Lemma 1: If for s, t) E w we define d s,t c k,m ρ) 20) k,m) E w λ k,m

10 where λ is the vector of Lagrange multipliers for the constraints on the wireless links in ˆP when the wireless link capacities x are c ρ), and denote d d s,t : s, t) E w, w W ), then d is the gradient direction of Ũρ) at ρ, i.e. Ũ ρ) d. It can be verified that the Lipschitz continuity condition holds true here, and therefore it follows from Proposition of [2] that there exists R such that for δ <, the sequence {ρ n) δ)}, which is generated using the procedures stated in 6) to 9), converges to a local optimum point of P. Since P and P are equivalent, a local optimum point in P is also a local optimum point in P. Therefore the following property holds true. Lemma 2: Denote ρ as the local optimum point for P, and ỹ is solved from 10), i.e. { } ỹ arg max logy ij ) y ij c i,ai) ρ ), y ij c j,aj) ρ ) y, 21) Ai),Aj)) Sl) ij c l, y ij 0 i,j) S then ρ, ỹ ) is the local optimum point for P. Proof: Assume that ρ, ỹ ) is not a local optimum point for P, then for any positive a > 0, there exists a point ρ, y ) within the ball centered at ρ, ỹ ) with radius a such that ρ, y ) gives a larger value in the objective of P than ρ, ỹ ) does. Note that ρ ρ, otherwise y ỹ and ρ, ỹ ) and ρ, y ) are the same point, since 21) gives unique y for each ρ. As ρ, y ) gives a larger value in the objective of P than ρ, ỹ ) does, it immediately follows from the definition of P and P that ρ gives a larger value in the objective of P than ρ does. Also note that, the distance between ρ and ρ is less than a since it is less than the distance between ρ, y ) and ρ, ỹ ). Therefore for any a > 0 there exists ρ such that ρ is within the ball centered at ρ with radius a and gives a larger value in the objective in P than ρ does. This contradicts with the fact that ρ is the local optimum for P. Therefore ρ, ỹ ) is the local optimum for P. This completes the proof. From Lemma 2 we conclude that, the stationary point ρ of the sequence {ρ n) δ)}, which is generated using the procedures stated in 6) to 9), and the corresponding y, which is calculated using 10), constitute a local optimum for P. Define z ij, r s,t and d l as the logarithmic values of the session rate y ij, transmission rate ρ s,t, and wired link capacity c l respectively, i.e. z ij logy ij ), r s,t logρ s,t ), and d l logc l ). Since both sides of the first three constraints in P are positive, and since the logarithmic function is strictly increasing, taking logarithmic values on both sides of the first three constraints, we have the following constraints which are equivalent to the constraints in P z ij r i,ai) log 1 k O Ai) e rai),k ) k I Ai) e rk,ai) z ij r Aj),j log 1 k O Aj) e raj),k ) k I Aj) e rk,aj) log Ai),Aj)) Sl) ezij ) dl 22) It is worth noting that, for a > 0 and x i R, i 1,..., n, the function log n i1 ex i a) is a convex function for x 1,..., x n ) see the proof in [10]). It then follows that the transformed constraints in 22) constitutes to a convex set. Therefore we have the following lemma. Lemma 3: The end-to-end proportionally fair rate control problem in a wired-cum-wireless network, as given in P, can be rewritten as the following convex programming problem P e max i,j) S z ij s.t. z ij r i,ai) log 1 k O Ai) e r Ai),k ) k I Ai) e r k,ai) z ij r Aj),j log 1 k O Aj) e r Aj),k ) k I Aj) e r k,aj) i, j) S i, j) S log Ai),Aj)) Sl) ez ij) dl l L where z ij, r s,t and d l are the logarithmic values of the session rate y ij, transmission rate ρ s,t, and wired link capacity c l respectively, i.e. z ij logy ij ), r s,t logρ s,t ), and d l logc l ). Note that the variables y ij, ρ s,t and c l in P have a one-to-one correspondence with the variables z ij, r s,t and d l in P e. Specifically, y, ρ) yields the same objective value in P as the corresponding z, r) does in P e, where 23)

