Scheduling, Routing and Power Allocation for Fairness in Wireless ...

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and transmission scheduling for wireless networks. We focus on throughput and fairness between end-to-end rates and formulate the associated cross-layer ...
Scheduling, Routing and Power Allocation for Fairness in Wireless Networks Mikael Johansson

Lin Xiao

Department of Signals, Sensors and Systems KTH, SE-100 44 Stockholm, Sweden Email: [email protected]

Information Systems Laboratory Stanford University, Stanford, CA 94305-9510 Email: [email protected]

Abstract— We consider the problem of finding the jointly optimal end-to-end communication rates, routing, power allocation and transmission scheduling for wireless networks. We focus on throughput and fairness between end-to-end rates and formulate the associated cross-layer design problem as a nonlinear mathematical program. We develop a specialized solution method, based on a nonlinear column generation technique, that applies to a wide range of media access schemes and converges to the optimal solution in a finite number of steps. The approach is applied to a large set of sample networks and the influence of power control, spatial reuse, routing strategies and variable transmission rates on network performance is discussed.

I. I NTRODUCTION Wireless ad-hoc networks is a promising technology for realizing the vision of ubiquitous communications. Such systems could allow rapid deployment with little planning or userinteraction and possibly coexist with a sparse fixed infrastructure. However, the design of radio resource management schemes that work reliably and efficiently in distributed and heterogeneous environments is a major engineering challenge, and it is not clear if ad-hoc networks will be a viable technology for achieving affordable and scalable communications. In this paper, we develop an approach for computing the performance that can be achieved by optimally coordinating all networks nodes as well as coordinating the operation of several layers of the networking stack. When developing such cross-layer optimization schemes, it is important to specify an appropriate network performance objective. Fairness is a key consideration, as simply maximizing the throughput of wireless networks typically leads to grossly unfair communication rates between source-destination pairs (cf. [1]). Since fairness often comes at the price of decreased network throughput, it is important to have methods that allow us to find fair resource allocations and that help us to understand the tradeoffs between fairness and throughput. In this paper, we extend our previous work on simultaneous routing and resource allocation in wireless networks [2], [3] to also address transmission scheduling. The research on optimal scheduling of transmissions in multi-hop radio networks has a long history (see, e.g., [4] and the references therein), and our effort is closely related to the recent work reported in [5], [6], [7]. Our method extends the previous approaches by

allowing nonlinear performance objectives (necessary to obtain proportional fairness), multi-path routing, and a much wider range of MAC schemes. The approach is based on a nonlinear column-generation technique and generates a sequence of feasible designs that converges to the optimum in a finite number of steps. For a given network configuration, our approach provides the optimal operation of transport, routing and radio link layers under several important medium access control schemes, as well as the optimal coordination across layers. This allows us to gain insight in the influence of power control, spatial reuse, routing strategies and variable transmission rates on the network performance, and provides a benchmark for alternative (heuristic, distributed) strategies. II. M ODEL AND ASSUMPTIONS We consider a communication network formed by a set of nodes located at fixed positions. Each node is assumed to have infinite buffering capacity and can transmit, receive and relay data to other nodes across wireless links. Our model tries to capture how end-to-end rates, routing, power allocation and scheduling influence the network performance. A. Network flow model Consider a connected communication network containing N nodes labeled n = 1, . . . , N and L directed links labeled l = 1, . . . , L. The topology of the network is represented by a node-link incidence matrix A ∈ RN ×L whose entry Anl is associated with node n and link l via   1, if n is the start node of link l −1, if n is the end node of link l Anl =  0, otherwise. We define O(n) as the set of outgoing links from node n and I(n) as the set of incoming links to node n. We use a multicommodity flow model for the routing of data flows (cf. [2]) that describes average data rates in bits per second. We identify the flows by their destinations, labeled d = 1, . . . , D, where D ≤ N . For each destination d, we define a source-sink vector s(d) ∈ RN , whose nth (n = d) (d) entry sn denotes the non-negative amount of flow injected into the network at node n (the and destined for node d  source) (d) (d) (the sink), where sd = − n=d sn . We also define x(d) ∈

