Impact of Energy Expenditure Rate Constraints on the
Performance of Multi-Hop Wireless Mesh Networks
A Technical Report
by
Nachiket Sahasrabudhe
Adviser: Prof. Joy Kuri
Centre for Electronics Design and Technology(CEDT)
INDIAN INSTITUTE OF SCIENCE
BANGALORE - 560 012, INDIA
March 2008
Contents
Abstract
2
1 Effect of energy expenditure rate constraints
3
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Contention Graph . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.3.2
Energy Expenditure Rate Constraint . . . . . . . . . . . . . . . .
10
Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4.1
Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.4.2
Solution to the relaxed problem . . . . . . . . . . . . . . . . . . .
14
1.4.3
Solution to the routing problem . . . . . . . . . . . . . . . . . . .
15
1.4.4
Solution to the scheduling problem . . . . . . . . . . . . . . . . .
16
1.4.5
Dual problem solution by sub-gradient method . . . . . . . . . . .
18
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.5.1
Single end-to-end flow . . . . . . . . . . . . . . . . . . . . . . . .
20
1.5.2
Multiple end-to-end flows . . . . . . . . . . . . . . . . . . . . . .
21
1.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.7
Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4
1.5
Bibliography
22
1
Abstract
In this article we study the problem of joint congestion control, routing and
MAC layer scheduling in multi-hop wireless mesh network, where the nodes
in the network are subjected to maximum energy expenditure rates. We
model link contention in the wireless network using the contention graph
and we model energy expenditure rate constraint of nodes using the energy
expenditure rate matrix. We formulate the problem as an aggregate utility
maximization problem and apply duality theory in order to decompose the
problem into two sub-problems namely, network layer routing and congestion
control problem and MAC layer scheduling problem. The source adjusts its
rate based on the cost of the least cost path to the destination where the cost
of the path includes not only the prices of the links in it but also the prices
associated with the nodes on the path. The MAC layer scheduling of the
links is carried out based on the prices of the links. We study the effects of
energy expenditure rate constraints of the nodes on the optimal throughput
of the network.
2
Chapter 1
Effect of energy expenditure rate constraints
1.1
Introduction
Resource allocation in wireless networks is considered more challenging compared to the
wireline networks because of the constrained resources in wireless networks. In a typical
multi-hop wireless network, there is no fixed infrastructure and every node acts as a
router, forwarding the packets of other nodes. Since the wireless medium is a shared
one, activity on one link can greatly affect the activity on others. In the simplest case,
reception at a node requires that other nodes in its vicinity remain silent for the duration
of the reception. So only a subset of a given collection of links can be active at the same
time. This additional constraint, called the wireless link contention constraint, makes
the resource allocation problem more challenging in wireless networks.
Resource allocation problems in wire-line networks have been extensively studied in
the literature [7], [8] and many cross layer optimization algorithms have been developed.
Researchers have also studied the problem of optimal throughput in wireless networks
by taking into account the wireless link interference constraint [6], [2], [4].
In this article we study the the problem of determining the optimum throughput
when the nodes in the network are constrained to have a maximum energy expenditure
rate. Wireless sensor networks provide a natural context for our problem. The constraint
on energy expenditure rates translates directly to increased network lifetimes. We apply
the cross layer optimization to determine the optimal throughput of the network. We
apply the theory of duality in convex optimization to solve the problem.
3
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
4
We observe the effects of variation of the energy expenditure rate constraint on the
nodes in the network. We notice that the optimal throughput is determined by the dominant of the two bounds, namely, the bound imposed by wireless link capacity constraint
and the one imposed by energy expenditure rate constraint.
The rest of this article is organized as follows. In Section 1.2 we have a brief discussion
on the related work. Section 1.3 describes the system model. In Section 1.4 we state the
problem as a convex optimization problem and discuss the corresponding dual problem.
In Section 1.5 we state the numerical results while Section 1.6 concludes the article.
1.2
Related Work
Here we give a brief survey of the work that has already been done in the area of crosslayer optimization. In a paper by Yuan Xue et al. [1] the authors consider the problem
of multi-hop flows on ad-hoc wireless networks by proposing a price based resource
allocation model in order to maximize the aggregate utility. They propose a maximal
clique associated shadow prices, rather than the link associated prices. In [3], Lijun
Chen el al. propose two algorithms which decompose vertically into protocol layers
where TCP and MAC jointly solve the system problem. They also provide a distributed
algorithm for the scheduling of the wireless links in the network. A similar approach is
presented in [2] by Francesco Lo Presti where a cross layer optimization is carried out
and a distributed algorithm for scheduling is suggested which is a combination of the
one suggested in [1]and [3]. [6] discusses the impact of interference on multi-hop wireless
network performance. There a problem of optimal throughput is handled and upper and
lower bounds on it are suggested.
In all these papers, the wireless nodes are assumed to have no constraints on the rate
of energy expenditure; essentially, this amounts to assuming that nodes have access to
unlimited energy. Our work is motivated by wireless sensor networks, and therefore we
impose upper limits on the rate at which energy can be spent by each node.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
1.3
5
System Model
We represent a network as a directed graph G = (N , L), where N represents the set of
nodes in the network and L represents the set of possible wireless links in the network.
The total number of nodes in the network is denoted by N = |N | and the total number
of links by L = |L|. A wireless link (i, j) ∈ L if nodes i and j are within transmission
range of each other i.e. direct communication between nodes i and node j is possible.
As in [2] we assume that if a link (i, j) ∈ L, then so does the link (j, i). We assume a
static topology and each link l ∈ L has a fixed finite capacity Cl , when active. Let F
denote the set of all end-to-end multi-hop flows in the network. For every flow f ∈ F ,
s(f ) and d(f ) represent the source and destination nodes, respectively. The data rate
associated with the flow f is represented by xf and yf l denotes the part of the flow f
that is carried by the link l. Let yf = (yf l )l∈L be a vector representing the part of a flow
f carried by each link in the network. Let h(l) and t(l) denote the head and the tail of
a link l. Then we define two sets H(n) and T (n) for every node n ∈ N as follows:
H(n) =
n
T (n) =
n
l : h(l) = n, l ∈ L
o
.
l : t(l) = n, l ∈ L
o
.
Then in order to satisfy the flow conservation equation, the net flow inserted by a node
n into the network shall be zero unless n is a source or destination node for some flow
f ∈ F . In other words, for every f ∈ F , n ∈ N ,



