226 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009
Joint Scheduling and Resource Allocation in
Uplink OFDM Systems for Broadband Wireless
Access Networks
Jianwei Huang, Vijay G. Subramanian, Rajeev Agrawal, and Randall Berry
Abstract—Orthogonal Frequency Division Multiplexing
(OFDM) with dynamic scheduling and resource allocation is
a key component of most emerging broadband wireless access
networks such as WiMAX and LTE (Long Term Evolution)
for 3GPP. However, scheduling and resource allocation in an
OFDM system is complicated, especially in the uplink due to
two reasons: (i) the discrete nature of subchannel assignments,
and (ii) the heterogeneity of the users’ subchannel conditions,
individual resource constraints and application requirements.
We approach this problem using a gradient-based scheduling
framework. Physical layer r esour ces (bandwidth and power) are
allocated to maximize the projection onto the gradient of a total
system utility function which models application-layer Quality
of Service (QoS). This is formulated as a convex optimization
problem and solved using a dual decomposition approach. This
optimal solution has prohibitively high computational complexity
but reveals guiding principles that we use to generate lower
complexity sub-optimal algorithms. We analyze the complexity
and compare the performance of these algorithms via extensive
simulations.
Index Terms—Orthogonal Frequency Division Multiplexing
(OFDM), scheduling, resource allocation, optimization, dual de-
composition, uplink communications
I. INTRODUCTION
O
RTHOGONAL Frequency Division Multiplexing
(OFDM) is the core technology for most recent
wireless data systems, including IEEE 802.16 (WiMAX),
IEEE 802.11a/g (Wireless LANs), and LTE for 3GPP. In
this paper, we analyze an uplink scheduling and resource
Manuscript received 15 January 2008; revised 15 August 2008. Part of this
work was done while J. Huang and V. G. Subramanian were at Motorola.
J. Huang is supported by the Competitive Earmarked Research Grants (Project
Number 412308) established under the University Grant Committee of the
Hong Kong Special Administrative Region, China, the Direct Grant (Project
Number C001-2050398) of The Chinese University of Hong Kong, and the
National Key Technology R&D Program (Project Number 2007BAH17B04)
established by the Ministry of Science and Technology of the People’s
Republic of China.
V. Subramanian is supported by SFI grant 03/IN3/I346.
R. Berry was supported in part by the Motorola-Northwestern Center for
Seamless Communications and NSF CAREER award CCR-0238382. The
work was partially presented at the 2007 Asilomar Conference on Signals,
Systems and Computers.
J. Huang is with the Department of Information Engineering, The Chinese
University of Hong Kong (e-mail: jwhuang@ie.cuhk.edu.hk).
V. G. Subramanian is with the Hamilton Institute, National University of
Ireland (e-mail: vijay.subramanian@nuim.ie).
R. Agrawal is with the Advanced Networks and Performance Dept.,
Motorola Inc., (e-mail: Rajeev.Agrawal@motorola.com).
R. Berry is with the Dept. of EECS, Northwestern University (e-mail:
rberry@ece.northwestern.edu).
Digital Object Identifier 10.1109/JSAC.2009.090213.
allocation problem for OFDM wireless access networks.
The specific problem is motivated by the WiMAX/802.16e
standard, where there is a centralized scheduler that knows
the QoS classes, and can estimate the queue-lengths on
each mobile device. The WiMAX/802.16e standard specifies
mechanisms for communicating this information to the
scheduler and for conveying the scheduling decisions to the
mobiles, both with low delays.
1
Our approach is motivated by our previous work on down-
link scheduling in CDMA systems [3] and OFDM systems [4].
As in [3], [4], we consider a gradient-based scheduling
framework, which is described in detail in Section II along
with our system model. In this framework, the time-varying
gradient of a utility function is used to guide the resource
allocation decisions and provide long-term Quality of Service
(QoS) guarantees. In particular, we maximize a weighted sum-
rate during each scheduling interval, where the weights are
time-varying. The optimization variables are the assignment
of OFDM subchannels to the users and the allocation of each
user’s power across the assigned subchannels. We highlight
two challenging aspects of this problem in the OFDM uplink
context. First, the discrete nature of subchannel assignments
in OFDM systems usually leads to difficult integer program-
ming problems. Second, the per-user power constraints in the
uplink make the problem even less tractable. We initially
relax the integer constraints and allow multiple users to
share one subchannel using orthogonalization (e.g. via time-
sharing
2
). In Section III we derive an optimal solution to this
relaxed problem via a dual decomposition. Due to the per-
user power constraints, the resulting algorithm has very high
computational complexity. However, this provides insights into
the structure of an optimal solution. In Section V we use
the insights gained from the optimal solution to propose a
family of sub-optimal algorithms that also take into account
the integer constraints on subchannel allocations. Finally, in
Section VI we these algorithms using simulation.
