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Scheduling, Routing and Power Allocation for Fairness in Wireless Networks Mikael Johansson Lin Xiao Department of Signals, Sensors and Systems Information Systems Laboratory KTH, SE-100 44 Stockholm, Sweden Stanford University, Stanford, CA 94305-9510 Email: mikaelj@s3.kth.se Email: lxiao@stanford.edu Abstract— We consider the problem of ﬁnding the jointly op- allowing nonlinear performance objectives (necessary to obtain timal end-to-end communication rates, routing, power allocation proportional fairness), multi-path routing, and a much wider and transmission scheduling for wireless networks. We focus on range of MAC schemes. The approach is based on a nonlinear throughput and fairness between end-to-end rates and formulate the associated cross-layer design problem as a nonlinear math- column-generation technique and generates a sequence of ematical program. We develop a specialized solution method, feasible designs that converges to the optimum in a ﬁnite based on a nonlinear column generation technique, that applies number of steps. For a given network conﬁguration, our to a wide range of media access schemes and converges to the approach provides the optimal operation of transport, routing optimal solution in a ﬁnite number of steps. The approach is and radio link layers under several important medium access applied to a large set of sample networks and the inﬂuence of power control, spatial reuse, routing strategies and variable control schemes, as well as the optimal coordination across transmission rates on network performance is discussed. layers. This allows us to gain insight in the inﬂuence of power control, spatial reuse, routing strategies and variable I. I NTRODUCTION transmission rates on the network performance, and provides Wireless ad-hoc networks is a promising technology for re- a benchmark for alternative (heuristic, distributed) strategies. alizing the vision of ubiquitous communications. Such systems could allow rapid deployment with little planning or user- II. M ODEL AND ASSUMPTIONS interaction and possibly coexist with a sparse ﬁxed infras- We consider a communication network formed by a set of tructure. However, the design of radio resource management nodes located at ﬁxed positions. Each node is assumed to have schemes that work reliably and efﬁciently in distributed and inﬁnite buffering capacity and can transmit, receive and relay heterogeneous environments is a major engineering challenge, data to other nodes across wireless links. Our model tries to and it is not clear if ad-hoc networks will be a viable technol- capture how end-to-end rates, routing, power allocation and ogy for achieving affordable and scalable communications. scheduling inﬂuence the network performance. In this paper, we develop an approach for computing the performance that can be achieved by optimally coordinating A. Network ﬂow model all networks nodes as well as coordinating the operation of Consider a connected communication network containing several layers of the networking stack. When developing such N nodes labeled n = 1, . . . , N and L directed links labeled cross-layer optimization schemes, it is important to specify an l = 1, . . . , L. The topology of the network is represented by appropriate network performance objective. Fairness is a key a node-link incidence matrix A ∈ RN ×L whose entry Anl is consideration, as simply maximizing the throughput of wire- associated with node n and link l via less networks typically leads to grossly unfair communication 1, if n is the start node of link l rates between source-destination pairs (cf. [1]). Since fairness Anl = −1, if n is the end node of link l often comes at the price of decreased network throughput, 0, otherwise. it is important to have methods that allow us to ﬁnd fair resource allocations and that help us to understand the trade- We deﬁne O(n) as the set of outgoing links from node n and offs between fairness and throughput. I(n) as the set of incoming links to node n. In this paper, we extend our previous work on simultaneous We use a multicommodity ﬂow model for the routing of data ﬂows (cf. [2]) that describes average data rates in bits per routing and resource allocation in wireless networks [2], [3] to second. We identify the ﬂows by their destinations, labeled also address transmission scheduling. The research on optimal d = 1, . . . , D, where D ≤ N . For each destination d, we scheduling of transmissions in multi-hop radio networks has deﬁne a source-sink vector s(d) ∈ RN , whose nth (n = d) (d) a