User-Level Performance of Channel-Aware
Scheduling Algorithms in Wireless Data Networks
Sem Borst
∗,†,‡
∗
Bell Laboratories, Lucent Technologies
P.O. Box 636, Murray Hill, NJ 07974, USA
†
CWI
P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
‡
Department of Mathematics & Computer Science
Eindhoven University of Technology
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Abstract— Channel-aware scheduling strategies, such as the
Proportional Fair algorithm for the CDMA 1xEV-DO system,
provide an effective mechanism for improving throughput per-
formance in wireless data networks by exploiting channel fluctua-
tions. The performance of channel-aware scheduling algorithms
has mostly been explored at the packet level for a static user
population, often assuming infinite backlogs. In the present paper,
we focus on the performance at the flow level in a dynamic setting
with random finite-size service demands. We show that in certain
cases the user-level performance may be evaluated by means of
a multi-class Processor-Sharing model where the total service
rate varies with the total number of users. The latter model
provides explicit formulas for the distribution of the number of
active users of the various classes, the mean response times, the
blocking probabilities, and the mean throughput. In addition we
show that, in the presence of channel variations, greedy, myopic
strategies which maximize throughput in a static scenario, may
result in sub-optimal throughput performance for a dynamic user
configuration and cause potential instability effects.
Index Terms— Channel-aware scheduling, elastic traffic, in-
sensitivity, Processor Sharing, Proportional Fair scheduling, re-
sponse time, stability, throughput optimization, wireless data
networks.
I. INTRODUCTION
Next-generation wireless networks are expected to support
a wide variety of data services. Data applications have funda-
mentally different traffic characteristics and different quality-
of-service requirements than traditional voice services, calling
for a significant departure from a conventional circuit-switched
operation. In particular, the relative delay tolerance of data
applications, combined with the bursty activity patterns, opens
up the possibility of scheduling transmissions so as to obtain
efficiency gains. An especially attractive approach, in fading
environments, is to use channel-aware scheduling strategies,
such as the Proportional Fair algorithm for the CDMA 1xEV-
DO system [4], [11], [20], which harness channel variations
so as to improve the throughput performance.
The performance of channel-aware scheduling algorithms
has mostly been investigated at the packet level for a static user
population, sometimes including packet-scale dynamics [3],
[17], but often assuming infinite backlogs [1], [8], [14], see
also [15], [19] for related results. The assumption of a static
user population is a reasonable modeling convention because
of the separation of time scales: the scheduling algorithms
operate at the packet level on which the user population
evolves only relatively slowly. However, when examining
throughput performance, and in particular comparing the
throughput allocation among elastic traffic users under various
strategies, it does not seem entirely satisfactory to assume
that the user population is independent of the throughput
characteristics and the parameter settings of the scheduling
algorithm. For example, a scheduling algorithm that provides
high throughput to users with favorable channel conditions,
will tend to satisfy the service demands of these users sooner.
As a result, the algorithm would tend to be left facing a user
population with a higher fraction of users with poor channel
conditions. Conversely, a scheduling algorithm that grants
reasonable throughput to users with poor channel conditions,
should to a certain degree benefit from that by seeing fewer
of these users.
In order to capture the above interdependence between
the scheduling algorithm and the user population, we move
away from a static scenario with a fixed ensemble of users
to a dynamic setting where elastic traffic users come and
go as governed by the arrival and completion of service
demands over time. The notion of finite-size service demands
additionally allows us to consider user-perceived performance
in terms of response times for file transfers for example, as
opposed to delays experienced by individual packets. We will
show that in certain cases the user-level performance may be
evaluated by means of a multi-class Processor-Sharing model
where the total service rate varies with the total number of
users. The latter model provides explicit formulas for the
distribution of the number of active users of the various
classes, the mean response times, the blocking probabilities,
and the mean throughput.
To put the above observations further into perspective, it is
helpful to make a comparison with a situation where the trans-
mission rates are possibly different across users but constant
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
over time. In that case, a standard work conservation argument
implies that the ‘amount of work’ in the system (measured in
transmission time rather than bits) is the same under any non-
idling scheduling rule. In that sense, the throughput allocation
among users corresponding to various scheduling strategies
will have an impact on the delay characteristics, but no
effect on the system throughput in case of finite-size service
demands.
