Cross-layer optimization of wireless networks using nonlinear column generation Mikael Johansson and Lin Xiao November 2003 IR–S3-REG-0302
ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems Automatic Control SE-100 44 STOCKHOLM
KUNGL TEKNISKA HÖGSKOLAN Institutionen för Signaler, Sensorer & System Reglerteknik 100 44 STOCKHOLM
Cross-layer optimization of wireless networks using nonlinear column generation Mikael Johansson∗ and Lin Xiao † November 27, 2003
Abstract We consider the problem of finding the jointly optimal end-to-end communication rates, routing, power allocation and transmission scheduling for wireless networks. In particular, we focus on finding the resource allocation that achieves fair end-toend communication rates. Using realistic models of several media access schemes, we show how this cross-layer optimization problem can be formulated as a nonlinear mathematical program. We develop a specialized solution method, based on a nonlinear column generation technique, that converges to the optimal solution in a finite number of steps. We present computational results from a large set of networks and discuss the insight that can be gained about the influence of power control, spatial reuse, routing strategies and variable transmission rates on network performance.
∗
Assistant Professor, Department of Signals, Sensors and Systems, Royal Institute of Technology (KTH),
SE-100 44 Stockholm, Sweden. Email:
[email protected] † PhD Candidate, Information Systems Laboratory and Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305. Email:
[email protected] This research was sponsored in part by the Swedish Science Foundation (VR), NSF grant No. 0140700, by AFOSR grant F49620-01-1-0365, by DARPA contracts F33615-99-C-3014 and MDA972-02-1-0004.
1
Introduction
Wireless ad-hoc networks is a promising access technology for realizing the vision of ubiquitous communications. Such systems could allow rapid deployment with little planning or user-interaction and possibly coexist with a sparse fixed infrastructure. However, the design of radio resource management schemes that work reliably and efficiently in such a distributed and heterogeneous environment is a major engineering challenge. Despite the complexity of establishing the information-theoretic capacity of wireless networks, recent contributions have derived asymptotic bounds on their performance (see, e.g., [GK00, GT02]) and established the achievable rate regions for (small) ad-hoc networks under variable transmission strategies [TG02]. In addition, several approaches for maximizing the throughput of wireless networks have been developed (see, e.g., [XJB03, CS03, VYB03, JXB03]). Fairness is a key consideration in allocating limited network resources such as transmit powers and time slot allocations within a network. It has been observed that maximizing the throughput of wireless networks can lead to grossly unfair communication rates between source-destination pairs [RB03]. To operate a wireless networks in a fair and efficient way, it is thus important to have methods that allow us to find fair resource allocations and that help us to understand the trade-offs between fairness and throughput. In this paper, we extend our previous work on simultaneous routing and resource allocation in wireless networks [XJB03, JXB03] to joint scheduling, routing and power control for fair allocation of network resources. For a given network configuration, our approach provides the optimal operation of transport, routing and radio link layers under several important medium access control (MAC) 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) strategies. The research on optimal scheduling of transmissions in multihop radio networks has a long history (see, e.g., [BWE82, NK85, HS88, EE02]), and our effort is closely related to the recent work reported in [TG02, CS03, VYB03]. Our contribution extends the previous approaches by allowing nonlinear performance objectives (necessary to obtain proportional fairness be1
tween connections) and multipath routing, and by incorporating several important MAC schemes from the wireless networking literature. The approach is based on a nonlinear column-generation method and generates a sequence of feasible resource allocations that converges to the optimum in a finite number of steps. Admittedly, the centralized optimization algorithms that we present in this paper are quite different from the distributed resource management schemes that are eventually needed for practical operation of wireless ad-hoc networks. Still, we believe that our approach is useful in many respects: it provides the limits of performance for any ad-hoc wireless networks operating under the MAC and routing schemes under consideration; the solutions to specific scenarios are useful benchmark for alternative strategies; and the solution method itself may potentially be used as an inspiration for distributed protocols. The paper is organized as follows: In Section 2, we describe our mathematical model of the wireless network. We formulate the simultaneous routing, resource allocation and scheduling as a nonlinear integer-programming problem in Section 3, and develop a specialized algorithm for solving the optimization problem in Section 4. In Section 5, we use the approach to investigate the benefits of various network configurations in terms of throughput and fairness. Final remarks and conclusions are collected in Section 7.
2
Model and assumptions
We consider a wireless communication network formed by a collection of nodes located at fixed positions in the plane. Each node is assumed to have infinite buffering capacity and can transmit, receive and relay data to other nodes across wireless links. In this section, we develop a model of how end-to-end rate selection, routing and resource allocation influence the network performance.
