A Tutorial on Cross-Layer Optimization
in Wireless Networks
Xiaojun Lin, Member, IEEE, Ness B. Shroff, Senior Member, IEEE and R. Srikant, Fellow, IEEE
Abstract— This tutorial paper overviews recent developments
in optimization based approaches for resource allocation prob-
lems in wireless systems. We begin by overviewing important
results in the area of opportunistic (channel-aware) scheduling
for cellular (single-hop) networks, where easily implementable
myopic policies are shown to optimize system performance. We
then describe key lessons learned and the main obstacles in
extending the work to general resource allocation problems for
multi-hop wireless networks. Towards this end, we show that a
clean-slate optimization based approach to the multi-hop resource
allocation problem naturally results in a “loosely coupled” cross-
layer solution. That is, the algorithms obtained map to different
layers (transport, network, and MAC/PHY) of the protocol
stack are coupled through a limited amount of information
being passed back and forth. It turns out that the optimal
scheduling component at the MAC layer is very complex and
thus needs simpler (potentially imperfect) distributed solutions.
We demonstrate how to use imperfect scheduling in the cross-
layer framework and describe recently developed distributed
algorithms along these lines. We conclude by describing a set
of open research problems.
I. INTRODUCTION
Optimization based approaches have been extensively used
over the past several years to study resource allocation prob-
lems in communication networks. For example, Internet con-
gestion control can be viewed as distributed primal or dual
solutions to a convex optimization problem that maximizes the
aggregate system performance (or utility). Such approaches
have resulted in a deep understanding of the ubiquitous
Transmission Control Protocol (TCP) and resulted in improved
solutions for congestion control [1]–[6].
The key question is whether such approaches can be applied
to emerging multi-hop wireless networks to enable a clean-
slate design of the protocol stack
1
. Indeed there are unique
challenges in the wireless context that do not allow a direct
application of such techniques from the Internet setting. In
particular, the wireless medium is an inherently multi-access
medium where the transmissions of users interfere with each
other and where the channel capacity is time-varying (due
to user mobility, multipath, and shadowing). This causes
Xiaojun Lin and Ness B. Shroff are with the Center for Wireless Sys-
tems and Applications (CWSA) and the School of Electrical and Computer
Engineering, Purdue University, West Lafayette, IN 47907, USA (email:
{linx,shroff}@ecn.purdue.edu).
R. Srikant is with the Department of Electrical and Computer Engineering
and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801-
2307, USA (e-mail: rsrikant@uiuc.edu).
1
The notion of a clean-slate design becomes especially attractive for multi-
hop wireless networks, where the burdens of legacy systems are far less than
for the Internet.
interdependencies across users and network layers that are
simply not present in their wireline counterparts. In spite of
these difficulties, there have been significant recent advances
that demonstrate that wireless resources across multiple layers
(such as time, frequency, power, link data rates and end-user
data rates), can be incorporated into a unified optimization
framework. Interestingly, as will be described in detail in
Section III, the solution of such an optimization framework
will itself exhibit a layered structure with only a limited degree
of cross-layer coupling.
We will illustrate the use of such an optimization approach
for two classes of cross-layer problems, namely, the oppor-
tunistic scheduling problem in cellular (or access-point based
single-hop networks), and the joint congestion-control and
scheduling problem in multi-hop wireless networks. We will
see that convex programming is an important tool for this
optimization approach; in particular, Lagrange duality is a
key tool in decomposing the otherwise complex optimization
problem into easily-solvable components. However, we will
also see that convex programming is often not enough. In
fact, unlike their wireline counterparts, the essential features
of many wireless cross-layer control problems are not convex.
For example, due to interference, wireless networks typically
require sophisticated “scheduling” mechanisms to carefully
select only a subset of links to be activated at each time.
In wireless networks, the capacity of each link depends on
the signal and interference levels, and thus depends on the
power and transmission schedule at other links. This rela-
tionship between the link capacity, power assignment, and the
transmission schedule is typically non-convex. Therefore, the
scheduling component needs to solve a difficult non-convex
problem, and usually becomes the bottleneck of the entire
solution.