The above-mentioned work conservation property does not
extend to a situation where the transmission rates vary over
time, and it will no longer be the case that any non-idling
scheduling strategy yields the same system throughput. As
it turns out, it is not so much maximizing the instantaneous
throughput in an absolute sense that determines stability then,
but serving users at the right time so as to extract the
maximum possible gains from the varying channel conditions.
In particular, we will show that greedy, myopic strategies
which maximize throughput in a static scenario, may result
in sub-optimal throughput performance for a dynamic user
configuration and cause potential instability phenomena. Of
course, (in)stability is to a certain extent a theoretical concept
that cannot occur in an actual system due to admission and
flow control mechanisms and the inherent finiteness of buffers.
However, it is plausible that instability effects will be reflected
in poor performance in terms of long delays in practical
circumstances as well.
The remainder of the paper is organized as follows. In
Section II we recapitulate some relevant results from the
literature for a static user population and state some prelimi-
nary facts. We extend the model to accommodate a dynamic
user configuration in Section III. We describe how in certain
symmetric cases the system behavior may be described by
means of a multi-class Processor-Sharing model where the
total service rate varies with the total number of users. We
present exact results for the distribution of the number of active
users of the various classes, the mean response times, the
blocking probabilities, and the mean throughput. In Section IV
we turn the attention to asymmetric scenarios and derive
some stochastic majorization properties. We examine stability
issues in Section V. In Section VI we discuss some numerical
experiments that we conducted to illustrate the results.
II. S
TATIC USER POPULATION
We first review some relevant results from the literature for
a static scenario with a population of M data users served by
a single base station. The base station transmits in slots of
some fixed duration. In each slot, the base station transmits to
exactly one of the users.
We assume that the feasible rates for the various users vary
over time according to some stationary discrete-time stochas-
tic process {R
1
(t),...,R
M
(t)}, with R
i
(t) representing the
feasible rate for user i in time slot t. In order to estimate the
feasible rates, the base station relies on feedback information
from the users on the instantaneous rates that can reliably be
supported, as is for instance the case in the CDMA 1xEV-
DO system (also known as HDR) [4]. The prediction of the
feasible rates should be reasonably accurate when the feedback
delay is relatively short compared to the fading frequency.
For convenience, we assume that the base station has perfect
knowledge of the feasible rate R
i
(t) for every user i at the
start of slot t, although the results may be extended to account
for possible prediction errors.
Let (R
1
,...,R
M
) be a random vector with as distribution
the joint stationary distribution of the feasible rates. We focus
on the case where the feasible rates (R
1
,...,R
M
) have a
discrete distribution on some finite set R⊆R
M
+
.Letp(r) be
the stationary probability that the instantaneous feasible rate
vector is r ∈R. With minor modifications, most of the results
extend to scenarios with a continuous rate distribution.
Let T
i
be the (long-term) throughput received by user i,
and let A⊆R
M
+
be the set of achievable throughput vectors.
The next proposition provides a characterization of the
set A [3], [8].
Proposition 2.1: The set of achievable throughput vec-
tors A may be characterized as
A = {T ∈ R
M
+
: z(T ) ≥ 1},
where z(T ) is the optimal value of the linear program
max z
sub z ≤ z
i
=
r∈R
p(r)x
i
(r)r
i
/T
i
i =1,...,M
M
i=1
x
i
(r) ≤ 1 r ∈R
x
i
(r) ≥ 0 i =1,...,M,r ∈R.
The variable x
i
(r) in the above linear program may be
interpreted as the fraction of time slots allocated to user i
in which the instantaneous rate vector is r. Thus, the term
r∈R
p(r)x
i
(r)r
i
represents the throughput received by user i,
and the variable z
i
measures the throughput as a fraction of
the target throughput T
i
.
The next proposition provides a characterization of the
optimal solution of the above linear program based on the
complementary slackness conditions [3], [8].
Proposition 2.2: There exists a vector w
∗
∈ R
M
+
such
that any optimal solution x
∗
i
(r) to the above linear program
satisfies
x
∗
i
(r)
w
∗
i
r
i
− max
j=1,...,M
w
∗
j
r
j
=0,
for all i =1,...,M, r ∈R.
