Resource allocation for mobile video conferencing
Chao Yang and Scott Jordan
University of California, Irvine
Email: [email protected], [email protected]
ep
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Abstract—We consider resource allocation for mobile videoconferencing applications. We represent the performance of a
session by a sigmoid utility function of the average bit rate
over one or more groups of pictures, subject to an outage
constraint. The goal is to maximize the total expected utility of all
active video-conferencing users. We propose that resources can
be allocated if the base station chooses a price per unit rate based
on total demand and the users respond by choosing rates. Since
video-conferencing is interactive, an outage constraint requires
that resources be assigned based on a user’s channel. We suggest
that prices should be set to statistically guarantee a minimum
rate for users with poor combined pathloss and shadowing.
I. M AJOR C ONTRIBUTIONS
ta
rin
s.t.
max
p∈A
T
K
1 ∑∑
Uk (Sk,t )
T t=1
K ∑
N
∑
(1)
k=1
pk,n,t ≤ P ∀t; pk,n,t ≥ 0 ∀k, n, t
k=1 n=1
′
P r(Sk,t < Sk ) ≤ P r ∀k
t
bs
We give a closed form representation of the solution that is
based on shadow costs for rate, {λk,t , ∀k, 1 ≤ t ≤ T }, and
shadow costs for outage, {νk , ∀k}. The key innovation here is
that the outage constraint can be satisfied by charging users an
average∑
price per unit rate over the previous group-of-pictures,
t+W −1
β k,t = τ =t βk,τ /W , where the price per unit rate βk,t is
comprised of the shadow cost for rate λk,t and a normalized
version of the shadow cost for outage νk .
Third, we pose and characterize the solution of a causal
optimization problem that maximizes average user utility over
T time slots through allocation of power and subcarriers. The
causal problem is created by transformation of the non-causal
problem. The challenge is that the non-causal outage price
requires knowledge of future channel information. Decompose
2
2
the channel gain |Hk,n,t |2 = αk,n,t
γk,t P Lk,t where αk,n,t
is fast fading, γk,t is slow fading and shadowing and P Lk,t
is pathloss. The key observation is that fast fading largely
averages out during a group-of-pictures. We thus propose
basing the average rate price β k,t on a user’s combined
pathloss and shadowing ψk = γk,t P Lk,t and ignoring fast
fading. To obtain a causal problem, we propose replacing the
decision variables β k,t by a set of functions {βk (ψk ), ∀k}. To
make solution of the optimization problem computationally
feasible, we propose partitioning the domain of ψk into M
slices, which reduces the determination of {βk (ψk ), ∀k} to
ct
ra
The existence of an outage constraint poses a key challenge
to resource allocation. For best-effort applications, commonly
represented by concave utility functions, outage is not an issue
and total utility is maximized by allocating few resources
to users with poor channels. For inelastic applications such
as voice, commonly represented by step utility functions,
outage constraints are satisfied by allocating resources to
try to maintain a minimum threshold rate. For semi-elastic
applications such as video conferencing, however, neither
approach is optimal. There is a benefit to allocating fewer
resources to users with poor channels, but not so few as to
violate the outage constraint. The key question is when and
how much to compensate for a user’s poor channel. This paper
has four major contributions.
First, we formulate power and subcarrier allocation as an
optimization problem with a metric of total user utility as
a sigmoid function of the average rate over each group-ofpictures, and with outage constraints. We consider a single
cell downlink Orthogonal Frequency-Division Multiplexing
(OFDM) system serving K users with N subcarriers, each of
bandwidth B. The rate of user
( k on subcarrier
)n in time slot
|H
|2
t is rk,n,t (pk,n,t ) = B log2 1 + pk,n,t σk,n,t
, where pk,n,t
2 +I
is the power allocated, |Hk,n,t |2 is the composite channel gain,
σ 2 is the noise power, and I is the interference∑power. The
N
total rate of user k in time slot t is thus Rk,t = n=1 rk,n,t .
Whereas best-effort applications are often judged by
throughput over a long time period, and voice applications
by instantaneous rate, we propose that video should be judged
by the average rate during the previous group-of-pictures
com∑t
prised of W time slots, i.e. by Sk,t = τ =t−W +1 Rk,τ /W .
Furthermore, whereas utility for best-effort applications is
commonly concave, utility for video is thought to be sigmoid.
We thus represent the utility of user k by a sigmoid function
Uk (Sk,t ), namely there exists an inflection point Skf such that
Uk is convex for Sk,t < Skf and concave for Sk,t > Skf .