These inherent non-convex features require that advanced
techniques in addition to convex programming be used to
satisfactorily solve the cross-layer control problem in wireless
networks. In this tutorial, we will see a few examples where
tools from convex programming, combinatorial optimization,
stochastic stability, graph theory, large deviations, and heavy-
traffic limits are used to obtain realistic and efficient solutions
to the cross-layer control problem.
We acknowledge that cross-layer optimization has become
a very active research area in the last few years. A compre-
hensive survey would be difficult due to the space constraints
in this special issue. Hence, this tutorial is by no means an
exhaustive survey of all subjects in cross-layer optimization.
Rather, our focus is to provide the readers with a sketch of
the main issues, challenges, and techniques in this area, and
also identify the main open problems to the community. For
another survey in this area see [7].
The rest of the tutorial is organized as follows. Section II
will begin with an exposition of the important problem of
scheduling in cellular networks. Here, the emphasis is on
incorporating physical layer channel information into the
scheduling decision. In Section III, we investigate the joint
congestion-control and scheduling problem in multihop wire-
less networks. The general formulation provided in Section III
elegantly decomposes the cross-layer problem into a conges-
tion control component and a scheduling component. However,
due to non-convexity, the perfect scheduling component is usu-
ally very complex and difficult to implement in real networks.
One approach to address this complexity is to use simpler
(and potentially distributed) imperfect scheduling components
in the cross-layer solution. However, the impact of these
imperfect scheduling policies on the overall solution must
be carefully studied. This is the subject of Section IV. In
Section V, we will describe recent developments in obtaining
imperfect distributed scheduling policies with provably achiev-
able throughput bounds. We then conclude with a set of open
problems.
II. OPPORTUNISTIC SCHEDULING FOR CELLULAR
WIRELESS NETWORKS
In this section, we focus on the opportunistic scheduling
problem in cellular networks (the results also apply to access-
point based single-hop wireless networks). Over the last few
years, this multi-user scheduling problem has received signif-
icant attention in both academia and industry. These schedul-
ing schemes have been motivated by the unique features in
wireless networks: scarce resources, mobile users, interference
from other users in the network, and time-varying channel
conditions (due to fading and mobility). Hence, good schedul-
ing schemes in wireless networks should opportunistically
seek to exploit channel conditions to achieve higher network
performance. For example, consider a cellular network that
consists of a base station and N users. Further, assume a time-
slotted system and downlink communications, i.e., from the
base-station to the users (receivers). Then the base-station can
determine which user(s) to transmit to, based on the channel
conditions. The idea is that transmissions to receivers with
favorable channel conditions (e.g., with higher SINR) allows
the base-station to transmit at a higher rate (using adaptive
modulation and coding schemes) for a given target bit-error-
rate. Thus, the base-station can opportunistically exploit the
channel conditions to achieve higher network performance. It
should be noted here that the idea of exploiting multi-user
diversity is in contrast to traditional methods (e.g., spread
spectrum, repetitive coding, power averaging, etc.), where the
goal is to smooth out channel variability rather than to exploit
it. Opportunistic scheduling achieves multi-user diversity gains
because when users experiencing good channels are selected,
it enables the system to potentially operate close to its peak
rather than average performance.
In [8], under an AWGN (additive white gaussian noise)
model, it has been shown that the sum capacity
2
of a wireless
system is maximized when only one user is selected to transmit
at any given time. This result can be shown for either uplink
or downlink assuming complete channel information at both
the receiver and the transmitter. The user with the best channel
condition is chosen for transmission. However, in a networking
context, the difficulty with such a solution is that while it
maximizes the overall throughput, it could result in significant
unfairness among the users. For example, under such a scheme
users that are close to the base-station may always be favored
over those that are further away, resulting in potentially poor
performance for certain users in the system. Such an approach
is especially troubling for high-data-rate wireless users that
may have stringent quality of service (QoS) requirements.
In order to address the above concerns, there have been sev-
eral approaches to ensure fairness/QoS in a wireless context.
For simplicity of presentation, we will focus on the downlink,
i.e., base-station to user communication. We will overview
opportunistic scheduling solutions that have been derived for
both infinite- and finite-backlogged cases.