2.1
Network flow model
Network topology We represent the topology of the network by a directed graph, with nodes labelled n = 1, . . . , N and links labelled l = 1, . . . , L. A link is represented by an ordered pair (i, j) of distinct nodes. The presence of link (i, j) means that the network is 2
able to send data from the start node i to the end node j. The network topology can be represented by a node-link incidence matrix A ∈ RN ×L , whose entries anl satisfy 1 if n is the start node of link l anl = −1 if n is the end node of link l 0 otherwise We define O(n) as the set of links that are outgoing from node n, and I(n) the set of links that are incoming to node n. Multicommodity network flows We use a multicommodity flow model for describing the routing of packets across the network (see, e.g., [BG91]). In this model, each node can send data to many destinations and receive data from many sources, but multicast is not considered. We assume that data flows are lossless across links and that they satisfy the flow conservation law at each node. We identify the flows by their destinations, i.e., flows with the same destination are considered as one single commodity regardless of their sources. We assume that the destination nodes are labelled d = 1, . . . , D where D ≤ N . For each destination d, we define (d)
a source-sink vector s(d) ∈ RN whose nth (n = d) entry, sn denotes the non-negative data rate injected to the network at node n (the source) destined for node d (the sink). In light (d) (d) of the flow conservation law, the sink flow at the destination is given by sd = n=d sn . (d)
For each link l, we let xl
be the amount of flow destined for node d. We call x(d) ∈ RL
the flow vector for destination d. At each node n, the components of the flow vector and the source-sink vector for the same destination satisfy the conservation law
(d)
xl −
l∈O(n)
(d)
xl
= s(d) n ,
d = 1, . . . , D
l∈I(n)
The flow conservation law across the network can compactly be written as Ax(d) = s(d) ,
d = 1, . . . , D
where A is the node-link incidence matrix defined above. Finally, we impose capacity constraints on the individual links. Note that the total amount of traffic on link l is given 3
by
d
(d)
xl . Thus, if c is the vector of link capacities, we require that D
x(d) c
d=1
In summary, our network flow model imposes the following group of constraints on the network flow variables x(d) and s(d) : Ax(d) = s(d) , x(d) 0, s(d) d 0, D d=1
x
(d)
d = 1, . . . , D
c
(1)
where denotes componentwise inequality, and d means componentwise inequality (d)
except for the dth component (the sink flow sd is always negative). It should be noted that this model describes the average behavior of data transmissions (that is, the average data rates) and ignores packet-level details of transmission protocols and forwarding mechanisms. The link capacity in practical communication systems should be defined appropriately, taking into account packet loss and retransmissions, so that the flow conservation law holds for the effective throughput (c.f. [BPSK97]). Network flow model for fixed routing 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, . . . , P , 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×P whose entries rlp are defined via 1 if the traffic between node pair p is routed across link l 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)
Although this formulation assumes that data for each source-destination pair is transmitted along a single path, our framework extends directly to systems where data is distributed over multiple paths. This extension is detailed in Section 6. 4
2.2
Communications model
In a wireless system, the capacities of individual links depend on the media access scheme and the allocation of communications resources, such as transmit powers or time-slot fractions, to the transmitters. In this paper, we will consider a CDMA system where all the communication links share the same frequency band. Radio propagation model
Let Glm denote the effective power gain between the trans-
mitter of link l and receiver of link m. We use a deterministic fading model Glm = Klm d−α lm where dlm is the distance between transmitter l and receiver m, α is a constant path loss exponent, and Klm is a normalization constant. The normalization constant depends on the radio propagation properties of the environment, but also allows us to account for the effects of coding gain, spreading gain, beam-forming, etc. (see, e.g., [TG02, OCJB03]) The details of the specific fading model that we use in our examples are given in Appendix A. Power control, transmission scheduling and link capacities Let Pl be the transmit power used by the transmitter node of link l. We assume that each transmitter is subject to a simple power limit 0 ≤ Pl ≤ Pl,max ,
l = 1, . . . , L
We let σl denote the thermal noise power at its receiver, and define the signal to interference and noise ratio (SINR) of link l as γl (P ) =
G P ll l σl + m=l Glm Pm
(3)
where P = P1 . . . PL denotes the vector of transmit powers. We assume 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 )) 5
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 (see [DDP98]). To be able to capture this effect, we will use the model (r)
(r)
cl = ctgt,l (0)
(0)
(r+1)
if γtgt,l ≤ γl (P ) < γtgt,l (r)
(r+1)
(r)
(4)
(r)
with ctgt,l = 0, γtgt,l = 0, and γtgt,l < γtgt,l . Here, ctgt,l and γtgt,l 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. Higher link capacities can often be achieved also allowing time-divisioning between fixed 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 ) | 0 ≤ Pl ≤ Pl,max }
(5)
where co denotes the convex hull. We use the compact notation c ∈ C(P ) to denote that any feasible rate vector must satisfy the constraints (4) and (5). Antenna constraints If nodes are equipped with omnidirectional antennas, we add the constraints that nodes can only communicate with one other node at a time.
2.3
Examples of media access schemes
We will consider three particular classes of media access control (MAC) schemes that are commonly used in the wireless networking literature. Further details of how they can be included in the optimization formulation are given in Section 4.