Most initial work on OFDM scheduling and resource alloca-
tion focused on the downlink case. The optimality conditions
and algorithms derived for the downlink, however, can not
be directly applied to the uplink due to differences in the
resource constraints (see Section IV for a detailed discussion).
1
Our model is also appropriate for LTE [29] for 3GPP, UMB [30] for
3GPP2 and the FLASH OFDM system [19] from Qualcomm Flarion.
2
While super-position coding would yield an even larger (and more
tractable) capacity region, we do not consider it as it is still not practical.
0733-8716/08/$25.00
c
2008 IEEE
HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 227
Recently, uplink OFDM resource allocation has received some
attention, including [21]–[28]. In [21], an iterative OFDM
resource allocation was proposed to find a Nash Bargaining
solution. A heuristic algorithm that tries to minimize each
user’s transmission power while satisfying the individual rate
constraints is given in [22]. In [23], an algorithm for maxi-
mizing the sum-rate assuming Rayleigh fading is given; this
is a special case of the problem considered here with equal
weights. In our case no assumption on the fading distribution
is made. In [24], an uplink problem with multiple antennas
at the base station is considered; here, we focus on single
antenna systems.
The work in [25]–[28] is closer to ours. In [25], a weighted
rate maximization problem is also considered for the uplink,
but with static weights. The results in [25] are generalized
in [26] to account for per-time slot fairness via a utility
function of the instantaneous rate. Per time-slot fairness is
also considered in [27]. Our work differs from these in that
by using a gradient-based scheduler, we can consider long-
term fairness, which depends on the average rate or queue
sizes. For elastic data applications, long-term QoS evaluation
is more reasonable than the short-term QoS evaluation during
each time slot. It not only more faithfully reflects users’ actual
perceived performance, but also gives the system more flexi-
bility in terms of exploiting multi-user diversity. Finally, [28]
proposed a heuristic algorithm based on Lagrangian relaxation,
which has high complexity due to a subgradient search of the
dual variables. Here we use Lagrangian relaxation to give an
optimal solution of the uplink problem when time-sharing is
allowed. Solving this problem provides both an upperbound
on the actual system performance as well as the intuition we
use to design good heuristic algorithms.
II. P
ROBLEM STATEMENT
We consider the problem of scheduling and resource al-
location for the uplink of a OFDM cell, where a set M =
{1,...,M} of users transmit to the same base station. The
total frequency band is divided into a set N = {1,...,N}
of subchannels (e.g., frequency bands). Let p
ij
be the power
user i allocates to subchannel j, which is subject to a per-user
power constraint:
j
p
ij
≤ P
i
, ∀i ∈M. (1)
Let x
ij
be the fraction of subchannel j allocated to user i,
where the total allocation across all users should be no larger
than 1, i.e.,
i
x
ij
≤ 1, ∀j ∈N. (2)
We use bold symbols to denote vectors and matrices of
these quantities, e.g., P = {P
i
, ∀i}, e = {e
ij
, ∀i, j}, p =
{p
ij
, ∀i, j},andx = {x
ij
, ∀i, j}.
Time is divided into equal length slots. At the beginning
of every time slot, the scheduler seeks to maximize a (time-
varying) weighted sum of the users’ rates over a (time-varying)
rate-region. We describe this rate-region next.
The scheduler is assumed to have knowledge of the nor-
malized received signal-to-interference plus noise ratio (SINR)
per unit transmit power, e
ij
, for each user i and subchannel
j.