A. Infinite-Backlog Case
The infinite-backlog case is often studied in communica-
tion systems to evaluate protocols and study their maximum
achievable performance. It is also simple and results in a
tractable solution that provides important insights. The ob-
jective in our context is to find a feasible scheduling policy
Q that maximizes the overall system performance for given
fairness/QoS requirements. A policy Q maps a vector
~
U =
[U
1
(x
1
), ..., U
N
(x
N
)] to Q(
~
U), which is the index of the user
selected for transmission. Here, x
i
is the data rate transmitted
to user i if it is selected for transmission, U
i
is the utility
function of user i, and U
i
(x
i
) measures the value or benefit to
user i of the receiving data rate x
i
. Note that x
i
is a function of
the channel condition and the coding and modulation scheme
used. There have been many scheduling schemes that address
this problem [9]–[13]. Interestingly, most of these approaches
result in an optimal solution that can be expressed in the form
of simple myopic index policies given by:
Q
∗
= argmax
i=1,...,N
(α
i
U
i
(x
i
) + β
i
), (1)
where α
i
and β
i
are constants and can be viewed as Lagrange
multipliers. For example, consider the following problem
studied in [12], [13]:
max
Q∈Θ
N
X
i=1
E (U
i
(x
i
))
subject to P {Q(
~
U) = i} ≥ r
i
, i = 1, 2, · · · , N, (2)
where Θ is the set of all stationary scheduling policies, and
r
i
is the minimum fraction of time-slots assigned to user i
2
The maximum total throughput that can be achieved in the system.
(i.e., fairness in time). Clearly, this is a non-linear optimization
problem. However, it can be readily shown that policy (1)
with α
i
= 0 and some appropriate choice of β
i
is optimal
(i.e., maximizes the expected system utility). The optimal
policy (1) would be executed at each time-slot and β
i
can
be obtained through a stochastic approximation algorithm, as
shown in [13]. Similarly, instead of temporal fairness, let us
now consider other forms of QoS:
• Fairness in utility: Each user receives at least a fraction
r
i
of the aggregate utility value [11], [13].
• Minimum data rate requirement: Each user receives a
minimum data rate of r
i
bits per second [13].
• Proportional Fairness: Here, the objective is to achieve
a solution that is proportionally fair, i.e., increasing the
mean throughput of one user from the optimal level by
x% results in a cumulative percentage decrease by greater
than x% of the mean throughput of other users [14].
It turns out that such a solution is achieved when the
optimization problem is to maximize the sum of the
logarithms of the expected rates (or the product of the
expected rates), i.e.,
max
Q∈Θ
N
X
i=1
(log(E(x
i
))) .
In each of these cases it can be shown that the optimal solution
will correspond to (1), now with β
i
= 0. Thus, these results
tell us that simple myopic index scheduling policies can be
used to opportunistically improve the system performance in
wireless networks.
B. Stability of Opportunistic Scheduling Schemes
The problem that we have described so far assumed an
infinitely backlogged system with the objective of maximizing
the aggregate system utility under QoS/fairness constraints.
Another important class of problems deals with the develop-
ment of opportunistic scheduling schemes with the intention
of accommodating the maximum possible offered load on the
system without violating stability or other QoS constraints.
Here, the problem moves from the maximization of utility
to stochastic stability. The work has largely been motivated
by the seminal work on throughput-optimal
3
scheduling [15].
This work shows that scheduling schemes that maximize the
queue-weighted sum of the rates are throughput optimal. For
the case of cellular networks, the scheduling scheme is of the
following form:
Q
∗
= argmax
i=1,...,N
q
i
x
i
, (3)
where q
i
is the queue length of user i, and x
i
is again the
data rate transmitted to user i. While this scheduling solution
does not account for fairness, it provides the important insight
that queue-length information is critical in developing through-
put optimal scheduling schemes. This idea has been further
developed into a general class of queue-length based (or,
3
A scheduling scheme is said to be throughput optimal if it stabilizes the
system whenever any other feasible scheduler can stabilize the system.
equivalently, delay-based) opportunistic scheduling schemes
that focus on stability and throughput optimality [16]–[20].
For example, in [16], [17], simple index scheduling policies
of the following form are shown to be throughput optimal:
Q
∗
= argmax
k=1,...,N
α
i
d
i
x
i
, (4)
where α
i
is a constant, d
i
is the head-of-the-line packet delay
at queue i, and as before x
i
is the data rate of user i. In
[17], [18], a related delay-based index policy that provides
exponential weight to the delay (the so-called exponential rule)
is shown to be throughput optimal.