long history (see, e.g., [4] and the references therein), and entry sn denotes the non-negative amount of ﬂow injected our effort is closely related to the recent work reported in into the network at node n (the source) and destined for node d (d) (d) [5], [6], [7]. Our method extends the previous approaches by (the sink), where sd = − n=d sn . We also deﬁne x(d) ∈ RL as the ﬂow vector for destination d, whose component III. T HE SRRAS PROBLEM (d) xl is the (non-negative) amount of ﬂow on link l destined Let unet (x, s) be a concave utility function of the end-to-end for node d. Finally, we let c be the vector of capacities of the individual links. Then our network ﬂow model imposes the rates s and link ﬂows x. The simultaneous routing, resource following group of constraints on the ﬂow variables x and s: allocation and scheduling (SRRAS) problem is then Ax(d) = s(d) , x(d) 0, s(d) d 0, d = 1, . . . , D maximize unet (x, s) (1) D d=1 x (d) c subject to Ax(d) = s(d) , x(d) 0, s(d) d 0, ∀d (3) (d) Here, means component-wise inequality and d means dx c, c ∈ C(P ) component-wise inequality except for the dth component. The The optimization variables are s, x and c, where c is the ﬁrst set of constraints are the ﬂow conservation laws for each long-term average rate resulting from power allocation and destination, while the last set of constraints are the capacity time-sharing (scheduling) which also needs to be found. The constraints for each link. SRRAS problem is very general and includes, among others, It is sometimes natural to keep the routes between the the following design problems for wireless networks. source-destination pairs ﬁxed, and only allow the source rates Maximum throughput and transport capacity One important to vary. We then label the source-destination pairs by integers performance metric for wireless data networks is the total p = 1, . . . , J, and let sp denote the data rate communicated throughput of the system. Finding the combined routing, between source-destination pair p. In place of the node-link resource allocation and scheduling that gives the maximum incidence matrix, we use the link-route incidence matrix R ∈ system throughput can be formulated as the SRRAS problem RL×J whose entries rlp are deﬁned via rlp = 1 if the trafﬁc between node pair p is routed across link l, and rlp = 0 (d) maximize n d=n sn otherwise. The vector of total trafﬁc across the links is given subject to constraints in (3) by Rs, and the ﬁxed routing model imposes the following set of constraints on the end-to-end rates s, The transport capacity (in bit-meters/s) can be computed similarly, by replacing the objective function above by Rs c, s 0 (2) L D (d) l=1 dl d=1 xl where dl is the length of link l. B. Communications model Proportionally fair SRRAS As illustrated in [1], throughput The capacities of individual wireless links depend on the maximization can lead to grossly unfair allocations of end-to- media access scheme and the allocation of communications end communication rates. An alternative is to use a maximum- (d) resources, such as transmit powers or bandwidths, to the utility formulation as follows. Let Un (·) be a concave and (d) (d) transmitters. We assume that the medium access, coding, and strictly increasing utility function, and let Un (sn ) for d = n (d) modulation schemes are ﬁxed and focus on the relationship represent the utility of node n for sending data at rate sn to between power allocation and link capacities, which we write destination d. Then, the maximum utility SRRAS problem is cl = cl (P ) (d) (d) maximize n d=n Un (sn ) Here, P = P1 . . . PL denotes the vector of transmit subject to constraints in (3) powers. The transmitters are subject to individual or network- This problem is closely related to fair allocation of end-to-end wide instantaneous power constraints, which we express as rates. More precisely, s is proportionally fair if and only if it P ∈ P. We postpone detailed examples of capacity formulas (d) solves the above problem with Un (·) = log(·) [8]. cl (P ) and power constraint sets P to Section V. Max-min fair SRRAS An alternative notion of fairness is the Higher link capacities can often be achieved by time-sharing so-called max-min fairness. An allocation s is called max-min between power allocations. Let c(P ) = [cl (P )] be the vector (d) fair if an increase in any component sn (n = d) of s must of link rates for a given communication scheme under the cause a decrease in an already smaller component. The max- power allocation P , and note that if P (1) and P (2) are two min