6
Scheme I: Fixed transmission rates and maximum transmit powers In this model, transmitters send with maximum power or stay silent. A collection of links can be active in the same time slot only if all active links exceed their signal-to-noise target γtgt,l . The associated link rate is given by (4). Note that the rate on each active link is constant, even if the link exceeds its SINR threshold. This is a common model for transmissions in 802.11, and has been used extensively in the literature (see, e.g., [RB03, VYB03]) Scheme II: Fixed transmission rates and SINR balancing The use of maximum transmit powers in the above scheme is inefficient for two reasons. First, the there is no increase in data rate once the SINR target is reached, so any power allocation that causes a link to exceed its SINR target consumes unnecessarily much energy. Secondly, high transmit powers lead to high interference levels and limit the number of links that can be activated at each time slot (cf. [EE02, SG02]). In the second scheme that we consider, the transmitters of the active links use the minimum total power necessary to reach their SINR targets. Scheme III: SINR balancing and discrete rate selection Wireless devices often support multiple data rates and mechanisms to switch between them are based on the channel conditions (see, e.g., [GC97, KRZ99, QC99, HVB01]). Such mechanisms have the potential to increase the data rates on links with good channel quality, hence increasing the total traffic that can be carried by the network. In the third scheme that we will consider, each link can choose between a finite set of transmission rates. Each transmission rate has an associated SINR target which must be met for the rate to be admissible.
7
3
The SRRAS problem
We can now formulate the simultaneous routing, resource allocation and scheduling (SRRAS) problem as follows maximize unet (x, s) subject to Ax(d) = s(d) , x(d) 0, s(d) d 0, (d) c dx
d = 1, . . . , D
(6)
c ∈ C(P ) where unet is a concave utility function. For fixed routing, the problem reduces to maximize unet (x, s) subject to Rs c,
(7)
s0
c ∈ C(P ) The SRRAS problem is very general and includes many important design problems for wireless networks. We conclude this section by detailing some of them. Maximum throughput and transport capacity One important performance metric for wireless data networks is the total throughput of the system. The combined routing, resource allocation and scheduling that gives the maximum system throughput can be put into the SRRAS form by letting maximize
n
(d)
d=n
sn
subject to constaints in (6) An important performance measures for wireless networks is the transport capacity [GK00]. The transport capacity of a given network satisfying our modelling assumptions can be computed by solving the optimization problem maximize
L l=1
dl
D d=1
(d)
xl
subject to constraints in (6) where dl is the physical length of link l (in meters). 8
Maximum utility SRRAS As illustrated in [RB03], throughput maximization can lead to grossly unfair allocations of end-to-end communication rates. An alternative is to use (d)
a maximum-utility formulation as follows. Let Un (·) be a concave and strictly increasing (d)
(d)
utility function, and let Un (sn ) for d = n represent the utility of node n for sending data (d)
at rate sn to destination d. Then, the maximum utility SRRAS problem is maximize
n
(d)
d=n
(d)
Un (sn )
subject to constraints in (6) This problem is closely related to fair allocation of end-to-end rates. Recall that a rate allocation s¯ is called proportionally fair if, for all other feasible allocations s (d) s¯(d) n − sn (d)
n
sn
d=n
≤0
It is well-known (see [KMT98]) that s is proportionally fair if and only if it solves the (d)
above problem with Un (·) = log(·). Max-min fairness An alternative notion of fairness is the so-called max-min fairness. An (d)
allocation s is called max-min fair if an increase in any component sn of s must cause a decrease in an already smaller component. The max-min fair allocation can be found by solving the following variation of SRRAS maximize τ (d)
n = d,
subject to τ ≤ sn ,
d = 1, . . . , D.
and the constraints in (6) A particular min-max fair solution is the maximum equal-rate allocation, where we seek the maximum end-to-end rate that can be sustained by all source-destination pairs simultaneously. We can formulate this problem as maximize ser subject to s(d) = ser e(d) and the constraints in (6) where e(d) is an N vector of ones, except for the dth element which equals 1 − N . 9
4
A Column Generation Approach to SRRAS
In this section, we will show how the SRRAS problem can be approached using a classical technique from mathematical programming known as column generation (c.f. [VYB03]). We will explain the technique on the SRRAS problem with fixed routing, and then extend the approach to the more general formulation (6). We conclude the section by deriving specific solution methods for the three MAC schemes described in Section 2.3.