Throughput optimal scheduling schemes have also been
derived in [21], where the authors also incorporate flow-level
dynamics into their model. In particular, the authors model
users arriving to the system with a random amount of workload
(e.g., a file size) and depart when this workload has been
transmitted. Recently, in [22], [23] the authors have attempted
to characterize the impact of different forms of scheduling on
stability and QoS using techniques from large-deviation and
heavy traffic limits. The key results from these works empha-
size the importance of queue-length based (QLB) scheduling
(e.g., in the form of (3) and (4)) for finite-backlogged systems
when there are delay constraints. Under certain conditions,
it can be shown that, when there are delay constraints, the
network throughput of QLB policies is larger than policies
for which queue-length information is not taken into account,
e.g., (1). Moreover, for a given delay violation constraint,
when the number of users in the system increases, the total
network throughput under policy (1) initially increases and
then eventually decreases to zero, but not so under the QLB
policy. This should not be entirely surprising since index
policies of the form (1) are agnostic to the delays incurred
for different users and may not serve users whose queues are
building up fast enough to remain within a delay violation
probability.
C. Limitations and Lessons Learned
Thus far, we have focused on opportunistic scheduling
solutions for cellular systems. For such systems, one can often
find simple myopic index policies that are optimal and easy to
implement. However, scheduling cannot address the problem
of ensuring that the system is operating in a stable or feasible
regime. Hence, while opportunistic scheduling expands the
capacity region over its non-opportunistic counterparts, it may
be difficult to utilize this gain if we are unable to operate
the system close to the boundary of the capacity region.
For example, if one were to make a conservative estimate
of the boundary of the capacity region, and traffic were
injected into the system based on this conservative estimate,
the opportunistic gains may never be realized. Thus, it is
imperative that one solves the problem of determining the
rates to be injected into the network (i.e., congestion control
to be discussed in Section III) jointly with which user(s) to be
scheduled for transmission (i.e., opportunistic scheduling).
Further, when we move from cellular to multi-hop wire-
less systems, we encounter other difficulties that need to be
addressed. For example, how should one determine the end-
to-end data rates for the users? When should a given link
be activated in the network? What should be the forwarding
rate of each link along with its power allocation, coding
and modulation schemes? How can one ensure that the rate
provided by the links is enough to support the end-to-end
rate of all users? The potential state space in a moderately
sized multi-hop network could be quite large, so can one
develop low-complexity solutions to these problems? Perhaps,
most importantly, how does one go about developing efficient
distributed solutions to these problems?
In the next few sections, we will describe some of the recent
research developments that have taken place in addressing the
above problems.
III. CROSS-LAYER CONGESTION CONTROL AND
SCHEDULING FOR MULTI-HOP WIRELESS NETWORKS
In this section, we study the following problem in multi-hop
wireless networks: how does one jointly choose the end-to-
end data rate of each user and choose the schedule for each
link? (Here, as in Section II, we will use the term schedule to
refer to the joint allocation of resources at MAC/PHY layers,
which include modulation, coding, power assignment and link
schedules, etc.) As we will see, the solution to this problem is
obtained by choosing an appropriate congestion control algo-
rithm to regulate the user data rates and a scheduling policy
which is a modification of the queue-length based scheduling
algorithm in the previous section. While congestion control
has been studied extensively for wireline networks [1]–[6],
these results cannot be applied directly to multihop wireless
networks because the the link capacity in multihop wireless
networks varies and depends on the scheduling policies at
the underlying layers. There have been attempts to solve this
cross-layer control problem using a “layered” approach [24]–
[27]. The approach is to find a feasible rate region that has a
simpler set of constraints similar to that of wireline networks
and then develop congestion controllers that compute the rate
allocation within this simpler rate region. Unfortunately, for
general network settings, it is not always possible to find such
a simpler rate region. Further, because the rate region reduces
the set of feasible rates that congestion control can utilize, the
layered approach results in a conservative rate allocation.
The general cross-layer solutions for jointly optimizing con-
gestion control and scheduling have recently been developed
by a number of researchers [20], [28]–[34]. In this section, we
will review two types of formulations and solutions that can
potentially be used for online implementation.
A. The Model
We consider a multi-hop wireless network with N nodes.