fair allocation can be found by solving the problem feasible power allocations with associated link rates c(P (1) ) and c(P (2) ), then time-divisioning allows us to achieve the maximize τ long-term rates αc(P (1) ) + (1 − α)c(P (2) ) for all values of α subject to (d) τ ≤ sn , n = D, d = 1, . . . , D. with 0 ≤ α ≤ 1. More generally, the set of link rates obtained and the constraints in (3) by combined power allocation and scheduling is given by A particular max-min fair solution is the maximum equal-rate C(P ) = co {c(P ) | P ∈ P} allocation (sometimes called the uniform capacity) where we where co denotes the convex hull. We use the compact notation seek the maximum end-to-end rate that can be sustained by all c ∈ C(P ) to denote the set of rates that can be sustained source-destination pairs simultaneously. The maximum equal- (d) by combining power control and time-sharing for the given rate SRRAS is obtained by replacing the inequalities τ ≤ sn (d) communications scheme. in the max-min fair formulation by equalities, i.e., τ = sn . IV. A C OLUMN G ENERATION A PPROACH TO SRRAS uupper − ulower serves as a measure of accuracy of the current We will now show how the SRRAS problem can be solved solution, and we consider (s, k∈K α(k) c(k) ) to be the optimal using a classical technique from mathematical programming solution to (4) if the difference drops below a predeﬁned called column generation (cf. [7]). We will ﬁrst describe the threshold. If the current solution does not satisfy the stopping technique on the SRRAS problem with ﬁxed routing and then criterion, we conclude that the vertices {c(k) }k∈K used in the extend the approach to the general multi-path formulation (3). restricted master problem do not characterize the relevant part of the capacity region sufﬁciently well, and that a new extreme A. Column Generation for SRRAS with Fixed Routing point should be added to the description before the procedure Consider the SRRAS problem with ﬁxed routing, is repeated. In particular, we add the vertex that solves maximize u(s) maximize λT c (9) subject to Rs c, s 0 (4) subject to c ∈ C(P ) c ∈ C(P ) We will call this problem the column generation subproblem. in the variables s and c. Since C(P ) is a convex polytope, any In our implementation, we solve the restricted master prob- element of C(P ) can be written as a convex combination of lem to optimality using a primal-dual interior-point method. its extreme points c(1) , . . . , c(K) . This allow us to re-write (4) It is then natural to use the optimal Lagrange multipliers λ (k) (k) as the following optimization problem in s and α(k) for the capacity constraint Rs k∈K α c in (6) when computing the upper bound g(λ) via (8). Since this computa- maximize u(s) tion includes solving the subproblem (9), the subproblem can subject to Rs c, s 0 (5) only return an extreme point c(k ) with k ∈ K if the restricted c = k α(k) c(k) , k α (k) = 1, α (k) ≥0 master problem solves the original problem exactly. As long as We refer to this problem as the full master problem and this is not the case, the algorithm will add one new extreme note that it is similar to the formulation used for investi- point of C to the restricted formulation, and the size of K gating the capacity of a number of small ad-hoc networks increases by one in each step. Since C has a ﬁnite number of in [5]. In general, however, this formulation is inconvenient vertices it follows that the algorithm has ﬁnite convergence. for several reasons. Firstly, C(P ) may have a very large B. Column Generation for the General SRRAS problem number of extreme points so explicit enumeration of all these The column generation method is directly applicable to the quickly becomes intractable as the size of the network grows. general SRRAS problem (3). In this case, we compute a lower Secondly, even when explicit enumeration is possible the bound ulower by solving formulation (5) may have a very large number of variables and can be computationally expensive to solve directly. Consider maximize unet (x, s) instead a subset {c(k) | k ∈ K} of extreme points of C(P ), subject to Ax(d) = s(d) , x(d) 0, s(d) d 0, ∀d (d) where K ⊆ {1, . . . , K}. The restriction of (5) to this subset is dx c c = k∈K α(k) c(k) k∈K α(k) = 1 α(k) ≥ 0 maximize u(s) while the upper bound is computed as subject to Rs c, s 0 (6) c= α(k) c(k) , α (k) = 1, α (k) ≥0 uupper = sup unet (x, s) − λT x(d) | Ax(d) = s(d) k∈K k∈K x(d) 0 d We will refer to (6) as the restricted master problem. Since this s(d) d 0 problem is a restriction of (5), its optimal solution provides