Cross-layer Congestion Control, Routing and
Scheduling Design in Ad Hoc Wireless Networks
Lijun Chen
†
,StevenH.Low
†
, Mung Chiang
‡
and John C. Doyle
†
†
Engineering & Applied Science Division, California Institute of Technology, USA
‡
Department of Electrical Engineering, Princeton University, USA
Abstract— This paper considers jointly optimal design of cross-
layer congestion control, routing and scheduling for ad hoc
wireless networks. We first formulate the rate constraint and
scheduling constraint using multicommodity flow variables, and
formulate resource allocation in networks with fixed wireless
channels (or single-rate wireless devices that can mask channel
variations) as a utility maximization problem with these con-
straints. By dual decomposition, the resource allocation problem
naturally decomposes into three subproblems: congestion control,
routing and scheduling that interact through congestion price.
The global convergence property of this algorithm is proved. We
next extend the dual algorithm to handle networks with time-
varying channels and adaptive multi-rate devices. The stability
of the resulting system is established, and its performance is
characterized with respect to an ideal reference system which
has the best feasible rate region at link layer.
We then generalize the aforementioned results to a general
model of queueing network served by a set of interdependent
parallel servers with time-varying service capabilities, which
models many design problems in communication networks. We
show that for a general convex optimization problem where a
subset of variables lie in a polytope and the rest in a convex set,
the dual-based algorithm remains stable and optimal when the
constraint set is modulated by an irreducible finite-state Markov
chain. This paper thus presents a step toward a systematic way
to carry out cross-layer design in the framework of “layering as
optimization decomposition” for time-varying channel models.
I. INTRODUCTION
We consider the problem of congestion control and resource
allocation (through routing and scheduling) over a multi-
hop wireless ad hoc network. Traditionally, network protocols
take a strictly layered structure and implement congestion
control, routing and scheduling independently at different
layers. However, wireless spectrum is a scarce resource, and
it is important to use the wireless channel efficiently. In order
to achieve high end-to-end throughput and efficient resource
utilization, congestion control, routing and scheduling should
be jointly designed while the architectural separation among
them is preserved.
The need for joint design across these three layers is moti-
vated by three observations. First, wireless channel is a shared
medium and interference-limited. Unlike in wireline networks
where links are disjoint resources with fixed capacities, in ad
hoc wireless networks the link capacities are “elastic” and
the contention among links provide a fundamental constraint
for resource allocation (see e.g. [3]), i.e., they determine the
feasible rate region at link layer.
Second, most routing schemes for ad hoc networks select
paths that minimize hop count (see e.g. [12], [ 25]). This
implicitly predefines a route for any source-destination pair
of a static network, independent of the pattern of traffic
demand and interference/contention among links. This may
result in congestion at some region while other regions are
under-utilized. To use the wireless spectrum more efficiently,
we should exploit multiple paths based on the pattern of
traffic demand and interference/contention among links. As
we will see below, routing is then determined from the rate
and scheduling constraints.
Lastly, TCP congestion control algorithms can be inter-
preted as distributed primal-dual algorithms over the Internet
to maximize aggregate utility, see e.g. [13], [20], [15]. This
series of work implicitly assumes a network where link
capacities are fixed and routes are pre-specified. Here, we
extend the basic utility maximization formulation with rate
constraints at nodes and additional constraints on scheduling
at link layer.
We model the contention relations between wireless links
as a conflict graph (see e.g. [11]). This construction indicates
which groups of links mutually interfere and cannot be active
simultaneously. The feasible rate region at link layer is the
convex hull of the corresponding rate vectors of independent
sets of the conflict graph. We introduce multi-commodity
flow variables to formulate rate constraint at the network
layer, and formulate resource allocation in wireless ad hoc
networks with fixed channel or single-rate devices as a utility
maximization problem with those constraints. We then apply
duality theory to decompose the system problem vertically
into congestion control s ubproblem and routing/scheduling
subproblem that interact through congestion prices. Based on
this decomposition, a distributed subgradient algorithm for
joint congestion control, routing and scheduling is obtained,
and proved to approach arbitrarily close to the optimum of
the system problem. This algorithm motivates a joint design
where the source adjusts its sending rate according to the
congestion price generated locally at the source node, and
backpressure from the differential price of neighboring nodes
is used for optimal scheduling and routing. We next extend the
dual subgradient algorithm to wireless ad hoc networks with
time-varying channels and adaptive multi-rate devices. The
stability of the resulting system is proved, and its performance
is characterized with respect to an ideal reference system.