Let L denote the set of links (i.e., node pairs) (i, j) such
that the transmission from node i to node j is allowed. Due
to the shared nature of the wireless media, the data rate r
ij
of a link (i, j) depends not only on the power P
ij
assigned
to the link, but also on the interference due to the power
assignments on other links. (In this paper we often refer to
P
ij
as the power assignment, however, the same formulation
would clearly apply if P
ij
represents other types of resource
control decisions at link (i, j), e.g., activation/inactivation, or a
random-access attempt-probability.) Let
~
P = [P
ij
, (i, j) ∈ L]
denote the power assignments and let ~r = [r
ij
, (i, j) ∈ L]
denote the data rates. We assume that ~r = u(
~
P ), i.e., the data
rates are completely determined by the global power assign-
ment. (One can also extend the model to incorporate channel
variations, see Section III-D.) The function u(·) is called the
rate-power function of the system. There may be constraints on
the feasible power assignment. For example, if each node has
a total power constraint P
i,max
, then
P
j:(i,j)∈L
P
ij
≤ P
i,max
.
Let Π denote the set of feasible power assignments, and let
R = {u(
~
P ),
~
P ∈ Π}. We assume that Co(R), the convex hull
of R, is closed and bounded.
There are S users and each user s is associated with a
source node f
s
and a destination node d
s
. Let x
s
be the rate
with which data is sent from f
s
to d
s
, over possibly multiple
paths and multiple hops. We assume that x
s
is bounded in
[0, M
s
]. Each user is associated with a utility function U
s
(x
s
),
which reflects the “utility” to the user s when it can transmit
at data rate x
s
. We assume that U
s
(·) is strictly concave, non-
decreasing and continuously differentiable on [0, M
s
]. The
use of such utility functions is common in the congestion
control literature to model fairness, since with different utility
functions the rate allocations x
s
that maximize the total system
utility can be mapped to a range of fairness objectives [35],
[36].
We assume that time is divided into slots. At each time
slot, the scheduling policy will select a power assignment
vector
~
P (or, equivalently, ~r = u(
~
P )), and select data to be
forwarded on each link. Given a user rate vector ~x = [x
s
, s =
1, ..., S], we say that a system is stable under a scheduling
policy if the queue length at each node remains finite. We
can then formulate the following joint congestion-control and
scheduling problem:
• The Congestion-Control Problem: Find the user rate
vector ~x that maximizes the sum of the utilities of all
users
P
s
U
s
(x
s
) subject to the constraint that the system
is stable under some scheduling policy.
• The Scheduling Problem: For any user rate vector ~x
picked by the congestion-control problem, find a schedul-
ing policy that stabilizes the system.
Define the capacity region Λ of the system as the largest
set of rate vectors ~x such that for any ~x ∈ Λ, there exists
some scheduling policy that can stabilize the network under
the offered-load ~x. Hence, the congestion control part of the
problem is simply
max
x
s
≤M
s
X
s
U
s
(x
s
) (5)
subject to ~x ∈ Λ.
In the sequel, we will review two different ways of stating
the capacity region Λ, which then lead to different solutions.
B. The Node-Centric Formualtion
In the node-centric formulation, a user rate vector ~x is in
the capacity region Λ if and only if there exists a link rate
vector ~r
d
associated with each destination node d, and the
vector
~
R = [~r
d
, d = 1, ..., N] satisfies [15], [37]:
r
d
ij
≥ 0 for all (i, j) ∈ L and for all d
X
j:(i,j)∈L
r
d
ij
−
X
j:(j,i)∈L
r
d
ji
−
X
s:f
s
=i,d
s
=d
x
s
≥ 0
for all d and for all i 6= d (6)
[
X
d
r
d
ij
] ∈ Co(R),
where r
d
ij
can be interpreted as the rate on link (i, j) that is
allocated for data towards destination d. These set of equations
simply represent a balance of incoming rates and outgoing
rates at each node. The convex-hull operator is due to a
standard time-averaging argument [15], [37]–[39].
The Solution: Although the rate-power function u(·) is
generally a non-convex function, the convex-hull operator in
fact makes the capacity region Λ a convex set. Hence, the
problem (5) has a dual such that there is no duality gap [28].