The above proposition shows that any feasible (non-
dominated) throughput vector can be achieved by some
weight-based strategy which allocates time slot t to a user i
∗
identified as
w
∗
i
∗
R
i
(t) = max
j=1,...,M
w
∗
j
R
j
(t),
augmented with a suitable tie-breaking rule. In particular, any
component-wise increasing function of the throughput vector
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is maximized by some weight-based strategy.
We now state some simple auxiliary results that will play
a crucial role in the further analysis.
Lemma 2.1: Any feasible throughput vector T ∈Asatisfies
M
j=1
α
j
T
j
≤ E{ max
j=1,...,M
α
j
R
j
} for any vector (α
1
,...,α
M
) ∈
R
M
+
.
Proof
Note that the throughput function
M
j=1
α
j
T
j
is maximized
by a weight-based strategy which assigns a weight w
i
= α
i
to user i (in fact, sample-path wise), and that the optimal
value equals E{ max
j=1,...,M
α
j
R
j
}. ✷
We now consider a scenario where the distribution of
the rate vector is symmetric in the sense that the relative
fluctuations in the feasible rates for the various users
around the respective time-average values are statistically
identical. Specifically, we assume that R
i
d
= C
i
Y
i
Z, where
C
i
:= E{R
i
} is the time-average rate of user i, Y
1
,...,Y
M
are independent and identically distributed copies, and Z
represents a possible correlation component with unit mean.
Define G(M):=E{ max
j=1,...,M
Y
j
}.
Lemma 2.2: In the case of a symmetric rate distribution as
described above, the weight-based strategy which assigns a
weight w
i
=1/C
i
to user i, and breaks ties between users at
random, provides each user a fraction G(M )/M of its time-
average rate.
Proof
Note that user i is selected when
R
i
C
i
= max
j=1,...,M
R
j
C
j
, i.e.,
Y
i
= max
j=1,...,M
Y
j
, and possible ties are broken to its advantage.
By symmetry considerations, user i thus receives a fraction
1/M of the time slots, and the expected rate when selected is
E{R
i
|
R
i
C
i
= max
j=1,...,M
R
j
C
j
} = E{C
i
Y
i
Z|Y
i
= max
j=1,...,M
Y
j
} =
C
i
E{Y
i
|Y
i
= max
j=1,...,M
Y
j
} = C
i
E{ max
j=1,...,M
Y
j
} = C
i
G(M).
✷
Remark 2.1: The assumption that the relative rate fluctu-
ations are statistically identical is roughly valid when the
users for example have Rayleigh fading channels and the
feasible rate is approximately linear in the SNR (signal-
to-noise ratio). The latter approximation is reasonably ac-
curate when the SNR is not too high. It is not necessary
that the Doppler frequencies are identical, since only the
instantaneous rate distribution affects the long-term average
throughput achieved under a weight-based strategy. Of course,
the Doppler frequencies do matter for the transient throughput
behavior and also affect the ability to predict the feasible rate.
Also, the assumption R
i
d
= C
i
Y
i
Z could be further relaxed.
For instance, a somewhat milder condition would be that
P{(R
1
/C
1
,...,R
M
/C
M
) ≤ (t
π (1)
,...,t
π ( M )
)} is invariant
under permutations π(1),...,π(M).
Remark 2.2: In certain cases, the Proportional Fair schedul-
ing algorithm for the CDMA 1xEV-DO system mentioned
earlier behaves approximately like a weight-based strategy. In
Proportional Fair scheduling, the weights w
i
are dynamically
adapted and are inversely proportional to the exponentially
smoothed throughputs W
i
of the users. Thus, the expected
rate of user i when selected is
E{R
i
|w
i
R
i
= max
j=1,...,M
w
j
R
j
} = E{R
i
|
R
i
W
i
= max
j=1,...,M
R
j
W
j
}.