RL as the flow vector for destination d, whose component (d) xl is the (non-negative) amount of flow on link l destined for node d. Finally, we let c be the vector of capacities of the individual links. Then our network flow model imposes the following group of constraints on the flow variables x and s: Ax(d) = s(d) , x(d)  0, D (d) c d=1 x

s(d) d 0,

d = 1, . . . , D

(1)

Here,  means component-wise inequality and d means component-wise inequality except for the dth component. The first set of constraints are the flow conservation laws for each destination, while the last set of constraints are the capacity constraints for each link. It is sometimes natural to keep the routes between the source-destination pairs fixed, and only allow the source rates to vary. We then label the source-destination pairs by integers p = 1, . . . , J, and let sp denote the data rate communicated between source-destination pair p. In place of the node-link incidence matrix, we use the link-route incidence matrix R ∈ 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 otherwise. The vector of total traffic across the links is given by Rs, and the fixed routing model imposes the following set of constraints on the end-to-end rates s, Rs  c,

s0

(2)

B. Communications model The capacities of individual wireless links depend on the media access scheme and the allocation of communications resources, such as transmit powers or bandwidths, to the transmitters. We assume that the medium access, coding, and modulation schemes are fixed and focus on the relationship between power allocation and link capacities, which we write cl = cl (P )  Here, P = P1 . . . PL denotes the vector of transmit powers. The transmitters are subject to individual or networkwide instantaneous power constraints, which we express as P ∈ P. We postpone detailed examples of capacity formulas cl (P ) and power constraint sets P to Section V. Higher link capacities can often be achieved by time-sharing between power allocations. Let c(P ) = [cl (P )] be the vector of link rates for a given communication scheme under the power allocation P , and note that if P (1) and P (2) are two feasible power allocations with associated link rates c(P (1) ) and c(P (2) ), then time-divisioning allows us to achieve the long-term rates αc(P (1) ) + (1 − α)c(P (2) ) for all values of α with 0 ≤ α ≤ 1. More generally, the set of link rates obtained by combined power allocation and scheduling is given by 

C(P ) = co {c(P ) | P ∈ P} where co denotes the convex hull. We use the compact notation c ∈ C(P ) to denote the set of rates that can be sustained by combining power control and time-sharing for the given communications scheme.

III. T HE SRRAS PROBLEM Let unet (x, s) be a concave utility function of the end-to-end rates s and link flows x. The simultaneous routing, resource allocation and scheduling (SRRAS) problem is then maximize subject to

unet (x, s) (d) Ax = s(d) , x(d)  0, s(d) d 0,  (d)  c, c ∈ C(P ) dx

∀d

(3)

The optimization variables are s, x and c, where c is the long-term average rate resulting from power allocation and time-sharing (scheduling) which also needs to be found. The SRRAS problem is very general and includes, among others, the following design problems for wireless networks. Maximum throughput and transport capacity One important performance metric for wireless data networks is the total throughput of the system. Finding the combined routing, resource allocation and scheduling that gives the maximum system throughput can be formulated as the SRRAS problem   (d) maximize n d=n sn subject to

constraints in (3)

The transport capacity (in bit-meters/s) can be computed similarly, by replacing the objective function above by D L (d) where dl is the length of link l. l=1 dl d=1 xl Proportionally fair SRRAS As illustrated in [1], throughput maximization can lead to grossly unfair allocations of end-toend communication rates. An alternative is to use a maximum(d) utility formulation as follows. Let Un (·) be a concave and (d) (d) strictly increasing utility function, and let Un (sn ) for d = n (d) represent the utility of node n for sending data at rate sn to destination d. Then, the maximum utility SRRAS problem is   (d) (d) maximize n d=n Un (sn ) subject to

constraints in (3)

This problem is closely related to fair allocation of end-to-end rates. More precisely, s is proportionally fair if and only if it (d) solves the above problem with Un (·) = log(·) [8]. Max-min fair SRRAS An alternative notion of fairness is the so-called max-min fairness. An allocation s is called max-min (d) fair if an increase in any component sn (n = d) of s must cause a decrease in an already smaller component. The maxmin fair allocation can be found by solving the problem maximize subject to

τ (d) τ ≤ sn ,

n = D,

d = 1, . . . , D.

and the constraints in (3)

A particular max-min fair solution is the maximum equal-rate allocation (sometimes called the uniform capacity) where we seek the maximum end-to-end rate that can be sustained by all source-destination pairs simultaneously. The maximum equal(d) rate SRRAS is obtained by replacing the inequalities τ ≤ sn (d) in the max-min fair formulation by equalities, i.e., τ = sn .