x
if n = s(f )

 f
X
X
yf l −
yf l =
−xf if n = d(f )


l∈H(n)
l∈T (n)

 0
otherwise
This can be represented in a matrix form with the help of the node-link incidence
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
6
matrix A, which we define as follows:



1
if link l originates at node n


Anl =
−1 if link l terminates at node n



 0
otherwise
Thus the flow conservation constraint can be represented as
Ayf = uf , ∀f ∈ F .
(1.1)
where uf = (uf n )n∈N , and uf n represents the amount of traffic of type f ∈ F injected
into the network by node n. Thus,
uf n =



x

 f
if n = s(f )
−xf if n = d(f )



 0
otherwise
Let y denote a vector of size L, the l th entry of which represents the aggregate flow
on the link l, i.e.
y=
X
yf
f ∈F
1.3.1
Contention Graph
We model contention among the wireless links in the network by the wireless link contention graph as in [1].
DEFINITION 1 A graph Gc = (Vc , Ec ) is a wireless link contention graph of network
G, if there exists a mapping function ϕ : L → Vc that satisfies
1. ϕ(l) ∈ Vc , if and only if l ∈ L;
2. (ϕ(l), ϕ(l0 )) ∈ Ec , if for l, l0 ∈ L, ∃l00 ∈ L, so that l ∩ l00 6= φ and l0 ∩ l00 6= φ.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
7
2
0
4
1
5
3
Figure 1.1: A network graph G
(1,2)
(2,4)
(0,1)
(4,5)
(1,3)
(3,4)
Figure 1.2: The wireless link contention graph Gc corresponding to the network graph G
in the Figure 1.1.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
8
Figure 1.1 shows an example of a wireless network and the corresponding link contention graph is shown in the Figure 1.2 . The contention graph models the interference
among the wireless links. Every link in the network graph G is represented as a node
in the link contention graph Gc . If there exists an edge between the two nodes in the
link contention graph, then the corresponding links in the network graph (which these
two nodes represent in the contention graph) can not be activated simultaneously. As
an example consider two links (i1 , j1 ) ∈ L and (i2 , j2 ) ∈ L in any arbitrary graph. Assume that the node j2 is in the transmission range of the node i1 . Then the signal
transmitted by the node i1 to the node j1 will reach the node j2 also. Thus the node
j2 can not transmit or receive data on the link (i2 , j2 ) as long as the node i1 is in the
transmission mode. Thus, in the link contention graph, an edge will be present between
the nodes representing the links (i1 , j1 ) and (i2 , j2 ) respectively. There are many ways
to incorporate these interference effects into the link contention graph. Here we assume
that a transmission by a node affects all the nodes within twohops from it. Thus there
exists an edge between two nodes in the link contention graph Gc if the corresponding
two links in the network graph G either share a node or share a common neighborhood
link. Figure 1.1 and Figure 1.2 illustrate this. The link (0, 1) and the link (1, 2) contend
for channel access as they share a common node, while the link (0, 1) and the link (2, 4)
contend for channel access as they share a neighborhood link (1, 2). As the contending
links can not be activated simultaneously the effective capacity of a link is less than
its actual capacity. All the links that do not interfere with each other can be scheduled simultaneously. These links correspond to the set of independent nodes in the link