11 y y ij, i, j) S), ρ ρ s,t, s, t) E w, w W ), z z ij logy ij ), i, j) S), and r r s,t logρ s,t ), s, t) E w, w W ). To show that the dual-based algorithm actually converges to the globally optimum values, we have the following property. Lemma 4: If yij, i, j) S and ρ s,t, s, t) E w, w W satisfy the first order necessary condition for P, then zij logy ij ), i, j) S and r s,t logρ s,t), s, t) E w, w W will also satisfy the first order necessary condition for P e. Conversely, if zij, i, j) S and r s,t, s, t) E w, w W satisfy the first order necessary condition for P e, then yij ez ij, i, j) S and ρ s,t e ρ s,t, s, t) Ew, w W also satisfy the first order necessary condition for P. Proof: We only prove the first part of the lemma, and the converse result follows the same line of analysis. Denote y yij, i, j) S), z zij logy ij ), i, j) S), ρ ρ s,t, s, t) E w, w W ), and r rs,t logρ s,t), s, t) E w, w W ). Let U be the objective function, then U i,j) S logy ij) i,j) S z ij. If y and ρ satisfy the first order necessary condition, then there exists λ i,ai), λ Aj),j 0 for i, j) S, and γl 0 for each wired link l L such that U λ i,ai) g ij λ Aj),j h ij γl f l 0 24) i,j) S I1 i,j) S I2 l L I y, where, g ij y ij c i,ai) ρ), h ij y ij c Aj),j ρ), f l y Ai),Aj)) Sl) ij c l, and S I1 {i, j) : g ij y, 0, i, j) S}, S I2 {i, j) : h ij y, 0, i, j) S}, L I {l : f l y, 0, l L} are the index set for the active constraints respectively. Therefore, for y ij, we have U g ij h ij γ f l l y ij y ij y ij y ij l L I y, i,j) S I1 λ i,ai) i,j) S I2 λ Aj),j 1 y ij I i,ai) λ i,ai) I Aj),jλ Aj),j l L I LAi),Aj)) γ l ) y, 0 where I i,ai) and I Aj),j are indicator function to show respectively if g ij and h ij are active, i.e I i,ai) is 1 if g ij is 0, and 0 otherwise, and I Aj),j is 1 if h ij is 0, and 0 otherwise. For ρ s,t, we have U g ij h ij y, i,j) S I1 λ i,ai) i,j) S I1 λ i,ai) c i,ai) i,j) S I2 λ Aj),j i,j) S I2 λ Aj),j c Aj),j 0 y, For P e, denote g ij z ij logc i,ai) r)) and h ij z ij logc Aj),j r)) for i, j) S, and denote f l log ez i,j) ) d Ai),Aj)) Sl) l for l L, where r r s,t logρ s,t ) : s, t) E w, w W ), ρr) e r s,t : s, t) E w, w W ), and c s,t r) c s,t ρr)). Denote S I1 {i, j) : g ij z,r 0, i, j) S}, SI2 {i, j) : h ij z,r 0, i, j) S}, LI {l : fl z,r 0, l L}. Obviously S I1 S I1, SI2 S I2, and L I L I. We take λ i,ai) λ i,ai) y ij for i, j) S I1, λ Aj),j λ Aj),j y ij for i, j) S I2, and γ l c l γl for l L I. 25) 26) Note that for l L I, fl z,r 0, and therefore c l Ai),Aj)) Sl) ez ij. For i, j) SI1, yij c i,ai)r ). For i, j) S I2, yij c Aj),jr ).

12 Therefore, for z ij, we have U z ij y ij For r s,t, we have g ij λ i,ai) z i,j) ij S I1 1 I i,ai) λ i,ai) I Aj),j λ Aj),j 1 y s I i,ai) λ i,ai) y si Aj),j λ Aj),j 1 y ij I i,ai) λ i,ai) I Aj),jλ Aj),j U g ij λ r i,ai) s,t r i,j) s,t S I1 U g ij λ ρ i,ai) s,t ρ i,j) s,t S I1 1 λ i,ai) c i,j) i,ai) r) S I1 i,j) S I1 λ i,ai) c i,ai) hij λ Aj),j γ l z i,j) ij S I2 l L I c i,ai) l L I LAi),Aj)) γ l l L I LAi),Aj)) l L I LAi),Aj)) hij λ Aj),j r i,j) s,t S I2 i,j) S I2 λ Aj),j hij ) ) f l z ij e z ij Ai),Aj)) Sl) ezij γ l ez ij) γ l ) ρ s,t r s,t 1 λ Aj),j c i,j) Aj),j r) S I2 ) ρ s,t 0 λ c Aj),j Aj),j ρ i,j) ij S I2 r s,t 0 c Aj),j ) ) ρ s,t r s,t 27) 28) Therefore we have U ) λ i,ai) g ij λ Aj),j h ij γl f l 0 29) i,j) S I1 i,j) S I2 l L I i.e. z and r satisfy the KKT condition for P e. Using the same line of analysis, we can prove the converse result. This completes the proof. Note that P e is a convex program and its KKT point is actually a globally optimum point. Since a point in P e is a KKT point if and only if its corresponding point is a KKT point in P, and since that a point in P e yields the same objective value as its corresponding points does in P, it immediately follows that the KKT point in P is actually a globally optimum point. Therefore the dual-based algorithm converges to the globally optimum solutions.