Now observe that both the instantaneous rate R
i
and the
exponentially smoothed throughput W
i
scale linearly with the
time-average rate C
i
in case the relative rate fluctuations are
statistically identical. As a result, the allocation of time slots
only depends on the relative rate fluctuations and not on the
time-average rates. Thus, each user receives a fraction 1/M
of the time slots, and we may write W
i
d
= C
i
V
i
, where
the random variables V
1
,...,V
M
are identically distributed
(but not independent). In addition, the exponentially smoothed
throughputs will not show any significant variation when
the time constant in the exponential smoothing is large, i.e.,
V
1
,...,V
M
≈ V for some constant V . Substituting R
i
d
=
C
i
Y
i
Z and W
i
≈ VC
i
in the above formula, we find that the
expected rate of user i when selected approximately equals
E{C
i
Y
i
Z|Y
i
= max
j=1,...,M
Y
j
} = C
i
G(M). In conclusion, in
case the relative rate fluctuations are statistically identical and
the time constant in the exponential smoothing is not too small,
the Proportional Fair scheduling algorithm roughly behaves
as the weight-based strategy which assigns a constant weight
w
i
=1/C
i
to user i. We refer to [13], [18] for a rigorous
justification of the above claims.
We would like to add that the above statements assume the
users to have infinite backlogs. In situations with packet-scale
dynamics, the Proportional Fair algorithm may be ill-behaved,
and the throughput performance be degraded by convergence
and fragmentation issues, giving rise to potential instability
phenomena [2].
III. D
YNAMIC USER CONFIGURATION
We now extend the model to accommodate a dynamic
configuration of users. The user dynamics result from finite-
size service demands that arrive randomly over time. We
assume that the duration of the time slots is short relative to the
size and arrival frequency of the service demands. Thus, the
scheduling strategy operates on an extremely fast time scale
compared to the user dynamics, making it natural to analyze
the user-level performance in continuous rather than discrete
time, and assume that the users are served simultaneously
rather than in a time-slotted fashion. The continuous-time
model naturally inherits its service characteristics from the
discrete-time model. Specifically, we assume that the set of
feasible service rate vectors in the continuous-time context for
a given user population coincides with the set of achievable
throughput vectors for that user population in a discrete-time
setting.
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For the latter model, we consider a scenario where the
relative fluctuations in the feasible rates for the various users
around the respective time-average values are statistically
identical as described in the previous section. Thus, we assume
that the instantaneous rate of user i with time-average rate C
i
is distributed as R
i
d
= C
i
Y
i
Z, where Y,Y
1
,Y
2
,... are inde-
pendent and identically distributed copies and Z represents a
possible correlation component with unit mean. According to
Lemma 2.2, we then have that under the strategy S
∗
which
assigns a weight w
i
=1/C
i
to a user i with a time-average
rate C
i
, each user is continuously served at a fraction G(n)/n
of its time-average rate whenever there are n users in the
system.
The above assumptions ignore the discrete nature of the time
slots and neglect the transient fluctuations in the throughput.
However, the law of large numbers suggests that these effects
should be negligible in some suitable asymptotic sense in a
limiting regime where the duration of the time slots shrinks
relative to the time scale of the user dynamics.
To describe the service demands, we assume that users
initiate file transfer requests randomly over time. We consider
a scenario with K user classes. Class-k users submit file
transfer requests as a Poisson process of rate λ
k
. We assume
that at most M users in total are admitted into the system
simultaneously (possibly M = ∞). Users which submit
requests when there are already M transfers in progress are
denied access and abandon. Let (C
k
,F
k
) be a pair of random
variables with as distribution the joint distribution of the time-
average transmission rate and the file size of an arbitrary
class-k user. We assume that the file size and time-average
transmission rate are independent across users, but we allow
for possible dependence between the file size and time-average
transmission rate of a given user. Let B
k
:= F
k
/C
k
be
the normalized service requirement of a class-k user, with
mean β
k
:= E{B
k
} = E{F
k
/C
k
}. The normalized service
requirement is the amount of time it would take to complete
the file transfer if a user were the only user in the system. Note
that the normalized service requirement encapsulates both the
file size and the time-average transmission rate of a user, and
is measured in transmission time rather than data volume.
Define ρ
k
:= λ
k
β
k
as the offered traffic associated with class-
k users. Denote by ρ :=
K
k=1
ρ
k
the total amount of offered
traffic. Let B
r
k
be a random variable representing the residual
lifetime of B
k
and B
r
k
(·) the associated distribution function,
i.e., B
r
k
(x):=P{B
r
k
<x} :=
1
β
k
x
y =0
P{B
k
>y}dy.