3
The time-varying subchannel quality vector at time t is
denoted by e
t
. As in [4], this model incorporates various
subchannelization schemes where the resource allocation is
performed in terms of subchannels (i.e., sets of tones); e
ij
represents a collective quality indicator for the subchannel,
e.g., the (harmonic/geometric/arithmetic) average across the
tones in the subchannel. This model also applies if resource
allocation is done with a granularity of multiple symbols in
the time domain.
We model the feasible rate region at time t by
R(e
t
)=
r : r
i
=
j∈N
f(x
ij
,p
ij
e
ij
(t))
, (3)
where (x, p) ∈X are chosen subject to (1) and (2) and the
set
X :=
(x, p) ≥ 0 :0≤ x
ij
≤ 1,p
ij
≤
x
ij
s
ij
e
ij
(t)
∀i, j
. (4)
Here, f(a, b)=a log(1 +
b
a
) so that r
i
is the achievable rate
of user i in a Gaussian Multiple Access Channel using time-
sharing (cf. [17, Section 15.3.6, pg. 547]). By continuity, we
assume that f(0,b)=0.Thevalueofs
ij
is a maximum
SINR constraint on subchannel j for user i, which can model
scenarios when users have limited choices of modulation and
coding schemes.
In practical OFDM systems, x
ij
is constrained to be an
integer, i.e., we have the additional constraint x
ij
∈{0, 1}
for all i, j. Initially, we ignore this constraint, and consider
a system in which users can share each tone. If resource
allocation is done on blocks of OFDM symbols, then fractional
values of x
ij
can be implemented by time-sharing the symbols
in a block. Alternatively, this can also be implemented by
frequency sharing (e.g., [31]), if there are a large number of
subchannels with roughly equal gains. We will re-introduce
these integer constraints in Sections IV and V.
Next we formulate the scheduling and resource allocation
problem. Our approach is based on the gradient-based schedul-
ing framework in [2], [10], [12]. Each user i is assigned
a utility function U
i
(W
i,t
,Q
i,t
) depending on their average
throughput W
i,t
up to time t and their queue-length Q
i,t
at
time t. This is used to quantify fairness and ensure stability of
the queues. At the beginning of each time slot t, the scheduler
chooses a r
t
∈R(e
t
) that maximizes a weighted sum of the
users’ rates, where the weights are determined by the gradient
of the sum utility across all users, i.e., it solves
max
r
t
∈R(e
t
)
∇
w
U(W
t
, Q
t
) −∇
q
U(W
t
, Q
t
)
T
r
t
, (5)
where U(W
t
, Q
t
)=
K
i=1
U
i
(W
i,t
,Q
i,t
). Further assuming
that for each user i, U
i
(W
i,t
,Q
i,t
)=u
i
(W
i,t
) −
d
i
p
(Q
i,t
)
p
,
then (5) is equivalent to
max
r
t
∈R(e
t
)
i
∂u
i
(W
i,t
)
∂W
i,t
+ d
i
(Q
i,t
)
p−1
r
i,t
, (6)
3
In both FDD and TDD systems this can be obtained using a combination
of measurements made on the uplink pilots as well as past transmissions from
the mobiles.
228 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009
where, u
i
(W
i,t
) is a increasing concave function used to
represent elastic data applications (e.g., [1], [8], [13], [18]),
d
i
≥ 0 is a QoS weight for user i’s queue length, and p>1
is a fairness parameter associated with the queue length.
The broad class of policies in (6) can be tuned to yield
good operating points by a proper choice of parameters. If
d
i
=0for all i ∈M, the resulting policy has been shown
to yield utility maximizing solutions (see [2], [10], [12]). If
u
i
(·) ≡ 0 with d
i
> 0 for all i ∈M, then the policy has
been shown to be stabilizing in a variety of settings (see [5]–
[7]). The weights can also be adapted so as to maximize sum
utility subject to stability [9] or (feasible) minimum throughput
constraints (see [11]).
More generally, the optimization in (6) can be written as
max
r
t
∈R(e
t
)
i
w
i,t
r
i,t
, (7)
where w
i,t
≥ 0 is a time-varying weight assigned to the ith
user at time t. Our focus is on solving such a problem for an
uplink OFDM system, i.e., when R(e
t
) is given by (3). Note
that (7) must be re-solved at each scheduling instant because
of changes in both the subchannel state, e
t
, and the weights.