We then extend the aforementioned results to a gener-
alized model of queueing network that is served by a set
of interdependent parallel servers with time-varying service
capabilities. This general technique leads to results regarding
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the stability and optimality of dual algorithm in face of time-
varying parameters, extending most of the earlier publications
in this area that assumes deterministic channel models. We
show that for a general convex optimization problem where a
subset of variables lie in a polytope and the rest in a convex
set, the dual-based algorithm remains stable and optimal when
the constraint set is modulated by an irreducible finite-state
Markov chain.
II. R
ELATED WORK
The work in [13], [15], [20], [21] provides a utility-based
optimization framework for Internet congestion control. The
same framework has been applied to study the congestion
control over ad hoc wireless networks (see, e.g., [4], [36], [35],
[3], [18]). In [3], the authors study joint congestion control
and media access control for ad hoc wireless network, and
formulate rate allocation as a utility maximization problem
with the constraints that arise from contention f or channel
access. This paper substantially extends [3] to include routing
and to study the network with time-varying channel and multi-
rate devices.
In [22], the authors use multi-commodity flow variables
to characterize the network capacity region for a wireless
network with time-varying channel, and propose a joint routing
and power allocation policy to stabilize the system whenever
the input rates are within this capacity region. In [11], the
authors study the impact of interference on multi-hop wireless
network performance. They model wireless interference using
the conflict graph, and show that there is an opportunity
for achieving throughput gains by employing an interference-
aware routing protocol. We use the same construction to model
the contention relations among wireless links. In [7], [14], the
authors use a similar model to study the problem of jointly
routing the flows and scheduling the transmissions to deter-
mine the achievable rates in multi-hop wireless networks. All
these works focus on the interaction between link and network
layers, and try to characterize the achievable rate region at
network layer. We include the end-to-end transport layer, and
as such, the network uses congestion control to automatically
explore the achievable rate region while optimizing some
global objective for the end users.
The stochastic Lyapunov function method is a powerful tool
to prove the stability of Markovian system [1], [29]. Especially,
Theorem 3.1 in [29] provides sufficient conditions for the
stability of general Markov chain. We combine convex analysis
with stochastic Lyapunov method to establish the stability and
optimality properties of networks with time-varying channels.
Our result is applicable to a variety of time-varying systems
that can be solved or modelled by dual algorithms. Similar
result is obtained in other contexts through different techniques
[28], [6].
Our goal is to present a systematic approach to cross-layer
design, not only to improve the performance, but more impor-
tantly, to make the interactions between different layers more
transparent. Motivated by the duality model of TCP/AQM,
which is an example of “horizontal” decomposition via dual
decomposition, researchers have extended the utility maxi-
mization framework to provide a general cross-layer design
methodology. As we will see in this paper, duality theory leads
to a natural “vertical” decomposition into separate designs
of different layers that interact through congestion price.
Recent publications along this line of “layering as optimization
decomposition” [5] includes [31], [8] for TCP/IP interaction,
[34] for routing and resource allocation, [4], [16] for TCP
and physical layer, and [3], [17], [18], [32] for joint TCP and
media access control or scheduling.
III. M
ODEL
Consider an ad hoc wireless network with a set N of nodes
and a set L of logical links. These links are directed, though
we assume connectivity to be symmetric, i.e., link (j, i) ∈ L
if and only if (i, j) ∈ L. We assume a static topology and
each link l has a fixed finite capacity c
l
bits per second when
active, i.e., we implicitly assume that the wireless channel
is fixed or some underlying mechanism is used to mask the
channel variation so that the wireless channel appears to have
a fixed rate. This assumption will be relaxed in Section V.
Wireless channel is a shared medium and interference-limited
where links contend with each other for exclusive access to
the channel. We will use the conflict graph to capture the
contention relations among links. The feasible rate region
at link layer is then a convex hull of the corresponding
rate vectors of independent sets of the conflict graph. We
will further introduce multi-commodity flow variables, which
correspond to the link capacities allocated to the flows towards
different destinations, to describe the rate constraint at network
layer. The resource allocation is then formulated as a utility
maximization problem with schedulability and rate constraints.