contention graph.
DEFINITION 2 A set of nodes V ⊆ Vc is said to be maximally independent if,
1. no pair of nodes vi ,vj ∈ V is connected.
2. every node v ∈ Vc \ V is connected to at least one node from the set V.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
Suppose there are K maximal independent sets in the link contention graph.
9
Let
I1 , I2 , I3 , . . . , IK be these maximal independent sets and let 0 ≤ ai ≤ 1, be the fraction of time an independent set Ii is activated. Then the fraction of time a link l is
active can be obtained by,
X
λl =
(
ak
k|l ∈ Ik
)
Thus the effective capacity of a link l is given by,
clef f = λl Cl
Let c represent a vector of size L, the lth entry of it representing the effective capacity
of a link l for the specified schedule. We call this vector an effective capacity vector. We
represent an independent set I by a column matrix rI defined as follows,

 C if a link l ∈ I
l
r Il =
 0 o.w.
Cl being the physical layer capacity of the link l. Also we define an L × K matrix MI
the columns of which are the vectors rI . Thus we can write the vector c as,
j=K
c =
X
aj rj
j=1
= MI a
where, aT = [ a1 , a2 , . . . , aK ]
DEFINITION 3 A link rate vector y is said to be schedulable if ∃ some schedule represented by a vector a such that
y ≤ MI a
(1.2)
10
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
1.3.2
Energy Expenditure Rate Constraint
We incorporate the effects of energy-expenditure-rate constraints of nodes into our network model. Let the amount of energy spent by each node while transmitting a bit be et ,
and the amount of energy spent while receiving a bit be er . The net data transmission
and reception rate for each node is restricted by the maximum allowable energy expenditure rate corresponding to that node, which we represent by γn for the node n. Thus
we have the following condition that has to be satisfied at each node in the network,
X
et yj +
j∈H(n)
X
er yj ≤ γn , ∀n ∈ N
(1.3)
j∈T (n)
We define the N × L energy matrix E as follows,
Enl =



e if l ∈ H(n)

 t
er if l ∈ T (n)



 0 otherwise
DEFINITION 4 A link rate vector y is said to be energy-feasible if ,
Ey ≤ Γ,
(1.4)
Γ being a column matrix of size N , the nth entry of which is γn .
It should be noted that, even though the vector y is schedulable, that does not imply
that it is energy feasible and vice versa.
The rest of this section introduces notations that we will be using often in this article.
Omn represents an all zero matrix of size m × n. Similarly 1mn represents an all one
matrix of size m × n. Im indicates an identity matrix of size m.
Let x̃ be a column matrix of size (|F | ∗ L + |F | + K), and for each flow f , we define
a column matrix vf of size N . Let w be a column matrix of size (N ∗ |F | + 1). Thus,
x̃ =
h
y1 x1 y2 x2 . . . y|F | x|F | a1 a2 . . . aK
i
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
11

 1 if i = N ∗ |F | + 1
wi =
 0 otherwise



−1 if i = s(f )


vf i =
1
if i = d(f )



 0
otherwise
For each flow f ∈ F , we define a N × (L + 1) matrix Af as:
Af =
h
A
i
vf
We define two matrices, a (N ∗ |F | + 1) × (L ∗ |F | + K + |F |) matrix B and a
(L + N ) × ((L + 1) ∗ |F | + K) matrix Me as follows:

A1


 ON (L+1)

..