4.1
Column Generation for SRRAS with Fixed Routing
Consider the SRRAS problem with fixed routing, maximize u(s) subject to Rs c,
s0
(8)
c ∈ C(P ) 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 c1 , . . . , cK . This allow us to re-write (8) as the following optimization problem in s and αk maximize u(s) subject to Rs c, s 0 c = k αk ck ,
(9)
k
αk = 1,
αk ≥ 0 k = 1, . . . , K
We refer to this problem as the full master problem, and note that it is similar to the formulation used for investing the capacity of a number of small ad-hoc networks in [TG02]. 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 (9) may have a very large number of variables and can be computationally expensive to solve directly. Consider instead a subset {ck | k ∈ K} of extreme points of C(P ), where K ⊆ {1, . . . , K}
10
The associated restriction of (9) to this subset is i.e., maximize u(s) subject to Rs c, s 0 c = k∈K αk ck ,
(10)
k∈K
αk = 1,
αk ≥ 0 k ∈ K
We will refer to (10) as the restricted master problem. Since this problem is a restriction of (9), its optimal solution provides a lower bound ulower to the SRRAS problem. An upper bound can be found by considering a dual formulation of original problem (8). If we dualize the capacity constraint in (8), we find the Lagrangian function L(s, λ) = u(s) − λT Rs + λT c Hence, for any λ 0, the value g(λ) = sup s0, c∈C
u(s) − λT Rs + λT c = sup u(s) − λT Rs + sup λT c s0
(11)
c∈C
provides an upper bound uupper to (8). Thus, by fixing a subset of extreme points of C and solving (10) and (11) we know that the optimal solution to the original problem lies between ulower and uupper . The difference uupper − ulower serves as a measure of accuracy of the current solution, and we consider (s, k∈K αk ck ) to be the optimal solution to (8) if the difference drops below a predefined threshold. If the current solution does not satisfy the stopping criterion, we conclude that C is not well enough characterized by the vertices {ck }k∈K 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 maximize λT c subject to c ∈ C(P )
(12)
We will call this problem the subproblem in our column generation method. The column generation algorithm is illustrated in Figure 1. 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 multi pliers λ for the capacity constraint Rs k∈K αk ck in (10) in computing the upper bound 11
g(λ) in (11), which includes solving the subproblem (12). In this way, the subproblem can only return an extreme point ck 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.
4.2
Column Generation for the General SRRAS problem
The column generation method is directly applicable to the SRRAS problem (6). In this case, we compute a lower bound ulower by solving maximize unet (x, s) d = 1, . . . , D subject to Ax(d) = s(d) , x(d) 0, s(d) d 0, (d) c dx αk ≥ 0, k ∈ K c = k∈K αk ck , k αk = 1, while the upper bound is computed as uupper =
sup x(d) 0, s(d) d 0
unet (x, s) −
λT x(d) | Ax(d) = s(d)
+
sup
λT c
c∈C(P )
d
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 (12) which appeared in the fixed-routing formulation. In all other respects, the column generation procedure proceeds as for the fixed-routing case.
4.3
Generating Feasible Link Rate Vectors
The nature of the feasible rate region and, hence, of the column generation subproblem (12) depends on the MAC scheme that we employ. We will now show how the subproblem (12) that appears in the column generation method can be solved for the communication schemes outlined in Section 2.3. For sake of readability, we assume that the power limits and SINR thresholds are equal for all links. 12
Scheme I: Fixed transmission rates and maximum transmit powers 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, for all active links we must have
Gll Pl ≥ γtgt σl + Glj Pj (13) j=l
Active transmitters use their maximal power Pmax and transmit at rate ctgt . To express this condition in a mathematical programming framework, introduce the boolean variables 1 if sender l is transmitting xl = 0 otherwise We can now write the interference constraints as
Gll Pmax xl + Ml (1 − xl ) ≥ γtgt
σl +
Glj Pmax xj
j=l
where Ml is a sufficiently large constant. We will use the value
Ml = γtgt σl + Glj Pmax + Gll Pmax j=l
which results in the transmission constraints
−1 σl + Glj Pmax xl + Glj Pmax xj ≤ Glj Pmax + γtgt Gll Pmax j=l
j=l
(14)
j=l
Thus, we suggest to generate transmission groups by solving the subproblems maximize λT x subject to (14),
xl ∈ {0, 1}, l = 1, . . . , L
(15)
Scheme II: Fixed transmission rates and SINR balancing Combined scheduling and power control can be performed similarly, by re-writing (13) as Gll Pl + (1 − xl )Ml ≥ γtgt (σl +
j=l
13
Glj Pj )
for a sufficiently large constant Ml . Using the value Ml = γtgt σl + j=l Glj Pmax , we find that the transmission constraints can be re-written as
Glj Pj − Gll Pl + γtgt σl + Pmax Glj xl ≤ γtgt Pmax Glj γtgt j=l
j=l
(16)
j=l
Maximizing λT x over these constraints finds the 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 λT x − 1T P subject to (16)
(17)
l = 1, . . . , L
0 ≤ Pl ≤ Pmax , xl ∈ {0, 1} l = 1, . . . , L 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 ) finds the power allocation of minimum total power among all allocations that support the most advantageous combination of active transmitters. Scheme III: SINR balancing and discrete rate selection The above approach is easily extended to the situation where nodes can transmit at a finite set of rates, depending on the achievable SINR levels. More precisely, we assume that link l can transmit at rate c(r) provided that
Gll Pl ≥
(r) γtgt
σl +
Glj Pj
j=l
Introducing the boolean variables 1 (r) xl = 0
if link l transmits at rate r otherwise
we can re-write the transmission constraint as
(r) (r) (r) (r) Glj Pj − Gll Pl + γtgt σl + Pmax Glj xl ≤ γtgt Pmax Glj γtgt j=l
j=l
14
j=l
(18)
Since each link can only transmit at a single rate, we also require that (r) xl ≤ 1 l = 1, . . . , L
(19)
r
Finally, we will have to account for the fact that links can transmit at different rates when solving the subproblem. In summary, we propose to solve maximize
l
λl
(r) (r) r ctgt xl
− 1T P
subject to (18), (19)
l = 1, . . . , L, ∀ r
(20)
0 ≤ Pl ≤ Pmax , xl ∈ {0, 1} l = 1, . . . , L Scheduling constraints for omnidirectional antennas In cases where nodes are equipped with omnidirectional antennas, one also needs to include the constraint that every node can only send or receive data on one link at a time. This constraint can be written as xl + xm ≤ 1 n = 1, . . . , N. l∈O(n)
m∈I(n)
These linear constraints are readily included in the subproblems (15), (17) and (20).
