Scheduling, Routing and Power Allocation for Fairness by flv13903

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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.
                                                                                                  R EFERENCES
   We have done the same investigations for a set of 60                                c
                                                                     [1] B. Radunovi´ and J.-Y. Le Boudec. Joint scheduling, power control and
sample networks and found that moving from maximum power                 routing in symmetric one-dimensional, multi-hop wireless networks. In
transmissions to SIR balancing gave performance increases of             Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks,
                                                                         pages 31–42, Sophia Antipolis, France, March 2003.
0.5% − 74.3%, with an average around 22.9%.                          [2] L. Xiao, M. Johansson, and S. Boyd. Simultaneous routing and resource
D. Fairness-throughput regions                                           allocation in wireless networks. IEEE Transactions on Communications,
                                                                         2003. To appear.
  As we have seen in Section VI-B, there is a clear tradeoff         [3] M. Johansson, L. Xiao, and S. Boyd. Simultaneous routing and power
between throughput and fairness in wireless networks. In this            allocation in CDMA wireless data networks. In IEEE International
                                                                         Conference on Communications, Anchorage, Alaska, May 2003.
section, we will try to shred some light on this tradeoff by         [4] T. ElBatt and A. Ephremides. Joint scheduling and power control for
solving the family of problems                                           wireless ad-hoc networks. In IEEE Infocom, volume 2, pages 23–27,
                                                                         New York, NY, June 2002.
                                 (d)              (d)   (d)
      maximize β n d=n sn + (1 − β)Un (sn )                          [5] S. Toumpis and A. Goldsmith. Capacity regions for wireless ad hoc
                                                                         networks. In IEEE International Conference on Communications, New
      subject to constraints in (3)                                      York, NY, May 2002.
Note that this problem reduces to throughput maximization            [6] R. L. Cruz and A. Santhanam. Optimal routing, link scheduling and
                                                                         power control in multi-hop wireless networks. In IEEE Infocom, San
when β = 1, and to utility maximization when β = 0. By                   Francisco, CA, April 2003.
solving the problem for all β ∈ [0, 1], we can trace out the                 a                           o
                                                                     [7] P. V¨ rbrand, D. Yuan, and P. Bj¨ rklund. Resource optimization of spatial
achievable combinations of throughput and total log-utility.             TDMA in ad hoc radio networks: A column generation approach. In
                                                                         IEEE Infocom, San Francisco, CA, April 2003.
Figure 3 shows a sample network (left) and the associated            [8] F. P. Kelly, A. K. Malulloo, and D. K. H. Tan. Rate control in
trade-off curves (right). As can be expected, the benefits of             communications networks: shadow prices, proportional fairness and
routing on the throughput when we can accept small values                stability. J. of the Operational Research Society, 49:237–252, 1998.
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of both SIR balancing and variable rate transmissions.                   Technology Conference – Fall, volume 3, pages 1653–1657, Amsterdam,
                                                                         The Netherlands, September 1999.
                     VII. C ONCLUSIONS                              [10] M. Johansson and L. Xiao. 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.

								
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