A. Schedulability and Rate Constraint
In this paper, we consider a network with primary inter-
ference: links that share a common node cannot transmit
or receive simultaneously, but links that do not share nodes
can do so. The same interference model has been used in
e.g. [14], [36]. It models a wireless network with multiple
channels available for transmission. For example, simultaneous
communications in a neighborhood are enabled by using
orthogonal CDMA or FDMA channels. Under this interference
model, we can construct a conflict graph [11] that captures
the contention relations among the links. In the conflict graph,
each vertex represents a link, and an edge between two vertices
denotes the contention between the two corresponding links:
these links cannot transmit at the same time. Fig.1 shows an
example of a wireless ad hoc network and its conflict graph.
1
246
35
5
31
2
64
ACBD
Fig. 1. Example of an ad hoc wireless network with 4 nodes and 6 logical
links and the corresponding conflict graph
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Given a conflict graph, we can identify all its independent
sets of vertices
1
. The links in an independent set can transmit
simultaneously. Let E denote the set of independent sets
with each independent set indexed by e. We represent an
independent set e as a |L|-dimensional rate vector r
e
, where
the lth entry is
r
e
l
:=
c
l
if l ∈ e
0 otherwise
The feasible rate region Π at the link layer is then defined as
the convex hull of these rate vectors
Π:=
r : r =
e
a
e
r
e
,a
e
≥ 0,
e
a
e
=1
(1)
Thus, given a link flow vector y, the schedulability constraint
says that y should satisfy y ∈ Π.
Let D denote the set of destination nodes of network layer
flows. Let f
k
i,j
≥ 0 denote the amount of capacity of link (i, j)
allocated to the flow to destination k. Then f
i,j
:=
k∈D
f
k
i,j
is the aggregate capacity on link (i, j). From the schedulability
constraint, f := {f
i,j
} should satisfy
f ∈ Π (2)
Let x
k
i
≥ 0 denote the flow generated at node i towards
destination k. Then the aggregate capacity for its incoming
flows and generated flow to the destination k should not exceed
the summation of the capacities for its outgoing flows to k:
x
k
i
≤
j:(i,j)∈L
f
k
i,j
−
j:(j,i)∈L
f
k
j,i
,i∈ N, k ∈ D, i = k (3)
Equation (3) is the rate constraint for resource allocation.
It is similar to multicommodity flow model for the routing
of data flows in the network, but we give multi-commodity
flow variables a different interpretation as the amount of link
capacities allocated to the flows of different destinations.
B. Problem Formulation
We use l ∈ L or alternatively node pair (i, j) ∈ N × N
to denote a link. We also stack up the entries of any tensor
t
i,j
(or t
j
i
) to construct a vector, denoted by {t
i,j
} (or {t
j
i
})
or just t. Assume the network is shared by a set S of sources
indexed by s. For notational simplicity, we assume that there
is at most one flow between any node and destination pair
[i, k]
2
, and use s or alternatively node pair [i, k] ∈ S × D to
denote a network layer flow.
Assume each source s attains a utility U
s
(x
s
) when it
transmits at rate x
s
packets per second. We assume U
s
is
continuously differentiable, increasing, and strictly concave.
Our objective is to choose source rates x
s
and allocated
capacities f
k
i,j
so as to solve the following global problem:
max
x
s
≥0,f
k
i,j
≥0
s
U
s
(x
s
) (4)
subject to x
k
i
≤
j:(i,j)∈L
f
k
i,j
−
j:(j,i)∈L
f
k
j,i
(5)
f ∈ Π (6)
1
An independent set of vertices is a set of vertices that has no edges between
each other.
2
The extension to the situation with multiple flows between any node-
destination pair is straightforward.
where i ∈ N, k ∈ D, i = k, and x
k
i
=0if [i, k] ∈ S × D.
Solving the system problem (4)-(6) directly requires coordi-
nation among possibly all sources and links, thus is impractical
in real network. Since (4) is a convex optimization problem
with strong duality, distributed algorithms can be derived by
formulating and solving its Lagrange dual problem. In the
next section, we will solve the dual problem and interpret the
resulting algorithm in the context of joint design of congestion
control, routing and scheduling.
IV. C
RO SS-LAYER DESIGN VIA DUAL DECOMPOSITION
A. Dual Algorithm
Consider the dual to the primal problem (4,5,6):
min
p≥0
D(p) (7)
with partial dual function
D(p)= max
x
s
≥0,f
k
i,j
≥0
s
U
s
(x
s
) −
i∈N,k∈D,i=k
p
k
i
(x
k
i
−
j:(i,j)∈L
f
k
i,j
+
j:(j,i)∈L
f
k
j,i
) (8)
subject to f ∈ Π (9)
where we relax only the constraint (5) by introducing Lagrange
multiplier p
k
i
for node i ∈ N and destination k ∈ D.The
maximization problem in (8) can be decomposed into the
following two subproblems
D
1
(p) = max
x
s
≥0
s
U
s
(x
s
) −
s
x
s
p
s
(10)
and
D
2
(p)= max
f
k
i,j
≥0
i,k
p
k
i
j
f
k
i,j
−
j
f
k
j,i
(11)
subject to f ∈ Π (12)
where we use p
s
to denote the multiplier p
k
i
if [i, k] ∈ S × D.