Let (N
1
,...,N
K
) be a random vector representing the
number of users of the various classes in the system under
strategy S
∗
at an arbitrary epoch in statistical equilibrium
(assuming it exists). Denote by N := N
1
+ ...+ N
K
the total
number of users in the system. Given that there are n
k
class-k
users in the system, let B
r
k,i
be the remaining normalized
service requirement of the i-th class-k user, i =1,...,n
k
,
k =1,...,K. Define G
∗
:= sup
M=1,2,...
G(M) = lim
M→∞
G(M).
Note that G
∗
= ∞ when the distribution of Y has infinite
support.
Proposition 3.1: Strategy S
∗
achieves stability for ρ<G
∗
or M<∞, in which case
P{N
k
= n
k
,B
r
k,i
≤ t
k,i
; i =1,...,n
k
,k =1,...,K} =
H
−1
n!ρ
n
φ(n)
K
k=1
1
n
k
!
ρ
k
ρ
n
k
n
k
i=1
B
r
k
(t
k,i
),
with n = n
1
+ ... + n
K
≤ M, φ(n):=
n
i=1
G(i), and
normalization constant
H :=
M
n=0
ρ
n
φ(n)
.
In particular,
P{N = n} = H
−1
ρ
n
φ(n)
,
E{N} = H
−1
M
n=1
nρ
n
φ(n)
,
and
E{N
k
} =
ρ
k
ρ
E{N}.
The blocking probability is given by
L = P{N = M }.
Proof
According to Lemma 2.2, each user is served at a fraction
G(n)/n of its time-average rate whenever there are n users
in the system. Thus, the normalized remaining service
requirement of each user is reduced at rate G(n)/n, which
means that the normalized remaining service requirements
evolve in a similar probabilistic fashion as the remaining
service requirements in a multi-class Processor-Sharing
system with arrival rates λ
k
, generic service requirements B
k
,
and service rate G(n) when there are n users in total
present. The statements then follow from results for the latter
system [9], [12]. ✷
Remark 3.1: Proposition 3.1 extends to the case where
users generate sessions consisting of multiple file requests
separated by arbitrarily distributed ‘think times’ [5], [7]. In that
case, the offered traffic should be calculated so as to include
the mean number of file requests per session.
Using Little’s law, we find that the mean transfer delay
experienced by a class-k user is given by
E{S
k
} =
β
k
ρ(1 − L)
E{N}.
The above formula reflects the celebrated insensitivity property
of the Processor-Sharing discipline, which shows that the mean
delay of a class-k user only depends on the service requirement
distribution of class k through its mean β
k
. In fact, it may be
shown that the conditional expected delay of any user with
actual service requirement b is given by
E{S|B = b} =
b
ρ(1 − L)
E{N}.
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Thus, the expected transfer delay incurred by a user is
proportional to its normalized service requirement, with
factor of proportionality E{N}/(ρ(1 − L)). The latter
property embodies a certain fairness principle, which
means that users with larger service requirements tend
to experience longer delays. Recall that the normalized
service requirement encapsulates both the file size and the
time-average transmission of a user, and is expressed in time
units rather than data bits.
Proposition 3.2: No strategy achieves stability for ρ>G
∗
.
Proof
Define the normalized amount of work as the sum of the
normalized remaining service requirements of all the users in
the system. Let A
m
and B
m
be the arrival epoch and the
normalized service requirement of the m-th arriving user, let
X
m
be the normalized amount of work in the system at time
t = A
m
, and let D
m
be the reduction in the normalized
amount of work between time epochs A
m
and A
m+1
.Ac-
cording to Lemma 2.1, taking α
i
=1/C
i
, no strategy is able
to reduce the normalized amount of work at a rate higher
than G(M) ≤ G
∗
when there are M users present. Hence,
X
m+1
= X
m
+ B
m
− D
m
≥ X
m
+ B
m
− G
∗
(A
m+1
− A
m
),
so that when ρ>G
∗
,
E{X
m+1
}≥E{X
m
} +
K
k=1
λ
k
λ
β
k
−
1
λ
G
∗
= E{X
m
} +
1
λ
[ρ − G
∗
] > E{X
m
},
with λ :=
K
k=1
λ
k
. Thus, the normalized workload process has
positive drift when ρ>G
∗
for any strategy. ✷
Propositions 3.1 and 3.2 combined imply that strategy S
∗
achieves stability whenever feasible. The heuristic explanation
is that the rate at which strategy S
∗
reduces the normalized
amount of work will approach the maximum possible value G
∗
as the number of users tends to infinity. In fact, the proof of
Proposition 3.2 shows that strategy S
∗
reduces the normalized
amount of work at a higher rate than any other strategy, given
the same number of users. (It is thus tempting to conjecture
that strategy S
∗
actually minimizes the normalized amount of
work among all strategies, but that does not appear to be true
without further assumptions.) In particular, a weight-based
strategy which assigns a weight F (C)/C to a user with
a time-average rate C reduces the normalized amount of
work at a rate
M
i=1
E{Y
i
I
{F (C
i
)Y
i
=max
j=1,...,M
F (C
j
)Y
j
}
} when
there are M users with time-average rates C
1
,...,C
M
.In
general, there is no guarantee that the latter quantity under
any circumstances approaches G
∗
when M tends to infinity.