5
Examples
In this section, we use our approach to gain some insight into how power control, spatial reuse, routing strategies and variable transmission rates influence the network performance. We will consider the case where every node transmits data to every other node in the network. As the results depend on both the traffic situation and the parameters of the radio link model, they should be seen as an indication of what type of questions that can be addressed using this framework rather than facts about wireless network performance. The precise details of our radio link model that we have used are given in Appendix A.
5.1
Performance objectives for wireless network optimization
Our first investigation considers the adequacy of various performance objectives in the network optimization. We will present results for the particular network shown in Figure 2(left). (The results are qualitatively similar for a large number of scenarios that we 15
have been studying.) In particular, we have experienced that throughput maximization is an inappropriate objective in the optimization. Maximum throughput solutions tend to activate a few (typically short and high-quality) links and allocates non-zero rates to the flows that only traverse these links. All other flows are set to zero. As an example, optimizing throughput for the network in Figure 2(left) generates a solution that only activates the two links illustrated in Figure 2(right). The problem can be avoided by optimizing with respect to logarithmic utility (proportional fairness) or equal end-to-end rate assignment (max-min fairness). The distribution of flow rates for the different approaches are shown in Figure 3. 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. Computing the throughput of the fair solutions, we see that the equal-rate allocation attains 37.2% of the achievable capacity, 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 trade-off between throughput and fairness. These observations are consistent with the findings in [RB03].
5.2
The influence of routing and MAC schemes
Next, we try to quantify the benefits of flexible 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 2. Table 1 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 the “transport efficiency” is the transport capacity divided by the average transmit power. 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 variable-rate MAC. The “transport 16
Throughput End-to-end rate Reuse
Transport efficiency
Max power
250.0
2.78
1.30
134932
SIR balancing
322.3
3.58
2.50
264629
Multiple rates
366.5
4.07
2.16
247331
Throughput
End-to-end rate Reuse
Transport efficiency
Max power
220.3
2.45
1.28
186139
SIR balancing
273.7
3.04
2.31
228350
Multiple rates
288.7
3.21
1.94
202790
Table 1: Max-min fair allocations for network in Figure 2 under free routing (top) and shortest-hop routing (bottom). 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 1, the influence of flexible routing scheme 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 2. 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 [TG02], where only very small improvements where obtained with SIR balancing. Throughput
End-to-end rates
Reuse
Transport efficiency
Max power
349.8
[ 1.38, 3.21, 9.81]
1.52
99959
SIR balancing
501.0
[ 1.88, 3.98, 20.31]
2.42
309446
Multiple rates
639.1
[ 1.97, 4.44, 46.42]
2.34
319558
Table 2: Proportionally fair allocations for network in Figure 2 under free routing. The end-to-end rates column give the minimal, median and maximal rates, respectively.
17
We have done the same investigations for a large number of networks, including the two networks shown in Figure 4. For a set of 60 sample networks, the performance increases from moving from maximum power transmissions to SIR balancing gave performance increases of 0.5% − 74.3%, with an average around 22.9%. For the larger 20-node network shown in Figure 4(right) we notice that the routing has a strong influence: the max-min rate under SIR-balancing is increased by 50% when we move from shortest-hop to free routing, and the reuse factor can be increased from 2.15 to 3.57.
5.3
Fairness-throughput regions
There is a clear tradeoff between throughput and fairness in wireless networks. As we have illustrated in Section 5.1, optimizing throughput typically results in a few short flows getting all the network resources while the majority of flows are not allowed to transmit at all. Fair end-to-end rate allocations, on the other hand, result in significantly decreased throughput. In this section, we will try to shred some light on this tradeoff using the network in Figure 4(left) as example. To trace out the region of achievable combinations of throughputs and total log-utility, we solve the family of problems maximize λ
n
(d)
d=n
(d)
(d)
sn + (1 − λ)Un (sn )
subject to constraints in (6) for a set of weight values λ ∈ [0, 1]. Note that this problem reduces to throughput maximization when λ = 1, and to utility maximization when λ = 0. The solutions obtained by solving this problem are Pareto optimal, in the sense that an increase in throughput must come at the expense of a decrease in log-utility, and vice versa ([BV04]). The resulting trade-off curves are shown in Figure 5. We can see that in this case, there is a clear benefit of both variable rate transmissions and SIR balancing. However, routing gives a relatively small performance increase. It is interesting to compare these results with what can be achieved by simply timesharing between the throughput-optimal and the proportionally fair allocation. Such an approach would, roughly speaking, reserve a fraction of the schedule to let a few highquality links could transmit at high rate (the throughput-optimal solution). Although this 18
approach is not optimal, plotting the set of achievable utility-throughput combinations by this approach and comparing this with the Pareto optimal surface reveals that the two curves are relatively close; see Figure 5(right).