If we interpret p
k
i
as the congestion price, the first subproblem
is congestion control [20], [21], and the second one is the joint
routing and scheduling since to solve it we need to determine
the amount of capacity f
k
i,j
that link (i, j) is allocated to
transmit the data flow towards destination k. Thus, by dual
decomposition, the flow optimization problem decomposes
into separate “local” optimization problems of transport and
network/link layers, respectively, and they interact through
congestion prices.
The congestion control problem (10) admits a unique max-
imizer
x
s
(p)=U
s
−1
(p
s
) (13)
which adjusts the source rate according to the congestion price
of the source node. In contrast to traditional TCP congestion
control where the source adjusts its sending rate according
to the aggregate price along its path, in our algorithm the
congestion price is generated locally at the source node.
Note that, since
i,k
p
k
i
j
f
k
i,j
−
j
f
k
j,i
=
i,j,k
f
k
i,j
p
k
i
− p
k
j
,
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problem (11)-(12) is equivalent to the following problem
D
2
(p) = max
f
i,j
≥0
i,j
f
i,j
max
k
p
k
i
− p
k
j
(14)
subject to f ∈ Π (15)
This motivates the following joint scheduling and routing
algorithm:
1) For each link (i, j), find destination k
∗
such that k
∗
∈
arg max
k
(p
k
i
− p
k
j
), and define w
i,j
= p
k
∗
i
− p
k
∗
j
.
2) Scheduling: choose
f
i,j
such that
f ∈ arg max
f ∈Π
(i,j)∈L
w
i,j
f
i,j
(16)
There may exist multiple maximizers, but we always pick
an extreme point maximizer
3
. An extreme point maximizer
corresponds to a maximal independent set of the flow con-
tention graph. The scheduling (16) is a difficult problem for
ad hoc wireless network. We will discuss its solution in detail
in Subsection IV-C.
3) Routing: over link (i, j), send an amount of bits for
destination k
∗
according to the rate determined by the above
scheduling.
The w
i,j
values represent the maximum differential congestion
price of destination k packets between nodes i and j.The
above algorithm uses back-pressure to do optimal scheduling
and find optimal routing
4
. Note that 1)–3) is equivalent to
solve the problem (11)-(12) by the following assignment
f
k
i,j
=
f
i,j
if k = k
∗
0 if k = k
∗
Now we come to solve the dual problem (7). Note that
the dual function D(p) is not differentiable, as D
2
(p) is a
piecewise linear function and not differentiable. Therefore, we
cannot use the usual gradient methods, we will instead solve
the dual problem using subgradient method.
Suppose (f
k
i,j
) is the solution from the above joint routing
and scheduling algorithm. It is easy to verify that
g
k
i
(p)=
j
f
k
i,j
(p) −
j
f
k
j,i
(p) − x
k
i
(p) (17)
is a subgradient
5
of dual function D(p) at point p. Thus,
by the subgradient method [26], [2], we obtain the following
algorithm for price adjustment for node destination pair (i, k)
p
k
i
(t +1) = [p
k
i
(t)+γ
t
( x
k
i
(p(t))
−(
j:(i,j)∈L
f
k
i,j
(p(t)) −
j:(j,i)∈L
f
k
j,i
(p(t))))]
+
(18)
where γ
t
is a positive scalar stepsize, and ‘+’ denotes the
projection onto the set
+
of non-negative real numbers.
3
A point in a convex set is an extreme point if it cannot be written as a
convex combination of other points in the convex set.
4
The above joint routing and scheduling recovers the DRPC policy in
reference [22], except that step 2 is scheduling here and power allocation
there, and data is routed based on destination here and “commodity” there. We
show that the DRPC policy follows mathematically from dual decomposition.
Similar decomposition result for the network with deterministic wireless
channel is also revealed in the journal version of [22] and [18].
5
Given a convex function f : R
n
→R, a vector d ∈R
n
is a subgradient
of f at a point u ∈R
n
if f(v) ≥ f(u)+(v − u)
T
d, ∀v ∈R
n
.