B = 
.


 ON (L+1)

O1(L+1)

IL OL1
Me = 
E ON 1
1.4
ON (L+1) . . . ON (L+1) ON K
A2
. . . ON (L+1) ON K
..
..
.
.
ON (L+1) . . .
O1(L+1)
...
IL OL1 . . .
E ON 1 . . .
A|F |
ON K
O1(L+1)
11K











IL OL1 −MI
E O N 1 ON K


Mathematical Formulation
Consider a wireless network having an arbitrary topology. Our aim is to send data on this
network between the source-destination pairs at the maximum possible rates, without
violating the capacity constraints of the wireless links, flow conservation constraints,
wireless link contention constraints and the energy expenditure rate constraints on the
nodes in the network. We associate a utility function with each flow f ∈ F [2]. Now our
aim is to maximize the aggregate utility associated with the network, which will be the
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
12
sum of the utilities of all the flows in the network. The utility function is assumed to be
concave, nondecreasing and differentiable everywhere. Thus the formal representation of
the problem of maximization of the source data-rates is as follows:
maximize:
X
U (xf )
f ∈F
Subject to: Ayf = uf , ∀f ∈ F
X
yf ≤ M I a
f ∈F
et
X
j∈H(n)
yj + e r
X
yj ≤ γn , ∀n ∈ N
j∈T (n)
K
X
ak = 1
k=1
a ≥ 0
yf ≥ 0, ∀f ∈ F
xf ≥ 0, ∀f ∈ F
This is a convex optimization problem with affine constraints. But to solve this problem,
a central entity is required that is aware of the network topology. We take the help of
duality theory in order to solve this problem, based on which a distributed algorithm can
be derived in future. Since Slater’s conditions for Strong Duality are satisfied (convex
optimization problem with affine constraints) [9], the optimal value of the dual problem
and that of the primal one will be the same. Before stating the dual problem we represent
the above problem in the standard form, with all the unknowns on one side.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
maximize: f (x̃) =
X
13
U (xf )
f ∈F
Subject to: Bx̃ = w
(1.5)
Me x̃ ≤ Γe
(1.6)
x̃ ≥ 0
(1.7)
where Γe is as follows,

 γ
(i−L) if L < i ≤ (L + N )
Γei =
 0
otherwise
All the matrices in the equations (1.5)-(1.7) are as defined in the previous section.
1.4.1
Dual Problem
While formulating the dual problem we partially relax the primal problem by relaxing
the inequality constraint given by the Equation (1.6) above. We represent a Lagrangian
function corresponding to this relaxation as:

L(x̃, p, q) = f (x̃) − 
p
q
T
 (Me x̃ − Γe )
where p (L × 1) represents the prices associated with the links and q (N × 1) represents
the prices associated with the nodes in the network. So the relaxed problem becomes:
max L(x̃, p, q)
subject to: Bx̃ − w = 0
(1.8)
x̃ ≥ 0
(1.9)
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
14
Thus the dual problem associated with the primal problem that we started with is
given by:
min D(p, q)
(1.10)
subject to: p ≥ 0
q ≥ 0
where,
D(p, q) = max L(x̃, p, q)
1.4.2
(1.11)
subject to: Bx̃ − w = 0
(1.12)
x̃ ≥ 0
(1.13)
Solution to the relaxed problem
The problem of maximizing the relaxed objective function can be split into two separate
problems which are illustrated below. We can notice that a solution to the one of them
is associated with the routing in the network while the solution to the other is associated
with the scheduling of the wireless links in the network. We define two functions D1 (p, q)
and D2 (p) (as in [3]) as follows:
D1 (p, q) = max(
X
(U (xf ) −
f ∈F
L
X
(pi + qh(i) et + pt(i) er )yf i ))
i=1
subject to: Ayf = uf , ∀f ∈ F
yf ≥ 0, ∀f ∈ F
xf ≥ 0, ∀f ∈ F .
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
15
D2 (p) = max(pT MI a)
K
X
subject to:
ak = 1
k=1
a≥0
So we have,
D(p, q) = D1 (p, q) + D2 (p) +
N
X
qi γ i .
(1.14)
i=1
It can be noticed that the function D2 (p) is a piecewise linear one and hence not differentiable everywhere, as a result of which the function D(p, q) is also not differentiable
everywhere.
1.4.3
Solution to the routing problem
The function D1 (p, q) is associated with the routing problem. The solution to this
problem is to route the traffic for each source destination pair along the least cost path
between the source and the destination where the cost of a link l is an effective cost
pef f (l) given by (pl + qh(l) et + qt(l) er ), which takes into account the cost associated with
the head and tail nodes of the link. Since there can be multiple paths with least total
effective-cost and traffic can be routed along these paths in more than one manner, the
solution to the routing problem may not be unique. Thus, for every flow f ∈ F , we have
a maximization problem,
max(U (xf ) −
xf ,yf
L
X
peff (l)yf l )
l=1
subject to: Ayf = uf .
yf ≥ 0.
xf ≥ 0.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
16
Solving this for every flow gives
xf = U 0−1 (p(f )), ∀f ∈ F
(1.15)
where p(f ) is the total effective-cost of the least effective-cost path from the source to
the destination for the flow f .
1.4.4
Solution to the scheduling problem
The function D2 (p) is associated with the MAC-layer scheduling problem. Here our aim
is to find a schedule a, that will maximize the product,
pT MI a.
DEFINITION 5 A solution to the scheduling problem will be said to satisfy the property P, if the schedule a is of the form,