5.4
Computational experience
Contrary to our previous approaches [XJB03, JXB03], which used convex formulations and can be solved by polynomial-time algorithms, the approach described in this paper has exponential complexity in the number of communication links. This has forced us to limit our investigations to networks with less than 20 nodes, with computation times ranging from a few seconds to several hours. Although much work could be done on improving the subproblem solvers, these problems are combinatorial in nature and computing exact solutions is, in general, NP-hard. In particular, even the simplified problem of scheduling maximum power transmissions (15) while disregarding antenna constraints is a multiconstraint knapsack problem, which is known to be NP-hard in general (see, e.g., [Shi79]). Despite such negative results, the practical performance of publically available mixed-integer linear programming solvers has allowed us to investigate a large number of non-trivial networks with a reasonable computational effort. To get a feel for how restrictive the antenna and interference constraints are, consider again the network in Figure 2. This network has 256 ≈ 7.2 · 1016 possible combinations of links that can be active in the same time-slot. Out of these, only 19253 satisfy the antenna constraints. Maximum power transmissions restricts the number of feasible link activations to a mere 178, while SIR balancing makes 16617 of these feasible. The column generation method itself has finite convergence, with a distinct “homing in/tailing off”-behavior, where initial progress is very steep but the convergence to a high precision solution is relatively slow; see Figure 6. For the ten-node networks that we have studied, our algorithm typically converges in around 30 − 70 iterations.
19
6
Extensions
There are several useful extensions to our model that can be dealt with using our approach at the cost of some additional notation. For clarity of presentation, we have postponed these extensions until now. Multiple fixed paths The fixed routing case can also be extended to allow for multiple paths between each source-destination pair. We then introduce one variable sp for the traffic on each path, and update the objective functions in the formulation (7) to account for the new meaning of the variables. Introducing the notation P(n, d) for the set of indices of the paths between start node n and destination node d, the maximum utility formulation reads maximize
n
Un(d)
d=n
sp
p∈P(n,d)
subject to Rs c c ∈ C(P ) This problem is readily solved using the approach developed in Section 4. Accounting for self-interference Our definition of signal-to-interference ratio (3) does not include self-interference that may appear from, for example, multipath fading. However, this can easily be done by considering the revised signal to interference and noise ratio γl (P ) =
Hll Pl σl + j Glj Pj
where Hll is the direct path gain and Gll is the self-interference gain. Now, the SINRconstraint for active links reads Hll Pl ≥ γtgt
σl +
Glj Pj
j
The only difference with the approach in Section 4.3 is that the direct path gain should be replaced by Hll − γtgt Gll . Following through the derivations for Scheme I, we find that the 20
transmission constraint (14) should be replaced by
−1 σl + Glj Pmax xl + Glj Pmax xj ≤ Glj Pmax + γtgt (Hll − γtgt Gll )Pmax j=l
j=l
j=l
The revisions for the other MAC schemes follow similarly. Extensions to other MAC schemes It is important to note that the method extends directly to other MAC schemes, as long as we can solve the weighted throughput optimization problem (12). For example, in ultra-wideband (UWB) systems, it is reasonable to assume a linear relationship between SIR and rates. The associated weighted throughput problem (disregarding antenna constraints) reads maximize
l
λl γl (P )
subject to 0 ≤ Pl ≤ Pmax It is well-known (see, e.g., [OW99]) that this function is convex in each variable Pl and that the optimal solution is for links to use their full power or to stay silent. The search for the optimal link activation can be performed using mixed-integer linear programming or by specialized search schemes. By incorporating such a method in our scheme, we could compute the practical limits of performance for UWB systems.
7
Conclusions
We have considered the problem of finding the optimal scheduling, routing and power allocation for wireless networks. Our objective has been to optimize throughput and (proportional) fairness in the network. We have shown how realistic models of several media access schemes can be incorporated in our model, and how the resulting optimization problem can be formulated as a nonlinear optimization problem. 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 21
performance, and provides a benchmark for alternative (heuristic) strategies. We have developed a specialized solution method based on Lagrange duality and column generation and demonstrated the approach on several examples. The computational aspects of the algorithm can be improved in many respects. This includes better choice of Lagrange multipliers in the column generation (c.f., [dMVDH99]) and faster methods for solving the column generation subproblem (see, e.g., [KRZ99, SG02]). It would also be interesting to see how the work in this paper could be combined with our previous approach for resource allocation in the high-SINR regime of the interference-limited channel [JXB03]. Acknowledgements The first author is grateful for fruitful discussions with the members of the Radio Systems Group at KTH and, in particular, Dr. Tim Giles for devising the model used to generate network scenarios.