While in the above examples, the weight w
i,t
is given by the
gradient of an utility function, our algorithms also apply to
other methods for generating these weights.
III. O
PTIMAL SOLUTION WITH FRACTIONAL
ALLOCATIONS
We now consider the optimal solution to (7) when R(e
t
)
is given by (3). Suppressing the time index, the problem is
max
(x,p)∈X
i∈M
w
i
j∈N
x
ij
log
1+
p
ij
e
ij
x
ij
(UL)
subject to the per subchannel assignment constraints in (2) and
the per user power constraints in (1), where X is given in (4).
It can be shown that Problem UL has no duality gap and so
we solve it via a dual formulation. We associate dual variables
λ =(λ
i
)
i∈M
with constraints (1) and μ =(μ
j
)
j∈N
with
constraints (2), resulting in the Lagrangian,
L(λ, μ, x, p):=
i,j
w
i
x
ij
log
1+
p
ij
e
ij
x
ij
+
i
λ
i
P
i
−
j
p
ij
+
j
μ
j
1 −
i
x
ij
. (8)
Therefore, the optimal solution to Problem UL is given by
min
(λ,μ)≥0
max
(x,p)∈X
L(λ, μ, x, p). (9)
We solve this problem by the following steps. First, we
analytically find the optimal p and x given fixed values of
the dual variables. We then show that the optimal μ is given
by a search for the maximum value of a per-user metric on
each subchannel. The final step is to numerically search for
the optimal value of λ.
The value of p which maximizes L(λ, μ, x, p) given x, μ
and λ is given bu
p
∗
ij
=
x
ij
e
ij
min
w
i
e
ij
λ
i
− 1
+
,s
ij
, (10)
where (·)
+
=max(·, 0). Substituting p
∗
into L(·, ·, ·, ·) yields
L(λ, μ, x)=
ij
x
ij
(w
i
h (λ
i
,w
i
e
ij
,s
ij
) − μ
j
)
+
j
μ
j
+
i
λ
i
P
i
, (11)
where we have used the function h(·, ·, ·) from [3]; namely,
h(a, b, c)=
⎧
⎪
⎨
⎪
⎩
0 if a ≥ b;
a
b
− 1 − log
a
b
if
b
1+c
≤ a<b;
log(1 + c) −
a
b
c if a<
b
1+c
,
(12)
where a ≥ 0, b>0 and c ≥ 0. Optimizing (11) over x such
that x
ij
∈ [0, 1] yields
L(λ, μ)=
j
μ
j
+
i
λ
i
P
i
+
ij
(w
i
h (λ
i
,w
i
e
ij
,s
ij
) − μ
j
)
+
, (13)
where the optimal subchannel allocation has the following
structure
x
∗
ij
(μ
j
)=
⎧
⎪
⎨
⎪
⎩
1, if w
i
h (λ
i
,w
i
e
ij
,s
ij
) >μ
j
;
[0, 1], if w
i
h (λ
i
,w
i
e
ij
,s
ij
)=μ
j
;
0, if w
i
h (λ
i
,w
i
e
ij
,s
ij
) <μ
j
.
(14)
Since the cost function in (13) is separable, by defining
μ
ij
(·):=w
i
h (·,w
i
e
ij
,s
ij
) as in [3], we can minimize
L(λ, μ) over μ for a given λ by setting μ
j
= μ
∗
j
(λ) given
by
μ
∗
j
(λ)=max
i
μ
ij
(λ
i
) . (15)
From (14) and (15), it is clear that x
∗
ij
(μ
∗
j
(λ)) ≡ 0 if
i ∈ arg max
i∈M
μ
ij
(λ
i
), i.e., users not maximizing a specific
subchannel metric are not allocated the subchannel. There will
be ties when multiple users achieve the value μ
∗
j
on subchannel
j. These can be broken arbitrarily for optimizing the dual
function. Substituting μ
∗
into L(λ, μ), and noticing that μ
∗
,
x
∗
, p
∗
are all functions of λ,wehave
L(λ):=
j
max
i
μ
ij
(λ
i
)+
i
λ
i
P
i
.