Associating a Lagrange multipler q
d
i
for each constraint in (6),
we can then obtain the following solution [28], [32], [33]:
• The data rates of the users are determined by
x
s
(t) = argmax
0≤x
s
≤M
s
h
U
s
(x
s
) − x
s
q
d
s
f
s
i
. (7)
• The schedule is determined by first solving the following
sub-problem:
~r(t) = argmax
~r∈u(
~
P ),
~
P ∈Π
X
(i,j)∈L
r
ij
max
d
(q
d
i
− q
d
j
). (8)
Each link then picks the corresponding power assignment
that achieves ~r(t), and computes the vectors ~r
d
(t) as
follows: For each link (i, j), let d
∗
(i, j) = argmax
d
(q
d
i
−
q
d
j
), and let r
d
ij
(t) = r
ij
(t) if d = d
∗
(i, j) and r
d
ij
(t) = 0,
otherwise.
• The Lagrange multipliers are updated by
q
d
i
(t + 1) =
q
d
i
(t) − h
t
X
j:(i,j)∈L
r
d
ij
(t)
−
X
j:(j,i)∈L
r
d
ji
(t) −
X
s:f
s
=i,d
s
=d
x
s
(t)
+
, (9)
where h
t
, t = 1, 2, ... is a sequence of positive stepsizes.
The physical interpretation of this set of equations is as
follows. The Lagrange multiplier q
d
i
can be viewed as a scalar
multiple of the queue length at node i for packets destined
to node d. Equation (7) corresponds to the congestion control
component for determining the data rate of each user. Equation
(8) corresponds to the scheduling component. The network
first computes the power assignment
~
P (t) that corresponds to
~r(t). Then, each link will route data destined to the destination
d
∗
that corresponds to the largest differential backlog q
d
i
− q
d
j
.
Note that given ~q, the congestion control decision and the
scheduling decision are made independently. Finally, Equation
(9) corresponds to the evolution of the queue length at each
node.
One can then show the following convergence result [28].
Proposition 1: If
h
t
→ 0 as t → ∞, and
P
t
h
t
= ∞,
then ~x(t) → ~x
∗
as t → ∞, where ~x
∗
is the unique optimal
solution to problem (5).
Alternatively, if h
t
does not approach zero, then as long as
it is small, one can still show that ~x(t) will converge to a small
neighborhood around ~x
∗
. Further, all queues will remain finite,
and hence the schedules ~r
d
(t) also stabilize the network.
C. Link-Centric Formulation
The link-centric formulation differs from the node-centric
formulation in that the capacity constraints are stated as
balance equations for each link. For simplicity, we focus on
the case where the routes for each user are pre-determined.
Let [H
ij
s
] denote the routing matrix, where H
ij
s
= 1 if traffic
of user s passes through link (i, j), H
ij
s
= 0 otherwise. Then
an end-to-end user rate vector ~x belongs to the capacity region
Λ if and only if there exists a link rate-vector ~r such that
S
X
s=1
H
ij
s
x
s
≤ r
ij
for all (i, j) ∈ L, and (10)
~r ∈ Co(R).
The Solution: The capacity region Λ is a convex set,
as in the case of the node-centric formulation. Associating
a Lagrange multiplier q
ij
for each constraint in (10), we can
obtain the following solution [28], [29]:
• The data rates of the users are determined by
x
s
(t) = argmax
0≤x
s
≤M
s
U
s
(x
s
) − x
s
X
(i,j)∈L
H
ij
s
q
ij
. (11)
• The schedule is determined by solving the following
subproblem:
~r(t) = argmax
~r∈u(
~
P ),
~
P ∈Π
X
(i,j)∈L
r
ij
q
ij
. (12)
Each link (i, j) then picks the corresponding power
assignment that achieves ~r(t).
• The Lagrange multipliers are updated by
q
ij
(t + 1) =
"
q
ij
(t) + h
t
S
X
s=1
H
ij
s
x
s
− r
ij
(t)
!#
+
.
(13)
where h
t
, t = 1, 2, ... is a sequence of positive stepsizes.
The above solution has a similar physical interpretation to
the node-centric solution. Further, a convergence result similar
to Proposition 1 can be shown [29]. In the case of cellular
networks, the node and link-centric formulations are identical,
a case which has been considered in [30].