References [BG91]
D. Bertsekas and R. G. Gallager. Data Networks. Prentice Hall, 1991.
[BPSK97]
H. Balakrishnan, V. N. Padmanabhan, S. Seshan, and R. H. Katz. A comparison of mechanisms for improving TCP performance over wireless links. IEEE/ACM Transactions on Networking, 6(5):756–769, December 1997.
[BV04]
S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
[BWE82]
D. J. Baker, J. Wieselthier, and A. Ephremides. A distributed algorithm for scheduling the activation of links in a self-organizing mobile radio network. In Proceedings of the IEEE International Conference on Communications, pages 2.F.6.1–2.F.6.5, Philadelphia, PA, June 1982.
[Cha95]
P. Chadwick. Sensitivity and range in WLAN receivers. In Proceedings of the IEE Colloquium on Radio LANs and MANs, April 1995.
22
[CS03]
R. L. Cruz and A. Santhanam. Optimal routing, link scheduling and power control in multi-hop wireless networks. In Proceedings of the 2003 Infocom, San Francisco, CA, April 2003.
[DDP98]
S. Dolinar, D. Divsalar, and F. Pollara. Code performance as a function of block size.
TDA Progress Report 42-133, Jet Propulsion Laboratory,
Pasadena, California, May 1998. [dMVDH99] O. du Merle, D. Villeneuve, J. Desroisers, and P. Hansen. Stabilized column generation. Discrete Mathematics, 194:229–237, 1999. [EE02]
T. ElBatt and A. Ephremides. Joint scheduling and power control for wireless ad-hoc networks. In Proceedings of the IEEE Infocom, volume 2, pages 23–27, New York, NY, June 2002.
[ETS98]
ETSI.
Universal mobile telecommunications system (UMTS): Selection
procedures of the choice of radio transmission technologies of the UMTS (UMTS 30.03 version 3.2.0). European Technical Standards Institute Technical Report TR 101 112 V3.2.0, April 1998. [GC97]
A. Goldsmith and S.-G. Chua. Variable-rate variable-power MQAM for fading channels. IEEE Transactions on Communications, 45:1218–1230, October 1997.
[GK00]
P. Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory, 46:388–404, March 2000.
[GT02]
M. Grossglauser and D. Tse. Mobility increases the capacity of ad hoc wireless networks. IEEE/ACM Transactions on Networking, 10(4):477–486, August 2002.
[HS88]
B. Hajek and G. Sasaki. Link scheduling in polynomial time. IEEE Transactions on Information Theory, 34(5):910–917, September 1988.
23
[HVB01]
G. Holland, N. Vaidya, and P. Bahl. A rate-adaptive MAC protocol for multi-hop wireless networks. In Proceedings of the ACM/IEEE International Conference on Mobile Computing and Networking, 2001.
[JXB03]
M. Johansson, L. Xiao, and S. Boyd. Simultaneous routing and power allocation in CDMA wireless data networks. In Proceedings of the 2003 IEEE International Conference on Communications, Anchorage, Alaska, May 2003.
[KMT98]
F. P. Kelly, A. K. Malulloo, and D. K. H. Tan. Rate control in communications networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society, 49:237–252, 1998.
[KRZ99]
S.-L. Kim, Z. Rosberg, and J. Zander. Combined power control and transmission rate selection in cellular networks. In Proceedings of the IEEE Vechicular Technology Conference – Fall, volume 3, pages 1653–1657, Amsterdam, The Netherlands, September 1999.
[NK85]
R. Nelson and L. Kleinrock. Spatial-TDMA, a collision-free multihop channel access protocol. IEEE Transactions on Communications, 33(9):934–944, September 1985.
[OCJB03]
D. O’Neill, M. Chiang, D. Julian, and S. Boyd. Optimal resource allocation with qos constraints in wireless cellular and ad hoc networks. Submitted to IEEE Transactions on Wireless Communications, 2003.
[OW99]
S.-J. Oh and K. Wasserman. Optimality of greedy power control in DSCDMA mobile networks. In MobiCom’99, 1999.
[QC99]
X. Qiu and K. Chawla. On the performance of adaptive modulation in cellular systems. IEEE Transactions on Communications, 47(6):884–895, June 1999.
[RB03]
B. Radunovi´c and J.-Y. Le Boudec. Joint scheduling, power control and routing in symmetric one-dimensional, multi-hop wireless networks. In WiOpt’03, pages 31–42, Sophia Antipolis, France, March 2003.
24
[SG02]
O. Somarriba and T. C. Giles. Transmission power control for spatial TDMA in wireless radio networks. In Proceedings of the 4th IEEE Conference on Mobile and Wireless Communication Networks, Stockholm, Sweden, September 2002.
[Shi79]
W. Shih. A branch and bound method for the multiconstraint zero-one knapsack problem. Journal of the Operational Research Society, 30:369–378, 1979.
[TG02]
S. Toumpis and A. Goldsmith. Capacity regions for wireless ad hoc networks. In Proceedings of the 2002 IEEE International Conference on Communications, New York, NY, May 2002.