 1 for exactly one 1 ≤ i ≤ K
ai =
 0 for the rest 1 ≤ i ≤ K
Thus, a solution to the scheduling problem satisfying property P is such that only
one independent set is activated repeatedly.
Claim 1 For an arbitrary price vector p, there will always be an optimal solution to the
scheduling problem which satisfies the property P.
Proof : A solution to this problem lies in the convex hull of the columns of the matrix
MI . As the objective is a linear function of the unknowns ai , 1 ≤ i ≤ K, from the theory
of linear programming we can say that the solution to this problem lies at the extreme
point of the convex hull of the columns of the matrix MI [11]. Since the extreme point
of this convex hull corresponds to one of the maximal independent sets of the contention
graph Gc , the resulting solution will schedule links that belong to this independent set
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
17
and will not activate the other independent sets, hence the schedule a will satisfy the
property P.
Claim 2 A feasible solution to the primal problem can not satisfy the property P, unless
a destination node is within a single-hop from the source node.
Proof : If we take a schedule satisfying property P, then only one independent set gets
scheduled repeatedly. This means that the effective capacity vector will be such that
there are few elements of capacity C, while the rest have zero capacity. So, in the
routing problem, we will get a disjoint set of links, and there will be no end-to-end path
between the source and the destination. Hence the claim.
Claim 3 There exists some optimal solution to the scheduling problem which does not
satisfy property P.
Proof : Since Slater’s conditions for Strong Duality are satisfied by the primal problem
(convex objective function with affine constraints), the duality gap is zero [10]. Which
implies that, optimal primal objective and optimal dual objective are equal. Hence there
exists an optimal solution to the dual problem, say p∗. Since a scheduling policy that
forms the part of the solution to the primal problem, does not satisfy the property P
(from the Claim 2 above), there must exist some optimal solution to the scheduling
problem, for p = p∗, which does not satisfy property P. Hence the claim.
Claim 4 If p∗ is an optimal solution to the Equation (1.10), then multiple columns r j
of matrix MI , satisfy the equation,
rj = arg max p∗T ri
1≤i≤K
(1.16)
Proof : Let an independent set given by an optimal schedule that satisfies the property
P be rm . Now from the Claim 3 there exists an optimal schedule a∗ (p∗ ) that does not
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
18
satisfy the property P. Thus we have,
p∗T rm = p∗T MI a∗ (p∗ )
K
X
=
p∗T ri a∗i (p∗ )
(1.17)
i=0
As a∗ (p∗ ) represents a schedule,
K
X
a∗i (p∗ ) = 1
(1.18)
i=0
From Equations (1.16),(1.17) and (1.18) we can say that the columns of MI corresponding
to the nonzero entries of the vector a∗ (p∗ ) shall satisfy,
p∗T ri = p∗T rm if a∗i (p∗ ) 6= 0
As multiple entries of a∗ (p∗ ) are non-zero, multiple columns of MI satisfy the Equation
(1.16).
1.4.5
Dual problem solution by sub-gradient method
Since the function D(p) is not differentiable everywhere, the usual gradient method is
not applicable. Hence we go for a sub-gradient method in order to solve the problem.
DEFINITION 6 [10] Given a convex function D(p) : <n → <, we say that a vector
h(p) ∈ <n is a sub-gradient of D(p) at point p ∈ <n if:D(p̄) ≥ D(p)+(p̄−p)T h(p), ∀p̄ ∈
<n
Now
h(p, q) = Γe − Me x̃(p, q)
(1.19)
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
0
1
3
4
2
19
7
6
5
Figure 1.3: The data flow is from the node 0 to the node 7 for a single source-destination
case.
is a sub-gradient of a dual function D(p) at point p.This can be shown as in [3]. Thus
using the sub-gradient algorithm the price vector p is updated as follows [10],
pj (t + 1) = [pj (t) − δhj (p(t), q(t))]+
(1.20)
qj (t + 1) = [qj (t) − δhL+j (p(t), q(t))]+
Where [x]+ = max(0, x) and δ is the step-size associated with the sub-gradient method.
It can be shown that if the norm of the sub-gradient is bounded i.e. there exists H, such
that ||h(p, q)||2 ≤ H, ∀p ≥ 0, q ≥ 0 then the sub-gradient algorithm converges within
the band of H 2 δ/2 around an optimal solution [3].
1.5
Numerical results
Here we give the results that we obtained after solving the dual problem by a sub-gradient
algorithm by implementing the equations (1.15),(1.19) and (1.20) . We illustrate the
results for one simple topology as shown in the Figure 1.3. We assume that all the links
in the network have the same capacity C = 10M bits/sec. Results are shown for a single
source-destination pair as well as multiple source-destination pairs.The utility function