Eq.(18) says that, if the demand x
k
i
(p(t)) for bandwidth at
node i for the flow to destination k exceeds the effective
capacity
j
f
k
i,j
−
j
f
k
j,i
, the price p
k
i
will rise, which will in
turn decrease the demand (see eq.(13)) and increases effective
capacity (see eq.(14)). Also, note that eq.(18) is distributed
and can be implemented by individual nodes using only local
information.
The above dual algorithm motivates a joint congestion
control, routing and scheduling design where at the trans-
port layer sources s individually adjust their rates accord-
ing to the local congestion price, and nodes i individu-
ally update their prices according to (18); and at the net-
work/link layer nodes i solve the scheduling (16) and route
data flows accordingly. In summary, we have the following
Algorithm 1: Joint Design Algorithm
At time t:
1) Each node
i implicitly updates its price with respect to
destination
k
p
k
i
(t +1) = [p
k
i
(t)+γ
t
( x
k
i
(p(t))
−(
j:(i,j)∈L
f
k
i,j
(p(t)) −
j:(j,i)∈L
f
k
j,i
(p(t))))]
+
,
and passes the price p
k
i
to all its neighbors. Note that p
k
i
(t) is
interpreted as the congestion price at the beginning of time
slot
t.
2) Congestion control: each source node
s adjusts its sending
rate for the period
t, according to local congestion price
x
s
(t)=U
s
−1
(p
s
(t))
3) Each node i collects congestion price information from
its neighbor j, finds destination k(t) such that k(t) ∈
arg max
k
(p
k
i
(t)−p
k
j
(t)), and calculates differential price w
i,j
(t)=
p
k(t)
i
(t) − p
k(t)
j
(t) and passes this information to its neighbors.
4) Scheduling: each node
i collects differential price infor-
mation from its neighbors in the previous period, and in the
beginning of period
t allocates a capacity
f
i,j
(t) over link (i, j)
such that
f(t) ∈ arg max
f ∈Π
(i,j)∈L
w
i,j
(t)f
i,j
5) Routing: over link (i, j), send an amount of bits for destina-
tion
k(t) according to the rate determined by the scheduling.
B. Convergence Analysis
In this subsection, we prove the convergence property of
Algorithm 1. Subgradient may not be a direction of descent,
but makes an angle less than 90 degrees with all descent
directions. Thus, the new iteration may not improve the dual
cost for all values of the stepsize. Using results on the
convergence of the subgradient method [26], [2], we show that,
for constant stepsize, the algorithm is guaranteed to converge
to within a neighborhood of the optimal value. For diminishing
stepsize, the algorithm is guaranteed to converge to the optimal
value. We would like a distributed implementation of the
subgradient algorithm, and thus a constant stepsize γ
t
= γ
is more convenient. Note that the dual cost usually will not
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monotonically approach the optimal value, but wander around
it under the subgradient algorithm. The usual criterion for sta-
bility and convergence is not applicable. We will use a similar
definition of convergence as in [3]. Let
p(t):=
1
t
t
τ =1
p(τ)
be the average price by time t.
Definition 1: Let p
∗
denote an optimal value of the dual
variable. Algorithm 1 with constant stepsize is said to converge
statistically to p
∗
, if for any δ>0 there exists a stepsize γ
such that lim sup
t→∞
D(p(t)) − D(p
∗
) ≤ δ.
Clearly, an optimal value p
∗
exists. The following theorem,
proved in the Appendix, guarantees the statistical convergence
of the subgradient method.
Theorem 2: Let p
∗
be an optimal price. If the norm of the
subgradients is uniformly bounded, i.e., there exists G such
that ||g(p)||
2
≤ G for all p, then Algorithm 1 converges
statistically to p
∗
.
The assumption of bounded norm for subgradient g(p) is
reasonable, since f is finite and we always have an upper
bound on x in practice. Note that D(p) ≥ D(p
∗
) always holds.
Since D(p) is a continuous function, Theorem 2 implies that
the congestion price p approaches p
∗
statistically when the
stepsize γ is small enough.
Let the primal function (the total achieved network utility)
be P (x) and achieve its optimum at x
∗
. Define x(t):=
1
t
t
τ =0
x(τ), the average data rate up to time t. As time goes
to infinity,
x(t) must be in the feasible rate region (determined
by eqs. (5)-(6)), otherwise
p(t) will go unbounded as time goes
to infinity, which contradicts Theorem 2.