Intuitively, unless the weights are set inversely proportional
to the time-average transmission rates, the relative rate
fluctuations are not maximally exploited. We will examine
these issues further in Section V.
Remark 3.2: As mentioned in Section II, strategy S
∗
may
be viewed as a proxy for the Proportional Fair scheduling
algorithm in case the relative rate fluctuations are statistically
identical and the time constant in the exponential smoothing
is not too small. The latter statement assumed a static user
population with infinite backlogs. With a dynamic user con-
figuration, we need to assume that the duration of the time
slots is relatively short compared to the backlog periods of the
users, so that the throughput performance of the Proportional
Fair algorithm is not substantially hampered by convergence or
granularity issues. Otherwise, when the weights are initialized
to zero, the algorithm may allocate time slots to arriving users
almost regardless of the channel conditions, and thus fail
to extract the maximum gains from the channel variations.
The Proportional Fair algorithm may then result in sub-
optimal throughput performance and potentially collapse into
instability.
IV. A
SYMMETRIC SCENARIOS
In the previous section we considered a scenario with
K user classes where the relative rate fluctuations in the fea-
sible rates are statistically identical for all users. We assumed
that the system is operated according to the weight-based
strategy S
∗
which assigns a weight w
i
=1/C
i
to a user i
with a time-average transmission rate C
i
.
We now consider a scenario where the relative fluctuations
in the feasible rates around the respective time-average val-
ues for all users of a given class are statistically identical
as before. However, we allow for the distributions of the
fluctuations to vary across user classes. Thus, we assume
that the instantaneous rate of a class-k user i is distributed
as R
k,i
d
= C
k,i
Y
ki
Z, where Y
k
,Y
k1
,Y
k2
,... are independent
and identically distributed copies and Z represents a possible
correlation component with unit mean.
The system is operated using a weight-based strategy S
α
which assigns a weight w
k,i
= α
k
/C
k,i
toaclass-k user i
with a time-average rate C
k,i
. The parameters α
k
allow for
differentiation among the various user classes. The differentia-
tion could be based on channel statistics, traffic characteristics,
or Quality-of-Service requirements.
With the heterogeneous user classes, the system loses the
symmetry properties of the ordinary Processor-Sharing disci-
pline which facilitated the analysis in the previous section. In
fact, asymmetric (discriminatory) versions of the Processor-
Sharing discipline have remained largely intractable so far,
even under exponentiality assumptions and when the service
rates are constant [10], [16]. Therefore, we will not aim for
full distributional results but focus on stochastic majorization
properties and stability issues.
Note that strategy S
α
allocates a time slot to a class-k user i
when w
k,i
R
k,i
= max
l=1,...,K
max
j=1,...,n
l
w
l,j
R
l,j
, i.e., α
k
Y
ki
=
max
l=1,...,K
max
j=1,...,n
l
α
l
Y
lj
. In order to avoid technicalities, we
assume that P{α
k
Y
k
= α
l
Y
l
} =0for k = l, so that there
are no tie-breaking issues between user classes. Ties between
users from the same class are broken at random.
Let y
k
:= inf{y : P{Y
k
>y} =0} be the maximum
value that Y
k
can achieve. We assume that the user classes are
0-7803-7753-2/03/$17.00 (C) 2003 IEEE IEEE INFOCOM 2003
![Fig. 1. CDF of SNR distribution from [4]](/figures/fig-1-cdf-of-snr-distribution-from-4-3vs5y91r.webp)