The solution to (9) is given by numerically minimizing
L(λ) over λ ≥ 0 . For this we use a subgradient-based search
and update λ by
λ
i
(t +1)=
λ
i
(t) − κ(t)
P
i
−
j
p
∗
ij
(t)
+
, ∀i ∈M.
where p
∗
ij
is given by (10) and x
ij
are given by (14) and addi-
tionally satisfy the feasibility constraint (2) in case of ties. The
algorithm will converge when κ(t) is chosen appropriately,
e.g., [20, Exercise 6.3.2]. Given an optimal λ
∗
, by duality,
L(λ
∗
) is the optimal objective value to Problem UL.
However, to implement this algorithm, the scheduler must
specify the corresponding optimal primal values of (x
∗
, p
∗
).
Here, as in [4], more care is required. Specifically, when ties
occur in (15), it is often needed to split the subchannel among
several users (i.e., allowing fractional values of x
∗
). Following
a similar approach as in [3] (for the downlink), the optimal
HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 229
fractional values can be found by solving a linear program
whose size increases with the number of users and tones
involved in each tie. As discussed below, this number can be
quite large in the uplink setting. Moreover, as noted earlier,
one is typically interested in an integer allocation in practice.
We consider this problem next.
IV. I
NTEGER SUBCHANNEL ALLOCATION BASED ON
OPTIMAL ALGORITHM
We now address the problem:
max
(x,p)∈X ,
x
ij
∈{0,1},∀i,j
i∈M
w
i
j∈N
x
ij
log
1+
p
ij
e
ij
x
ij
, (UL-Int)
subject to per user power constraints (1). Initially, consider the
following heuristic for Problem UL-Int: (i) Solve Problem UL
as in the previous section; and (ii) “break” any ties on all
subchannels, i.e., whenever there is a fractional x
∗
ij
value,
choose one user in the tie and allocate subchannel j to that
user only. Clearly, if there are no ties, this algorithm gives the
optimal solution to Problem UL-Int. After all ties are broken,
we can then re-optimize the power allocation for each user
using a finite-time water-filling algorithm as in [4].
In [4] a similar procedure is used for a downlink OFDM
scheduling problem. However, there are several major differ-
ences between the uplink and downlink settings that make this
approach less appealing for the uplink. First, in the downlink
case there is a single power constraint
i,j
p
ij
≤ P for the
base station instead of the per-user constraints in (1). Hence, in
the downlink L(λ) is a function of only a single dual variable
λ, which simplifies the numerical search for the optimizer. In
the uplink setting, the convergence of the subgradient search
is too slow to be useful for scheduling on a fast time-scale.
Second, even if the optimal λ can be found, breaking ties is
more difficult than in the downlink case. Scalar subgradients
of L(λ) in the downlink case can be used to devise simple
tie-breaking rules [4], while in the uplink case, the subgra-
dients are vectors, complicating such an approach. Also, the
uplink case can be more sensitive to how ties are resolved.
Forexample,iftwousers,i and l, have the same weights
(w
i
= w
l
) and the same gains on subchannel j (e
ij
= e
lj
),
then allocating subchannel j to either yields the same total
weighted rate and the same total power usage in the downlink
case. On the other hand, different allocations lead to different
individual power consumptions in the uplink case, and thus
may lead to totally different solutions.
Finally, the number of ties is typically much larger in the
uplink case than in the downlink case. Consider a simple
scenario with two users and two subchannels. Each user has
the same gain over both subchannels, i.e., e
i1
= e
i2
= e
i
for
i =1, 2, and P = P
1
= P
2
, where P is the total power
constraint in the downlink case. Assume user 2 has a much
better subchannel than user 1 so that in the downlink case,
the unique optimal solution is to allocate both subchannels to
user 2, and there is no tie. However, in the uplink case, it can
be shown that at the optimal dual solution, λ
1
and λ
2
will
satisfy μ
1j
(λ
1
)=μ
2j
(λ
2
) for j =1and 2, i.e., there is a tie
in each subchannel and four possible subchannel allocations
must be considered to determine how to break the tie. This can
be easily extended to M users and N subchannels, with each
user having the same gain over all its subchannels, resulting
in M
N
ties even in this simplistic setting.