IV. A C OLUMN G ENERATION A PPROACH TO SRRAS We will now show how the SRRAS problem can be solved using a classical technique from mathematical programming called column generation (cf. [7]). We will first describe the technique on the SRRAS problem with fixed routing and then extend the approach to the general multi-path formulation (3). A. Column Generation for SRRAS with Fixed Routing Consider the SRRAS problem with fixed routing, maximize subject to

u(s) Rs  c, s  0 c ∈ C(P )

u(s) Rs  s0  c, (k) c = k α c(k) , k α(k) = 1, α(k) ≥ 0

(5)

We refer to this problem as the full master problem and note that it is similar to the formulation used for investigating the capacity of a number of small ad-hoc networks in [5]. In general, however, this formulation is inconvenient for several reasons. Firstly, C(P ) may have a very large number of extreme points so explicit enumeration of all these quickly becomes intractable as the size of the network grows. Secondly, even when explicit enumeration is possible the formulation (5) may have a very large number of variables and can be computationally expensive to solve directly. Consider instead a subset {c(k) | k ∈ K} of extreme points of C(P ), where K ⊆ {1, . . . , K}. The restriction of (5) to this subset is maximize subject to

u(s) Rs  c, s  0 α(k) c(k) , α(k) = 1, α(k) ≥ 0 c= k∈K

(6)

k∈K

We will refer to (6) as the restricted master problem. Since this problem is a restriction of (5), its optimal solution provides a lower bound ulower to the SRRAS problem (4). An upper bound can be found by considering a dual formulation of original problem (4). If we dualize the capacity constraint in (4), we find the Lagrangian function L(s, λ) = u(s) − λT Rs + λT c Hence, for any λ  0, the value  g(λ) = sup u(s) − λT Rs + λT c = s0, c∈C   = sup u(s) − λT Rs + sup λT c s0

maximize λT c subject to c ∈ C(P )

(4)

in the variables s and c. Since C(P ) is a convex polytope, any element of C(P ) can be written as a convex combination of its extreme points c(1) , . . . , c(K) . This allow us to re-write (4) as the following optimization problem in s and α(k) maximize subject to

of accuracy of the current uupper − ulower serves as a measure  solution, and we consider (s, k∈K α(k) c(k) ) to be the optimal solution to (4) if the difference drops below a predefined threshold. If the current solution does not satisfy the stopping criterion, we conclude that the vertices {c(k) }k∈K used in the restricted master problem do not characterize the relevant part of the capacity region sufficiently well, and that a new extreme point should be added to the description before the procedure is repeated. In particular, we add the vertex that solves

(7) (8)

c∈C

provides an upper bound uupper to (4). Thus, by solving (6) and (8) we know that the optimal solution to the original problem lies between ulower and uupper . The difference

(9)

We will call this problem the column generation subproblem. In our implementation, we solve the restricted master problem to optimality using a primal-dual interior-point method. It is then natural to use the optimal  Lagrange multipliers λ for the capacity constraint Rs  k∈K α(k) c(k) in (6) when computing the upper bound g(λ) via (8). Since this computation includes solving the subproblem (9), the subproblem can  only return an extreme point c(k ) with k  ∈ K if the restricted master problem solves the original problem exactly. As long as this is not the case, the algorithm will add one new extreme point of C to the restricted formulation, and the size of K increases by one in each step. Since C has a finite number of vertices it follows that the algorithm has finite convergence. B. Column Generation for the General SRRAS problem The column generation method is directly applicable to the general SRRAS problem (3). In this case, we compute a lower bound ulower by solving maximize subject to

unet (x, s) (d) Ax = s(d) , x(d)  0, s(d) d 0, ∀d  (d) dx   c (k) = 1 α(k) ≥ 0 c = k∈K α(k) c(k) k∈K α

while the upper bound is computed as

 uupper = sup λT x(d) | Ax(d) = s(d) unet (x, s) − x(d) 0 s(d) d 0

+ sup



d

λT c

c∈C(P )

Note that in computing the upper bound, the first part requires the solution of an uncapacitated network flow problem, while the second subproblem is identical to (9) which appeared in the fixed-routing formulation. In all other respects, the column generation procedure proceeds as for the fixed-routing case. C. Generating Feasible Link Rate Vectors Note that the column generation approach can be applied to any wireless network that operates under a MAC scheme for which we can solve the associated subproblem (9). In many cases (such as those considered in the rest of this paper), the subproblem is the computational bottleneck of the approach and the size of networks that we can consider is limited by our ability to solve the weighted maximum throughput problem.