[VYB03]
¨ P. V¨arbrand, D. Yuan, and P. Bjorklund. Resource optimization of spatial TDMA in ad hoc radio networks: A column generation approach. In Proceedings of the 2003 Infocom, San Francisco, CA, April 2003.
[XJB03]
L. Xiao, M. Johansson, and S. Boyd. Simultaneous routing and resource allocation in wireless networks. IEEE Transactions on Communications, 2003. To appear.
25
A
Generating sample network scenarios
To evaluate our methodology, we construct a set of sample networks by randomly placing nodes on a square and letting nodes communicate with every other node that is within a threshold distance. Although the methodology is independent of the precise details of the propagation model, we have tried to use parameters that correspond to a (hypothetical) high-speed indoor wireless LAN using the entire 2.4000–2.4835 GHz ISM band. Radio Link Model We assume that all transmitters share the full 83.5 MHz band, are subject to a instantaneous power limit of Pmax = 100mW, and equipped with single omnidirectional antennas. For the deterministic fading model Glm = Kd−α lm we use the parameters α = 3 and K = G t Gr L where Gt and Gr are the transmit and receiver antenna gains, and L is the path loss at a distance of one meter from the transmitter. We assume unit antenna gains, Gt = Gr = 1 and let L = 2 · 10−4 . These parameters correspond broadly to the UMTS indoor scenario [ETS98] with mobiles placed on a single floor but where we have neglected the log-normal fading term. The background noise power can be approximated as σ = f kT W , where f is the receiver noise figure, k is Boltzmann’s constant, T is the absolute temperature of the receiver circuitry, and W is the communications bandwidth. We assume a noise figure of 10 to compensate for the increased background noise in an office environment [Cha95], let T = 290K and arrive at σ ≈ 3.34 · 10−12 . We will use the Shannon capacity formula to relate target SINR-levels and the associated capacities. Specifically, we let (r)
(r)
ctgt = W log2 (1 + γtgt ) where W is the communications bandwidth of 83.5MHz. Somewhat arbitrarily, we will use the SINR-target γtgt = 10 to generate a base rate. This gives the transmission rate 26
288.9MBps. For the multiple-rate scenarios, we will assume that the system can also offer half and double this rate (with associated SINR targets of 3.46 and 120, respectively). Network topology To generate the network topology, we place nodes randomly on a square a side of d meters. Links are introduced between every pair of nodes that can sustain the target SINR when all other transmitters are silent. In our model, this corresponds to the distance d0 =
Pmax K σγtgt
1/α
which evaluates to 84.2 m. We then adjust the dimension of our square, so that the randomly placed nodes form a network with desired connectivity properties. In particular, we will require that the generated networks are fully connected (i.e., that there is at least one path between any pair of nodes) and that the connectivity o=
L N (N − 1)
(i.e., the average number of node pairs that are connected by direct links) matches a desired target number.
27
start a set of initial extreme points ck , incidence matrix R, and > 0 optimal primal variables s, c, α
restricted master problem max. u(s) s.t. s 0, Rs c αk = 1 c = αk ck , αk ≥ 0, k ∈ K
augmented set of extreme points ck
optimal dual variable λ
scheduling subproblem
network subproblem max. u(s) − λT Rs s.t. s 0
max. λT c s.t. c ∈ C(P ) fscheduling
fnetwork +
cnew uupper
test duality gap
ulower
no
uupper − ulower < ?
column generation add cnew to the set of extreme points
yes exit solutions s, c, α, extreme points ck and associated power allocations P Figure 1: Flow chart for the column generation method 28
150 m
150 m
Active links
150 m
150 m
Figure 2: Topology of sample network (left) and the two active links for throughputoptimal solution under maximum power transmissions shown as shaded(right).
No. flows
Throughput optimal
Max−min fair
Proportionally fair
80
80
80
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
Figure 3: Flow distributions for throughput optimization (left), max-min fair solution (middle) and proportionally fair solution (right). The results are for the network in Figure 2 under free routing and SIR balancing.
29
150 m
250 m
150 m
250 m
Figure 4: Additional network scenarios with 10 nodes (left) and 20 nodes (right)
600
1100 Variable-rate
1000
500
900 SIR balancing 800
400 600
Throughput
Throughput
700 Maximum power
500 400
300
200
300 200
100
100 0 250
200
150
100
50 Log Utility
0
50
100
150
0 −250
−200
−150
−100
−50 Log Utility
0
50
100
150
Figure 5: Achievable combinations of log-utility and throughput for different MAC and routing schemes (left). The right figure shows a comparison of the Pareto optimal solutions (full) and the performance of a simple time-sharing between throughput optimal and proportionally fair allocations (dashed).
30
30
20
Primal and dual objective
10
0
−10
−20
−30
−40
−50
−60
0
10
20
30
40
50
60
70
Iteration
Figure 6: Primal and dual values as function of iteration number; the solution shows a “homing-in, tailing-off” behavior where initial progress is steep, but many iterations are required to reach the optimal solution.
31