V. L
OW COMPLEXITY SUBOPTIMAL ALGORITHMS
Due to the issues discussed in the previous section, finding
the optimal dual solution to Problem UL and breaking any
ties to get an integer allocation is computationally difficult,
even for a moderately sized system. Thus, we now present
a family of sub-optimal algorithms (SOAs), which try to
reduce this complexity using the structure revealed by the
optimal algorithm while enforcing an integer-tone allocation
and exhibiting good performance.
In the optimal algorithm, given the optimal λ
∗
, the optimal
subchannel allocation up to any ties is determined by sorting
the users on each tone according to the metric μ
ij
(λ
∗
) as
in (14). Given an optimal subchannel allocation, the optimal
power allocation is given by a per-user water-filling allocation
as in (10). In each SOA, we use the same two phases but
modify them to reduce the complexity. Specifically, we begin
with a subChannel Allocation (CA) phase which assigns each
subchannel to at most one user. Two different implementations
of the CA phase are given. In SOA1, instead of using μ
ij
(λ
∗
),
we consider metrics based on a constant power allocation over
all subchannels assigned to a user. In SOA2, we again use a
dual based approach, but here we first determine the number
of subchannels assigned to each user and then match specific
subchannels and users. After the CA phase in both SOAs, we
execute the Power Allo cation (PA) phase in which each user’s
power is allocated across the assigned subchannels as in the
optimal algorithm.
A. CA in SOA1: Progressive Subchannel Allocation based on
Metric Sorting
In this family of SOAs, subchannels are assigned sequen-
tially in one pass based on a per user metric for each
subchannel. Let K
i
(n) denote the set of subchannels assigned
to user i after the nth iteration. Let g
i
(n) denote user i’s metric
during the nth iteration and let l
i
(n) be the subchannel index
that user i would like to be assigned if he/she is assigned
the nth subchannel. The resulting CA algorithm is given in
Algorithm 1. Note that all the user metrics are updated after
each subchannel is assigned.
Algorithm 1 CA Phase for SOA1
1: Initialization: set n =0and K
i
(n)=∅ for each user i.
2: while n<N do
3: n = n +1.
4: Update subchannel index l
i
(n) for each user i.
5: Update metric g
i
(n) for each user i.
6: Find i
∗
(n) = arg max
i
g
i
(n) (break ties arbitrarily).
7: Assign the nth subchannel to user i
∗
(n):
K
i
(n)=
K
i
(n − 1) ∪{l
i
(n)} , if i = i
∗
(n);
K
i
(n − 1) , otherwise.
8: end while
230 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009
We consider several variations of Algorithm 1 which corre-
spond to different choices for steps 4 and 5. The choices for
step 4 are:
(4A): Sort the subchannels based on the best channel con-
dition among all users. This involves two steps. First, for each
subchannel j, find the best channel condition among all users
and denote it by ˜μ
j
:= max
i
e
ij
. Second, find a subchannel
permutation {α
j
}
j∈N
such that ˜μ
α
1
≥ ˜μ
α
2
≥···≥ ˜μ
α
N
,and
set l
i
(n)=α
n
for each user i at the nth iteration. Each max
operation has complexity of O(M), and the sorting operation
has a complexity of O(N log(N)). The total complexity is
O (NM + N log N). We note that this is a one-time “pre-
processing” that needs to done before the CA phase starts.
During the subchannel allocation iterations, the users just
choose the subchannel index from the sorted list.
(4B): Sort the subchannels based on the channel condi-
tions for each individual user. For each user i at the nth
iteration, set l
i
(n) to be the subchannel index with the
largest gain among all unassigned subchannels, i.e., l
i
(n)=
arg max
j∈N \∪
i
K
i
(n−1)
e
ij
. This requires M sorts (one per
user) that only need to be performed once, since each sub-
channel assignment does not change a user’s ordering of the
remaining subchannels. The total complexity of the M sorting
operations is O (MN log N), which is higher than that in (4A).
Let k
i
(n)=|K
i
(n)|. The choices for step 5 are:
(5A): Set g
i
(n) to be the total increase in user i’s utility if
assigned subchannel l
i
(n), assuming constant power alloca-
tion over all assigned subchannels, i.e.,
g
i
(n)=w
i
j∈K
i
(n−1)∪{l
i
(n)}
log
1+
P
i
e
ij
k
i
(n − 1) + 1
−
j∈K
i
(n−1)
log
1+
P
i
e
ij
k
i
(n − 1)
. (16)
(5B): Set g
i
(n) to be user i’s gain from only subchannel
l
i
(n), assuming constant power allocation, i.e.,
g
i
(n)=w
i
log
1+
P
i
k
i
(n − 1) + 1
e
i,l
i
(n)
.