a + sup λT c lower bound ulower to the SRRAS problem (4). c∈C(P ) An upper bound can be found by considering a dual Note that in computing the upper bound, the ﬁrst part requires formulation of original problem (4). If we dualize the capacity the solution of an uncapacitated network ﬂow problem, while constraint in (4), we ﬁnd the Lagrangian function the second subproblem is identical to (9) which appeared in the ﬁxed-routing formulation. In all other respects, the column L(s, λ) = u(s) − λT Rs + λT c generation procedure proceeds as for the ﬁxed-routing case. Hence, for any λ 0, the value C. Generating Feasible Link Rate Vectors g(λ) = sup u(s) − λ Rs + λ c = T T (7) Note that the column generation approach can be applied to s 0, c∈C any wireless network that operates under a MAC scheme for = sup u(s) − λT Rs + sup λT c (8) which we can solve the associated subproblem (9). In many s 0 c∈C cases (such as those considered in the rest of this paper), the provides an upper bound uupper to (4). Thus, by solving subproblem is the computational bottleneck of the approach (6) and (8) we know that the optimal solution to the origi- and the size of networks that we can consider is limited by our nal problem lies between ulower and uupper . The difference ability to solve the weighted maximum throughput problem. V. SRRAS FOR A CLASS OF CDMA SYSTEMS Scheme II: Fixed rates and SINR balancing In the second In this section, we will consider three particular MAC scheme, active transmitters use SINR balancing to minimize schemes that have been suggested in the wireless networking interference and power consumption. The associated subprob- literature (see e.g., [1], [9]) and show how the associated lem can be formulated by re-writing (11) as subproblems in SRRAS can be formulated and solved as Gll Pl + (1 − xl )Ml ≥ γtgt (σl + Glj Pj ) (14) mixed integer-linear programs. j=l Let Pl be the transmit power used by the transmitter node of for a sufﬁciently large constant Ml (such as Ml = γtgt σl + link l. We assume that each transmitter l is subject to a simple γtgt j=l Glj Pmax ). Maximizing λT x over these constraints power limit 0 ≤ Pl ≤ Pmax (i.e., that P = {P | 0 ≤ Pl ≤ ﬁnds a power allocation that allows the most advantageous Pmax , l = 1, . . . , L}) We deﬁne the signal to interference and transmission group to be active during the time slot. As there noise ratio (SINR) of link l as are typically many power allocations that achieve this goal, Gll Pl we suggest to solve the subproblem γl (P ) = σl + m=l Glm Pm maximize ctgt λT x − 1T P where Glm is the effective power gain from the transmitter of subject to (14), xl ∈ {0, 1} l = 1, . . . , L (15) link m to the receiver of link l, and σl is the thermal noise 0 ≤ Pl ≤ Pmax l = 1, . . . , L power at the receiver of link l. where is a sufﬁciently small positive constant. In particular, We assume all transmitters share the same frequency band, let λ+ be the smallest strictly positive component of λ. Then, min that data is coded separately for each link and that receivers do solving the subproblem with = λ+ /(2LPmax ctgt ) ﬁnds min not decode third-party data (hence treat it as noise). Each link the allocation of minimum total power that supports the most can then be viewed as a single-user Gaussian channel with advantageous combination of active transmitters. Shannon capacity cl = W log (1 + γl (P )) where W is the Scheme III: SINR balancing and discrete rate selection The system bandwidth. In practice, however, most communication approach extends directly to the case where nodes can transmit schemes will achieve signiﬁcantly lower rates, in particular at a ﬁnite set of rates depending on the achievable SINR level. when the coding block size is limited. To be able to capture We assume that link l can transmit at rate ctgt,r if this effect, we will use the model cl = ctgt,r if γtgt,r ≤ γl (P ) < γtgt,r+1 (10) Gll Pl ≥ γtgt,r σl + Glj Pj j=l with ctgt,0 = 0, γtgt,0 = 0, and γtgt,r < γtgt,r+1 . Here, ctgt,r Introducing boolean variables xlr = 1 if link l transmits at and γtgt,r denote the rth discrete rate level and the associated rate r and xlr = 0 otherwise we can write the transmission SINR target, respectively. Thus, each transmitter may choose constraints as between several transmission rates depending on what SINR Gll Pl + (1 − xlr )Mlr ≥ γtgt,r (σl + Glj Pj ) (16) level it can sustain. j=l Scheme I: Fixed rates and maximum power transmissions In Similarly to above, we suggest to use the value Mlr = this scheme, a collection of links can transmit data simulta- γtgt,r σl + γtgt,r j=l Glj