associated with the flows in the network is U (xf ) = log(xf ). Assume that et , the energy
spent by a node for transmitting a single bit is 0.75 units and er , the energy spent by
a node for receiving a single bit is 0.5 units. We assume that the energy expenditure
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
20
rate constraint is the same for all the nodes, we represent this by γ. We take a constant
step-size δ = 0.001 for the sub-gradient method. Along with the numerical evaluation,
we also compute the time averages of the data-rates of the respective flows. For both the
cases(i.e. single end-to-end flow and multiple end-to-end flows.) plots are shown below.
It is worth noticing that the time averages of the source data-rates and the data-rates
obtained by the algorithm converge to the same value.
1.5.1
Single end-to-end flow
Results from the subgradient algorithm
Here the end-to-end flow exists between node 0 and node 7. The graph in Figure 1.4
shows variations in source rate as the iterations proceed, for γ = 3 and γ = 10 and the
case of without γ. It can be noted that, as γ is varied from 3 to 10, the data-rate also
varies from around 4M bits/sec to 5M bits/sec and keeps oscillating around this value
of 5M bits/sec. It can be noticed that this limiting data-rate is the optimal throughput
when there is no energy rate expenditure constraint. Thus we see that for smaller values
of γ, the energy expenditure rate constraint is the dominating one, while as the value of
γ increases the wireless link capacity constraint becomes the dominating one. This can
be verified by noting that the curve corresponding to γ = 10 and the one without the
γ restriction merge with one another. Thus for larger γ, the behavior of the system is
same as that of without energy expenditure constraint. The oscillations in data-rate are
due to the constant step size, that we have used while implementing the sub-gradient
method [3]. We observe that, the path (node-0, node-1, node-2, node-4, node-7), which
is a four-hop path, is often selected over a three-hop path (node-0,node-1,node-4,node-7),
which differs from it by a single node while routing the traffic from the source to the
destination.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
21
Simulation Results
In the simulation, the routing and scheduling are carried out in the centralized manner,
according to the subgradient algorithm. The link and node prices are made available
to the source from which it determines the least cost route to the destination and the
data-rate at which it should operate over that route in each iteration. The source keeps
accumulating the bits according to this data-rate till any of the links associated with the
source is activated by the scheduling algorithm. On activation of a link associated with
the source, the source transmits over that link at maximum of CM bps. So the source is
idle whenever links associated with it are not activated and it transmits at maximum of
CM bps whenever a link associated with it is activated. Thus in the Figure 1.4 xf −simu (i)
indicates the time average of the data-rate, where average is obtained over i iterations.
1.5.2
Multiple end-to-end flows
Here there are three end-to-end flows. From node 0 to node 7, node 5 to node 4 and node
6 to node 1. The Figure 1.5 shows the variation in the net utility with the variation in
gamma. We note that as γ increases the net-utility with energy expenditure constraints
approaches the net-utility without energy expenditure constraints.The average net-utility
net−utisimu is obtained in the similar manner as xf −simu , as explained in the single source
destination case. Figure 1.6,Figure 1.7,Figure 1.8 show the plots of data-rates that are
present between the node pairs (0, 7),(5, 4) and (6, 1) respectively.
It is worth noting that, the source data rate is determined based on the total price of
the least cost path from the source to destination, where the cost of the path includes not
only the prices of the links, but also the prices associated with the nodes along the path.
Thus the source adjusts its rate as in the case of a network without energy expenditure
constraints, with a modified definition of a link price, which we call an effective link
price.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
1.6
22
Conclusion
We have presented a solution to the problem of joint congestion control, routing and MAC
layer link scheduling in the multi-hop wireless networks where the nodes in the network
are subjected to a maximum energy expenditure rate constraint. The problem is solved
by dual decomposition and the sub-gradient algorithm. We also study the effects of
maximum allowable energy expenditure rates of the nodes, on the maximum throughput