Finally, whereas outage is usually defined for voice whenever
instantaneous rate falls below a threshold, we propose that
′
′
outage for video occurs whenever Sk,t < Sk , where Sk =
arg maxSk,t Uk (Sk,t )/Sk,t is the rate at the maximum average
utility. Evaluation of utility over sliding windows is novel. The
introduction of an outage constraint in such a setting is also
novel.
Second, we pose and characterize the solution of a noncausal optimization problem that maximizes average user
utility over T time slots through allocation of power and
subcarriers. The decision variables are p = {pk,n,t , ∀k, n, t},
and the feasible set is A = {p s.t. ∀t, n, pk,n,t > 0 for at
most one user k}. The optimization problem is:
max
m
K ∑
M
∑
{βk ,∀k,m}
s.t.
Uk (Eα2 ,ψ−k Skm )qm
(2)
58
56
54
Total Utility
a choice of a finite set {βkm , ∀k, m}. Denote the resulting
average rate for user k while in slice m by Skm . The power
and rate for user k is expressed as an expectation over fast
fading α2 and the combined pathloss and shadowing for all
other users ψ−k . The quantized causal optimization problem
thus becomes:
k=1 m=1
K
M
N
∑∑∑
W=133, n>0
Expected Utility
W=1, n>0
50
(
)
Eα2 ,ψ−k pm
k,n qm ≤ P
k=1 m=1 n=1
M
∑
52
48
′
32
P rα2 ,ψ−k (Skm < Sk )qm ≤ P r, ∀k
34
36
38
40
42
Number of User
m=1
Fig. 1. Total Utility versus Number of Users
II. P ERFORMANCE E VALUATION
58
56
54
52
ta
rin
The paper also examines the performance of the proposed
iterative algorithms via simulation. We adopt the parameters
of an LTE scenario. A group-of-pictures comprises W = 133
time slots. All users have the same utility function, given by:
{
a(Sk,t /4)2 , if Sk,t < 240kbps
Uk (Sk,t ) =
c(Sk,t /4 + b)1/3 , else
60
Total Utility
ep
Pr
Finally, we propose an iterative algorithm that determines
power and subcarrier allocations based on quantization of
combined pathloss and shadowing. We illustrate how this
approach can maximize utility subject to long-term average
outage constraints, how the length of a group-of-pictures impacts the solution, and how these policies differ from alternate
resource allocation policies.
48
32
34
36
38
40
42
Number of User
Fig. 2. Utility under Various Policies
t
bs
W = 1, a user evaluates application performance based on the
instantaneous rate of one time slot. Thus, when the channel in
a time slot is poor, the system must decide whether to allocate
a very large amount of power and subcarriers. In contrast
when W = 133, when a channel is poor the system may
examine past achieved rates and likely future rates within the
time window, and often will allocate power and subcarriers
in a more moderate and efficient manner. We also compare
the utility for W = 133 with outage pricing to the utility that
would be earned from a policy that allocates resources using
a window W = 133 but without outage pricing, i.e. ν = 0.
Use of outage pricing results in users with poor channels who
′
are in danger of achieving rates less than Sk during a time
window receiving improved performance, but this comes at
the cost of decreased performance for users with rates above
′
Sk . Together, this results in lower utility but decreased outage.
ct
ra
where a = 4/5 ∗ (2/5)1/3 /(12/5)2 , b = −2, c = 4/5 and Sk,t
is expressed in units of 100kbps. The rate at the maximum
′
average utility Sk = 300 kbps. We allow 3% outage, i.e.
P r = 0.03. All users move at a constant speed of 10km/h,
with direction determined by a random walk. We simulate 45
minutes of real time.
The total utility as a function of the number of users K is
shown in Fig. 1. Total utility is an increasing concave function
of the number of users within the considered range. We also
plot the expected utility in the optimization metric in (2),
labelled in the figure as “Expected Utility”. The real utility
is slightly above the expected utility, which shows that only
small errors are introduced by the statistical average model
using quantization. The outages of this scenario with different
total user numbers are 0.028, 0.029, 0.022, 0.026, 0.028, 0.027
respectively. The proposed algorithms with outage price νk
can guarantee outage performance.
For comparison, in Fig. 2, we plot the utility that would be
earned from a policy that allocates resources without regard to
the time window, i.e. it attempts to maximize average utility
of instantaneous rate. This is labelled in the figure by W = 1.
The difference between the W = 1 and W = 133 curves
represents the increase in utility gained by allocating resources
in a more flexible manner within a group-of-pictures. When
W=133, n=0
W=133, n>0
W=1 , n>0
50