V. SRRAS FOR A CLASS OF CDMA SYSTEMS In this section, we will consider three particular MAC schemes that have been suggested in the wireless networking literature (see e.g., [1], [9]) and show how the associated subproblems in SRRAS can be formulated and solved as mixed integer-linear programs. Let Pl be the transmit power used by the transmitter node of link l. We assume that each transmitter l is subject to a simple power limit 0 ≤ Pl ≤ Pmax (i.e., that P = {P | 0 ≤ Pl ≤ Pmax , l = 1, . . . , L}) We define the signal to interference and noise ratio (SINR) of link l as γl (P ) =

G P  ll l σl + m=l Glm Pm

where Glm is the effective power gain from the transmitter of link m to the receiver of link l, and σl is the thermal noise power at the receiver of link l. We assume all transmitters share the same frequency band, that data is coded separately for each link and that receivers do not decode third-party data (hence treat it as noise). Each link can then be viewed as a single-user Gaussian channel with Shannon capacity cl = W log (1 + γl (P )) where W is the system bandwidth. In practice, however, most communication schemes will achieve significantly lower rates, in particular when the coding block size is limited. To be able to capture this effect, we will use the model

Scheme II: Fixed rates and SINR balancing In the second scheme, active transmitters use SINR balancing to minimize interference and power consumption. The associated subproblem can be formulated by re-writing (11) as  Glj Pj ) (14) Gll Pl + (1 − xl )Ml ≥ γtgt (σl + j=l

for asufficiently large constant Ml (such as Ml = γtgt σl + γtgt j=l Glj Pmax ). Maximizing λT x over these constraints finds a power allocation that allows the most advantageous transmission group to be active during the time slot. As there are typically many power allocations that achieve this goal, we suggest to solve the subproblem maximize ctgt λT x − 1T P subject to (14), xl ∈ {0, 1} 0 ≤ Pl ≤ Pmax

l = 1, . . . , L l = 1, . . . , L

(15)

where  is a sufficiently small positive constant. In particular, let λ+ min be the smallest strictly positive component of λ. Then, solving the subproblem with  = λ+ min /(2LPmax ctgt ) finds the allocation of minimum total power that supports the most advantageous combination of active transmitters.

(10)

Scheme III: SINR balancing and discrete rate selection The approach extends directly to the case where nodes can transmit at a finite set of rates depending on the achievable SINR level. We assume that link l can transmit at rate ctgt,r if

 Gll Pl ≥ γtgt,r σl + Glj Pj

with ctgt,0 = 0, γtgt,0 = 0, and γtgt,r < γtgt,r+1 . Here, ctgt,r and γtgt,r denote the rth discrete rate level and the associated SINR target, respectively. Thus, each transmitter may choose between several transmission rates depending on what SINR level it can sustain.

Introducing boolean variables xlr = 1 if link l transmits at rate r and xlr = 0 otherwise we can write the transmission constraints as  Glj Pj ) (16) Gll Pl + (1 − xlr )Mlr ≥ γtgt,r (σl +

Scheme I: Fixed rates and maximum power transmissions In this scheme, a collection of links can transmit data simultaneously if their signal to interference and noise ratios exceed their target values. In other words, active links must satisfy  Glj Pj (11) Gll Pl ≥ γtgt σl + γtgt