Pmax . Since each link can only neously if their signal to interference and noise ratios exceed transmit at a single rate, we also require that their target values. In other words, active links must satisfy xlr ≤ 1 l = 1, . . . , L (17) Gll Pl ≥ γtgt σl + γtgt Glj Pj (11) r j=l In summary, we propose to solve Active transmitters use maximum power Pmax and transmit r ctgt,r xlr − 1 P T at rate ctgt . To express this condition in a mathematical maximize l λl programming framework, introduce the boolean variables xl = subject to (16), (17), xlr ∈ {0, 1} ∀ (l, r) (18) 1 if sender l transmits, and xl = 0 otherwise. We can now 0 ≤ Pl ≤ Pmax ∀l write the interference constraints as Accounting for omni-directional antennas When nodes are Gll Pmax xl + Ml (1 − xl ) ≥ γtgt (σl + Glj Pmax xj ) (12) equipped with omni-directional antennas, one also needs to j=l include the constraint that every node can only send or receive where Ml is a sufﬁciently large constant. We suggest to use data on one link at a time. This constraints can be written as the value Ml = γtgt σl + j=l Glj Pmax + Gll Pmax , and to generate transmission groups by solving the subproblems xlr + xmr ≤ 1 T r maximize ctgt λ x l∈O(n) m∈I(n) (13) subject to (12), xl ∈ {0, 1}, l = 1, . . . , L These constraints are readily included in (13),(15) and (18). VI. E XAMPLES ﬂows are set to zero. The throughput-optimal solution for the In this section, we use our approach to gain some insight network in Figure 1(left), for example, activates only the two into how power control, spatial reuse, routing strategies and links shown in Figure 1(right). The problem can be avoided by variable transmission rates inﬂuence network performance in optimizing with respect to proportional or max-min fairness. a high-speed indoor wireless LAN scenario. The distribution of ﬂow rates for the different approaches are shown in Figure 2. As one can see, both fair approaches A. Generating sample networks allocate non-zero rates to all ﬂows; the proportionally fair We construct a set of sample networks using a radio link solution can allocate relatively large rates to some ﬂows model that broadly corresponds to a hypothetical high-speed at the expense of a slight decrease in the rates for a few indoor wireless LAN using the entire 2.4000–2.4835 GHz small ﬂows. The equal-rate allocation attains 37.2% of the ISM band (see [10] for details). We use the deterministic achievable throughput while the proportional fair solution fading model Glm = Klm d−α , where dlm denotes the distance lm results in a throughput of 57.8% of the maximum achievable. between the transmitter on link m and the receiver on link l, The results have been qualitatively similar for a large number Klm = 2 × 10−4 and α = 3. We let Pmax = 0.1W, of networks that we have considered: the equal rate allocation σ = 3.34 × 10−12 W, and use the Shannon capacity formula results in a large decrease in total throughput, while the (r) (r) ctgt = W log2 (1 + γtgt ) to relate target SINR-levels to rates. proportionally fair allocation makes a more balanced trade- Using W = 83.5 × 106 and a SINR-target of γtgt = 10, we off between throughput and fairness. These observations are ﬁnd the base rate 288.9 MBps. For multiple-rate scenarios, we consistent with the ﬁndings in [1]. assume that the system can also offer half and double this rate (with associated SINR targets of 3.46 and 120, respectively). Throughput optimal Max−min fair Proportionally fair To generate the network topology, we place nodes randomly on 80 80 80 a square and introduce links between every pair of nodes that No. flows 60 60 60 can sustain the base target SINR when all other transmitters are 40 40 40 silent (in our model, this corresponds to a distance of 84.2 m). 20 20 20 We then adjust the dimension of the square so that the nodes 0 0 0 0 100 200 300 0 2 4 0 10 20 form a network that is fully connected and that L/N (N − 1) End−to−end rate (i.e., the average number of node pairs that are connected by direct links) matches a desired target number. We will only Fig. 2. Flow distributions for throughput optimization (left), max-min fair solution (middle) and proportionally fair solution (right). The results are for consider the trafﬁc situation where every node always has the network in Figure 1 under free (multi-path) routing and SIR balancing. some data to transmit to every other node in the network. B. Performance objectives for wireless network optimization C. The inﬂuence of routing and MAC schemes Our ﬁrst investigation considers the adequacy of various per- Next, we try to