supported by the network. We observe that as the maximum energy expenditure rate
increases, the maximum throughput supported by the network also increases up-to the
certain value and it then remains constant irrespective of the increment in the maximum
energy expenditure rate.
Our results show the maximum allowable data rates under different energy expenditure constraints. In our ongoing work, we are trying to address the problem of finding a
distributed scheduling policy that will support this maximum allowable data rate.
1.7
Acknowledgment
This work is partially supported by the Defence Research and Development Organization
(DRDO), Ministry of Defence, Government of India, under a research grant on wireless
sensor networks (DRDO 571, IISc).
The authors are grateful to the anonymous reviewers for pointing out several inaccuracies in the version submitted for review.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
23
7
Data-rate in Mbps
6
5
4
3
xf-algo,γ=10
xf-simu,γ=10
xf-algo,γ=3
xf-simu,γ=3
xf-algo, w/o γ
xf-simu, w/o γ
2
1
0
0
0.0 X ⋅10
7.5 X ⋅10
4
1.5 X ⋅10
5
2.3 X ⋅10
5
3.0 X ⋅10
5
Iteration
Figure 1.4: Source data-rate for different values of γ for a single source-destination case.
Node 0 is the source node and node 7 is the destination node.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
24
4.00
3.00
2.00
Net Utility
1.00
0.00
-1.00
net-utialgo, γ=10
net-utisimu,γ=10
net-utialgo,γ=3
net-utisimu,γ=3
net-utialgo, w/o γ
net-utisimu, w/o γ
-2.00
-3.00
-4.00
-5.00
0
0.0 X 10
7.5 X 10
4
1.5 X 10
5
2.3 X 10
5
3.0 X 10
5
Iteration
Figure 1.5: Aggregate utility for different values of γ for a multiple source-destination
case. Data flow is between node pairs (0, 7),(5, 4) and (6, 1).
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
5
Data rate in Mbps
4
3
25
xf-algo-0,γ=10
xf-simu-0,γ=10
xf-algo-0,γ=3
xf-simu-0,γ=3
xf-algo-0,w/o γ
xf-simu-0,w/o γ
2
1
0
0
0.0 X ⋅10
7.5 X ⋅10
4
1.5 X ⋅10
5
2.3 X ⋅10
5
3.0 X ⋅10
5
Iteration
Figure 1.6: Source rate variation with γ as iterations proceed for a node pair (0, 7) in
the presence of multiple end-to-end flows.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
5
Data rate in Mbps
4
3
26
xf-algo-0,γ=10
xf-simu-0,γ=10
xf-algo-0,γ=3
xf-simu-0,γ=3
xf-algo-0,w/o γ
xf-simu-0,w/o γ
2
1
0
0
0.0 X ⋅10
7.5 X ⋅10
4
1.5 X ⋅10
5
2.3 X ⋅10
5
3.0 X ⋅10
5
Iteration
Figure 1.7: Source rate variation with γ as iterations proceed for a node pair (5, 4) in
the presence of multiple end-to-end flows.
CHAPTER 1. EFFECT OF ENERGY EXPENDITURE RATE CONSTRAINTS
5
Data rate in Mbps
4
3
27
xf-algo-0,γ=10
xf-simu-0,γ=10
xf-algo-0,γ=3
xf-simu-0,γ=3
xf-algo-0,w/o γ
xf-simu-0,w/o γ
2
1
0
0
0.0 X ⋅10
7.5 X ⋅10
4
1.5 X ⋅10
5
2.3 X ⋅10
5
3.0 X ⋅10
5
Iteration
Figure 1.8: Source rate variation with γ as iterations proceed for a node pair (6, 1) in
the presence of multiple end-to-end flows.
Bibliography
[1] Y. Xue, B. Li,K. Nahrstedt. ”Price-based Resource Allocation in Wireless Ad
Hoc Networks”,in Proc. of the 11th International Workshop on Quality of Service
(IWQoS), also Lecture Notes in Computer Science, ACM Springer-Verlag, Vol. 2707,
pp. 79-96, Monterey, CA, June, 2003.
[2] F.L. Presti, ”Joint Congestion Control, Routing and Media Access Control Optimization via Dual Decomposition for Ad Hoc Wireless Networks,” In the Proc. of
the 8-th International Symposium on Modeling, Analysis and Simulation of Wireless
and Mobile Systems (MSWiM 2005) , Montreal, Canada, October 2005.
[3] L. Chen,S. Low,J. Doyle ,”Joint Congestion Control and Media Access Control
Design for Ad Hoc Wireless Networks”in Proc. of IEEE Infocom 2005,March 2005.
[4] L. Chen,S. Low,M. Chiang and J.C. Doyle, ”Cross-Layer Congestion Control,Routing and Scheduling Design in Ad Hoc Wireless Networks”. in Proc. of
IEEE Infocom 2006,April 2006.
[5] X. Lin,N.B. Shroff,R. Srikant ”A Tutorial on Cross-Layer Optimization in Wireless
Networks”IEEE Journal on Selected Areas in Communication,vol.24 no. 8,August
2006.
[6] K. Jain, J. Padhye, V.N. Padmanabhan, L. Qiu ”Impact of Interference On Multihop Wireless Network Performance” in Proc. of IEEE MobiCom 2003, September
2003.
28
BIBLIOGRAPHY
29
[7] S.H. Low and D.E. Lapsley, ”Optimization flow control,I:Basic algorithm and convergence ,”IEEE/ACM Trans. Netw.,pp. 861-875,Dec 1999.
[8] F.P. Kelly,A. Maullo, and D. Tan, ”Rate control in communication networks:shadow
prices,proportional fairness and stability,”J.Oper.Res.Soc.,vol. 49,pp. 237-252,1998.
[9] S. Boyd and L. Vandenberghe, ”Convex Optimization”,Cambridge University Press
[10] D. Bertsekas ”Nonlinear Programming”, Athena Scientific, 2003.
[11] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, ”Nonlinear Programming,Theory and
Algorithms” John Wiley and Sons, Inc. 2004.