Similarly to above,  we suggest to use the value Mlr = γtgt,r σl + γtgt,r j=l Glj Pmax . Since each link can only transmit at a single rate, we also require that  xlr ≤ 1 l = 1, . . . , L (17)

cl = ctgt,r

if γtgt,r ≤ γl (P ) < γtgt,r+1

j=l

Active transmitters use maximum power Pmax and transmit at rate ctgt . To express this condition in a mathematical programming framework, introduce the boolean variables xl = 1 if sender l transmits, and xl = 0 otherwise. We can now write the interference constraints as  Glj Pmax xj ) (12) Gll Pmax xl + Ml (1 − xl ) ≥ γtgt (σl + j=l

where Ml is a sufficiently large constant. We

suggest to use  the value Ml = γtgt σl + j=l Glj Pmax + Gll Pmax , and to generate transmission groups by solving the subproblems maximize ctgt λT x subject to (12), xl ∈ {0, 1},

l = 1, . . . , L

(13)

j=l

j=l

r

In summary, we propose to solve   T maximize l λl r ctgt,r xlr − 1 P subject to (16), (17), xlr ∈ {0, 1} 0 ≤ Pl ≤ Pmax

∀ (l, r) ∀l

(18)

Accounting for omni-directional antennas When nodes are equipped with omni-directional antennas, one also needs to include the constraint that every node can only send or receive data on one link at a time. This constraints can be written as       xlr + xmr  ≤ 1 r

l∈O(n)

m∈I(n)

These constraints are readily included in (13),(15) and (18).

A. Generating sample networks We construct a set of sample networks using a radio link model that broadly corresponds to a hypothetical high-speed indoor wireless LAN using the entire 2.4000–2.4835 GHz ISM band (see [10] for details). We use the deterministic fading model Glm = Klm d−α lm , where dlm denotes the distance between the transmitter on link m and the receiver on link l, Klm = 2 × 10−4 and α = 3. We let Pmax = 0.1W, σ = 3.34 × 10−12 W, and use the Shannon capacity formula (r) (r) ctgt = W log2 (1 + γtgt ) to relate target SINR-levels to rates. Using W = 83.5 × 106 and a SINR-target of γtgt = 10, we find the base rate 288.9 MBps. For multiple-rate scenarios, we assume that the system can also offer half and double this rate (with associated SINR targets of 3.46 and 120, respectively). To generate the network topology, we place nodes randomly on a square and introduce links between every pair of nodes that can sustain the base target SINR when all other transmitters are silent (in our model, this corresponds to a distance of 84.2 m). We then adjust the dimension of the square so that the nodes form a network that is fully connected and that L/N (N − 1) (i.e., the average number of node pairs that are connected by direct links) matches a desired target number. We will only consider the traffic situation where every node always has some data to transmit to every other node in the network. B. Performance objectives for wireless network optimization Our first investigation considers the adequacy of various performance objectives in the network optimization. We present specific results for the network shown in Figure 1(left), but have found qualitatively similar results for all scenarios that we have considered. Our experience from this exercise is 150 m

150 m

flows are set to zero. The throughput-optimal solution for the network in Figure 1(left), for example, activates only the two links shown in Figure 1(right). The problem can be avoided by optimizing with respect to proportional or max-min fairness. The distribution of flow rates for the different approaches are shown in Figure 2. As one can see, both fair approaches allocate non-zero rates to all flows; the proportionally fair solution can allocate relatively large rates to some flows at the expense of a slight decrease in the rates for a few small flows. The equal-rate allocation attains 37.2% of the achievable throughput while the proportional fair solution results in a throughput of 57.8% of the maximum achievable. The results have been qualitatively similar for a large number of networks that we have considered: the equal rate allocation results in a large decrease in total throughput, while the proportionally fair allocation makes a more balanced tradeoff between throughput and fairness. These observations are consistent with the findings in [1]. Throughput optimal

No. flows

VI. E XAMPLES In this section, we use our approach to gain some insight into how power control, spatial reuse, routing strategies and variable transmission rates influence network performance in a high-speed indoor wireless LAN scenario.

Proportionally fair

80

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60

60

60

40

40

40

20

20

20

0 0

100

200

300

0 0

2 4 End−to−end rate

0 0

10

20

Fig. 2. Flow distributions for throughput optimization (left), max-min fair solution (middle) and proportionally fair solution (right). The results are for the network in Figure 1 under free (multi-path) routing and SIR balancing.