Theorem 3: Let x
∗
be the optimal source rates. Under the
same assumption of Theorem 2, the following inequality holds
lim inf
t→∞
P (x(t)) ≥ P (x
∗
) −
γG
2
2
. (19)
Note that P (x) ≤ P (x
∗
) holds for any x in the feasible
rate region. Since P (x) is continuous, Theorem 3 implies that
the average source rate approaches the optimal x
∗
when γ is
small enough.
C. Scheduling over Ad Hoc Networks
We now come to the scheduling problem (16). Scheduling
over ad hoc network is a difficult problem and in general NP-
hard. To see this, note that problem (16) is equivalent to a
maximum weight independent set problem over the conflict
graph, which is NP-hard for general graphs. However, with
the primary interference model we show that problem (16)
can be reduced to the maximum weighted matching problem
6
,
which is polynomial time solvable. As one of the extensions in
Subsection VI-A, we will see a NP-hard scheduling problem.
The scheduling problem (16) is to maximize the weighted
sum of the link capacities with the schedulability constraint. It
is defined on a weighted digraph whose link weights w
i,j
can
take negative value. To see how it is related to the maximum
weighted matching problem, first note that w
j,i
> 0 if w
i,j
< 0
and vice versa. Second, note that links (i, j) and (j, i) mutually
6
A matching in a graph is a subset of links, no two of which share a
common node. The weight of a matching is the total weight of all its links.
A maximum weighted matching in a graph is a matching whose weight is
maximized over all matchings of the graph.
interfere and have the same interference/contention relations
with other links. Corresponding to each directed link pair
(i, j), (j, i) ∈ L, define an undirected link i, j with weight
w
i,j
= max{w
i,j
c
i,j
,w
j,i
c
j,i
}.
Let L
denote the set of undirected links and W
the corre-
sponding set of weights, the scheduling problem (16) is then
equivalent to the maximum weighted matching problem on the
weighted graph G
=(N,L
,W
). Note that an (maximal)
independent set in the conflict graph will correspond to a
(maximal) matching in this undirected graph.
Maximum weighted matching problem can be computed in
polynomial time (see, e.g., [23]), but this requires centralized
implementation. If implemented over an ad hoc network,
each node needs to notify the central node of its weight and
local connectivity information such that the central node can
reconstruct the network topology as a weighted graph. This
will lead to an immense communication overhead which is
expensive in time and resources. There also exist simpler
greedy sequential algorithms to compute a weighted matching
at most a factor of 2 away from the maximum (see. e.g.,
[24]). But they also require centralized implementation. We
seek a distributed algorithm where each node participates in
the computation itself using only local information.
A few distributed approximation algorithms exist for max-
imum weighted matching problem, see e.g. [30], [33], [9].
In [9], the author presents a simple distributed algorithm to
compute a weighted matching at most a factor of 2 away from
the maximum in linear running time O(|L
|). This algorithm
is a distributed variant of the sequential greedy algorithm
presented in [24]. We utilize this algorithm to solve our
scheduling problem (16) distributedly, as summarized below.
Algorithm 2: Distributed Scheduling Algorithm
Each node i carries out the following steps:
1) Calculate weight
w
i,j
=max{w
i,j
c
i,j
,w
j,i
c
j,i
} for each
directed link pair
(i, j), (j, i) ∈ L incident upon it. Ties are
broken randomly.
2) Find node
j
∗
such that w
i,j
∗
is maximized over all links
i, j∈L
with free neighbors j:
–If having received a
matching request from j
∗
, t hen link i, j
∗
is a matched link. Node i sends a matched reply to j
∗
and a drop
message to all other free neighbors.
–Otherwise, node
i sends a matching request to node j
∗
.
3) Upon receiving a matching request from neighbor j:
–If
j = j
∗
, then link i, j is a matched link. Node i sends a
matched reply to node j and a drop message to all other free
neighbors.
–If
j = j
∗
, node i just stores the received message.
4) Upon receiving a
matched reply from neighbor j, node i
knows link i, j is a matched link, and send a drop message to
all other free neighbors.
5) Upon receiving a
drop message from neighbor j, node i
knows that j is in a matched link, and excludes j from its free
neighbors set.
6) If node
i is in a matched link or has no free neighbors, no
further action is taken. Otherwise, it will repeat steps 2)–5).
7) Matched links are allowed to transmit. Nodes
i, j in a
matched link i, j will schedule the directed link, which gives
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the Proceedings IEEE Infocom.
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