quantify the beneﬁts of multi-path routing formance objectives in the network optimization. We present and MAC schemes on our sample networks. We focus on speciﬁc results for the network shown in Figure 1(left), but the fair rate allocation problems, and start out by analyzing have found qualitatively similar results for all scenarios that the solutions for the network in Figure 1. Table I shows the we have considered. Our experience from this exercise is maximum equal rate allocations that can be achieved using 150 m 150 m various MAC schemes under ﬁxed (shortest-path) and free routing. The entry “reuse” gives the average number of links that are active in each time-slot, while “efﬁciency” is transport capacity divided by average transmit power. Active links Throughput Rate Reuse Efﬁciency Max power 250.0 2.78 1.30 134932 SIR balance 322.3 3.58 2.50 264629 Multi-rate 366.5 4.07 2.16 247331 150 m 150 m Throughput Rates Reuse Efﬁciency Fig. 1. Topology of sample network (left) and the two active links for Max power 220.3 2.45 1.28 186139 throughput-optimal solution under maximum power transmissions (right). SIR balance 273.7 3.04 2.31 228350 Multi-rate 288.7 3.21 1.94 202790 that throughput maximization appears to be an inappropriate TABLE I objective in the optimization. Maximum throughput solutions M AX - MIN FAIR ALLOCATIONS FOR NETWORK IN F IGURE 1 UNDER FREE tend to activate a few (typically short) links and allocate non- ROUTING ( TOP ) AND SHORTEST- HOP ROUTING ( BOTTOM ). zero rates to the ﬂows that only traverse these links. All other 150 m As we can see, SIR balancing and variable rate selection 1100 Variable-rate give throughput increases of 24.2% and 31.1%, respectively. 1000 900 SIR balancing The SIR balancing gives a good increase in the average reuse 800 700 factor, with a somewhat smaller increase for the variable- Throughput 600 Maximum power rate MAC. The “efﬁciency” is increased by 22.7% when SIR 500 400 balancing is introduced, but then decreased under variable 300 rate transmissions. This is due to the large increase in power 200 100 necessary for sustaining the higher transmission rates. As can 0 250 200 150 100 50 0 50 100 150 150 m Log Utility be seen in Table I, the inﬂuence of multi-path routing is quite signiﬁcant for this network: combined variable-rate MAC and Fig. 3. Sample network (left) and achievable combinations of log-utility and free routing results in a performance increase of 66.4% over throughput for different MAC and routing schemes (right). maximum power transmissions and shortest-hop routing. The corresponding results for proportional fair rate allocation under mission scheduling for wireless networks. Our objective has free routing are shown in Table II. A substantial increase in been to optimize throughput and fairness. We have shown throughput compared to the equal-rate assignment has been how realistic models of several media access schemes can achieved at the expense of a relatively slight decrease in the be incorporated in the model, and how the resulting op- smaller rates. Note that the general results of this section are timization problem can be formulated as a nonlinear opti- quite different from the ﬁndings in [5], where only very small mization problem. We have developed a specialized solution improvements where obtained with SIR balancing. method based on Lagrange duality and column generation and Throughput Rates Reuse Efﬁciency demonstrated the approach on several examples. For a given Max power 349.8 (1.4, 3.2, 9.8) 1.52 99959 network conﬁguration, our approach provides the optimal SIR balance 501.0 (1.9, 3.0, 20) 2.42 309446 operation of the transport, routing and radio link layers under Multi-rate 639.1 (2.0, 4.4, 46) 2.34 319558 several important medium access control schemes, as well TABLE II as the optimal coordination across layers. Finally, we have P ROPORTIONALLY FAIR ALLOCATIONS FOR NETWORK IN F IGURE 1 demonstrated how the method can be used to evaluate the UNDER FREE ROUTING . T HE RATES COLUMN GIVE THE MINIMAL , MEDIAN inﬂuence of power control, spatial reuse, routing strategies and AND MAXIMAL RATES , RESPECTIVELY. variable-rate transmissions on network performance. 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Corss-layer optimization of wireless net- works using nonlinear column generation. Technical report, Department We have considered the problem of ﬁnding the optimal of Signals, Sensors and Systems, KTH, Stockholm, Sweden, November end-to-end rate selection, routing, power allocation and trans- 2003. Available from http://www.s3.kth.se/ mikaelj.