C. The influence of routing and MAC schemes Next, we try to quantify the benefits of multi-path routing and MAC schemes on our sample networks. We focus on the fair rate allocation problems, and start out by analyzing the solutions for the network in Figure 1. Table I shows the maximum equal rate allocations that can be achieved using 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

150 m

Max−min fair

80

Max power SIR balance Multi-rate

Throughput 250.0 322.3 366.5

Rate 2.78 3.58 4.07

Reuse 1.30 2.50 2.16

Efficiency 134932 264629 247331

Max power SIR balance Multi-rate

Throughput 220.3 273.7 288.7

Rates 2.45 3.04 3.21

Reuse 1.28 2.31 1.94

Efficiency 186139 228350 202790

150 m

Fig. 1. Topology of sample network (left) and the two active links for throughput-optimal solution under maximum power transmissions (right).

that throughput maximization appears to be an inappropriate objective in the optimization. Maximum throughput solutions tend to activate a few (typically short) links and allocate nonzero rates to the flows that only traverse these links. All other

TABLE I M AX - MIN FAIR ALLOCATIONS FOR NETWORK IN F IGURE 1 UNDER FREE ROUTING ( TOP ) AND SHORTEST- HOP ROUTING ( BOTTOM ).

150 m

Max power SIR balance Multi-rate

Throughput 349.8 501.0 639.1

Rates (1.4, 3.2, 9.8) (1.9, 3.0, 20) (2.0, 4.4, 46)

Reuse 1.52 2.42 2.34

Efficiency 99959 309446 319558

TABLE II P ROPORTIONALLY FAIR ALLOCATIONS FOR NETWORK IN F IGURE 1 UNDER FREE ROUTING . T HE RATES COLUMN GIVE THE MINIMAL , MEDIAN AND MAXIMAL RATES , RESPECTIVELY.

1100

Variable-rate 1000 900

SIR balancing

800 700 Throughput

As we can see, SIR balancing and variable rate selection give throughput increases of 24.2% and 31.1%, respectively. The SIR balancing gives a good increase in the average reuse factor, with a somewhat smaller increase for the variablerate MAC. The “efficiency” is increased by 22.7% when SIR balancing is introduced, but then decreased under variable rate transmissions. This is due to the large increase in power necessary for sustaining the higher transmission rates. As can be seen in Table I, the influence of multi-path routing is quite significant for this network: combined variable-rate MAC and free routing results in a performance increase of 66.4% over maximum power transmissions and shortest-hop routing. The corresponding results for proportional fair rate allocation under free routing are shown in Table II. A substantial increase in throughput compared to the equal-rate assignment has been achieved at the expense of a relatively slight decrease in the smaller rates. Note that the general results of this section are quite different from the findings in [5], where only very small improvements where obtained with SIR balancing.

600

Maximum power

500 400 300 200 100 0 250

200

150 m

150

100

50 Log Utility

0

50

100

150

Fig. 3. Sample network (left) and achievable combinations of log-utility and throughput for different MAC and routing schemes (right).

mission scheduling for wireless networks. Our objective has been to optimize throughput and fairness. We have shown how realistic models of several media access schemes can be incorporated in the model, and how the resulting optimization problem can be formulated as a nonlinear optimization problem. We have developed a specialized solution method based on Lagrange duality and column generation and demonstrated the approach on several examples. For a given network configuration, our approach provides the optimal operation of the transport, routing and radio link layers under several important medium access control schemes, as well as the optimal coordination across layers. Finally, we have demonstrated how the method can be used to evaluate the influence of power control, spatial reuse, routing strategies and variable-rate transmissions on network performance. R EFERENCES

We have done the same investigations for a set of 60 sample networks and found that moving from maximum power transmissions to SIR balancing gave performance increases of 0.5% − 74.3%, with an average around 22.9%. D. Fairness-throughput regions As we have seen in Section VI-B, there is a clear tradeoff between throughput and fairness in wireless networks. In this section, we will try to shred some light on this tradeoff by solving the family of problems   (d) (d) (d) maximize β n d=n sn + (1 − β)Un (sn ) subject to constraints in (3) Note that this problem reduces to throughput maximization when β = 1, and to utility maximization when β = 0. By solving the problem for all β ∈ [0, 1], we can trace out the achievable combinations of throughput and total log-utility. Figure 3 shows a sample network (left) and the associated trade-off curves (right). As can be expected, the benefits of routing on the throughput when we can accept small values of log-utility is quite marginal, while there is a clear benefit of both SIR balancing and variable rate transmissions. VII. C ONCLUSIONS We have considered the problem of finding the optimal end-to-end rate selection, routing, power allocation and trans-

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