Compared with (5A), this metric ignores the change in user
i’s utility due to the decrease in power allocated to any
subchannels in K
i
(n − 1).
The complexity of either of these choices over N iterations
is O(NM),andso the total complexity for the CA phase
4
is O (NM + N log N) (if (4A) is chosen) or O (MN log N)
(if (4B) is chosen).
B. CA in SOA2: subchannel Number Assignment & subchan-
nel User Matching
As summarized in Algorithm 2, SOA2 implements the
CA phase through two steps: subchannel number assignment
(CNA) and subchannel user matching (CUM).
CN A Step: The CNA step determines the number of sub-
channels n
i
assigned to each user i ∈Mbasedonthe
4
We note that SOA1 with (4B) and (5B) is similar to the algorithms
proposed in [25]. In Section VI we show that other variations of SOA1 ((4B)
and (5A)) and SOA2 can achieve better performance with similar or slightly
higher complexity.
Algorithm 2 CA Phase of SOA2
1: CNA step: determine the number of subchannels n
i
allo-
cated to each user i such that
i∈M
n
i
≤ N .
2: CUM step: determine the subchannel assignment x
ij
∈
{0, 1} for all i and j, such that
j∈N
x
ij
= n
i
, for each
i.
approximation that each user sees a flat wide-band fading
subchannel. This step does not specify which subchannel is
allocated to which user; such a mapping is left to the CUM
step. The CNA step is further divided into two stages: a basic
assignment stage and an assignment improvement stage.
Stage 1, Basic Assignment : Here, we model each user i
as having a normalized SINR
e
i
=
1
N
j∈N
e
ij
over all
subchannels, and then determine the number of subchannels
assigned n
i
to user i by solving:
max
{n
i
≥0,i∈M}
i∈M
w
i
n
i
log
1+
P
i
n
i
e
i
subject to:
i∈M
n
i
≤ N. (SOA2-CNA)
It can be shown that Problem SOA2-CNA is a standard con-
cave maximization problem over a convex set with a unique
and possibly non-integer solution; we use a dual relaxation
method to find this solution. The optimal Lagrange multiplier
for the subchannel constraint and any intermediate optimal
n
i
allocation can be found by a line-search, over ranges
[0, max
i
(w
i
log(1 + P
i
e
i
))] and [0,N], respectively. Hence,
the worst case complexity of solving each subproblem with a
fixed dual variable is independent of M or N . Since we need
to determine the value of n
i
for every user, the complexity of
the basic assignment step is O(M ). If the resultant channel
allocations contain non-integer values, we will approximate
with an integer solution that satisfies
i∈M
n
i
= N.
Since each user is allocated only a subset of the subchan-
nels, the normalized SINR
e
i
=
1
N
j∈N
e
ij
is typically a
pessimistic estimate of the averaged subchannel conditions
over the allocated subset. This motivates the following as-
signment improvement stage of CNA.
Stage 2, Assignment Improvement: Here, we iteratively
solve the following variation of Problem SOA2-CNA:
max
{n
i
(t)≥0,i∈M}
i∈M
w
i
n
i
(t)log
1+
P
i
n
i
(t)
e
i
(t)
subject to:
i∈M
n
i
(t) ≤ N, (SOA2-CNA-t)
for t =1, 2,.... During the t-th iteration,
e
i
(t) is a refined
estimate of the normalized SINR based on the best n
i
(t − 1)
subchannels of user i (n
i
(0) = N). The iterative procedure
stops when the subchannel allocation converges or the maxi-
mum number of iterations allowed is reached. At the end an
integer approximation will be performed, if needed.
The complete algorithm for the CNA phase of SOA2 is
given in Algorithm 3. In order to perform the assignment
improvement, we need to perform M sorting operations once,
with a total complexity O(MN log(N)). Step 4 of each itera-
tion has complexity of O(M) due to solving M subproblems
