IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 9, NO. 6, OCTOBER 2007 1227
Media Flow Rate Allocation in Multipath Networks
Dan Jurca, Student Member, IEEE, and Pascal Frossard, Senior Member, IEEE
Abstract—We address the problem of joint path selection and
source rate allocation in order to optimize the media specific
quality of service in streaming of stored video sequences on mul-
tipath networks. An optimization problem is proposed in order
to minimize the end-to-end distortion, which depends on video
sequence dependent parameters, and network properties. An
in-depth analysis of the media distortion characteristics allows
us to define a low complexity algorithm for an optimal flow rate
allocation in multipath network scenarios. In particular, we show
that a greedy allocation of rate along paths with increasing error
probability leads to an optimal solution. We argue that a network
path shall not be chosen for transmission, unless all other available
paths with lower error probability have been chosen. Moreover,
the chosen paths should be used at their maximum available
end-to-end bandwidth. Simulation results show that the optimal
flow rate allocation carefully adapts the total streaming rate
and the number of chosen paths, to the end-to-end transmission
error probability. In many scenarios, the optimal rate allocation
provides more than 20% improvement in received video quality,
compared to heuristic-based algorithms. This motivates its use in
multipath networks, where it optimizes media specific quality of
service, and simultaneously saves network resources at the price
of a very low computational complexity.
Index Terms—Multipath networks, path selection, rate alloca-
tion, video distortion.
I. INTRODUCTION
W
ITH the development of novel network infrastructures
and increasing available bandwidth, multimedia appli-
cations over the Internet become attractive for both businesses
and home users. Fast deployment of broadband last-mile con-
nections, increase in wireless coverage of remote living areas,
and the long awaited debut of 3G wireless services offer as many
inter-operable communication solutions.
However, the viability of a streaming application mostly de-
pends on its ability to meet stringent requirements (e.g., con-
trolled error rate and low delay) and on medium and long term
stability of the transport infrastructure. As the Internet is still
far from providing any widely deployed guarantee of service
solution, efficient media streaming strategies have to be devised
to get the best out of the network infrastructure. Lately, multi-
path streaming emerged as a valid solution to overcome some
of the lossy Internet path limitations [1], [2]. It allows for an in-
crease in streaming bandwidth, by balancing the load over mul-
Manuscript received March 13, 2006; revised March 18, 2007. This work was
supported by the Swiss NSF under Grant PP002-68737. The associate editor
coordinating the review of this manuscript and approving it for publication was
Dr. Deepak S. Turaga.
The authors are with the Ecole Polytechnique Fédérale de Lausanne
(EPFL), Signal Processing Institute, CH-1015 Lausanne, Switzerland (e-mail:
dan.jurca@epfl.ch; pascal.frossard@epfl.ch).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMM.2007.902852
tiple network paths between the media server and the client. It
also provides means to limit packet loss effects, when combined
with error resilient streaming strategies and scalable encoding
capabilities of the latest encoding standards [3]–[6]. Most of the
research work dedicated to multipath streaming focuses on the
process itself (media caching and scheduling aspects), but gen-
erally not towards finding which paths should ideally be used for
the streaming application, for a given network topology between
a streaming server and a client. These works rely on classic
routing algorithms that find the best path (or set of paths) given
some established network metrics. While this may be optimal
in terms of network utilization, it is certainly suboptimal from
the viewpoint of the media streaming application. In 30–80% of
the cases, the best paths found by classic routing algorithms are
suboptimal from a media perspective [7].
This work proposes to address the problem of streaming
path allocation in a multipath network, which takes into ac-
count media aware metrics during the decision process. The
early work in [8] derives a few empirical rules on what paths
should be considered by the streaming application, based on
experimental data. These rules consider network metrics (e.g.,
available bandwidth, loss rate, and hop distance), and other
media aware metrics (e.g., link jointness/disjointness, video
distortion). Our work provides a more general framework for
the analysis of joint path selection and flow rate allocation in
multipath streaming, driven by media-specific metrics. We con-
sider a multipath network model that supports the partitioning
of a media sequence into multiple media flows. We further
assume that the streaming server that can perform simple
adaptation of the streaming rate of pre-encoded packet media
streams (by packet filtering, or by taking advantage of scalable
coding, for example). A generic video distortion metric is
proposed, which encompasses both the source distortion that is
mostly driven by the streaming rate, and the channel distortion
that depends on the loss probability.
Finding the optimal flow rate allocation in multipath networks
is a very complex problem in generic scenarios. However, in our
specific scenario, we show that a careful analysis of the video
distortion evolution allows to derive a linear complexity algo-
rithm for the joint optimal path selection, and flow rate alloca-
tion. In other words, our main objective is to jointly find i) the
optimal streaming rate for a given, pre-encoded video packet
stream so that the quality at receiver is maximized and, ii) which
network paths should be used for relaying the video stream to
the client. Interestingly enough, our conclusions demonstrate
that the answer to these two questions is represented by a careful
tradeoff among available network bandwidth (translated into
video streaming rate), transmission loss process, and number
of utilized paths. And, in contrary to the commonly admitted
opinion, flooding the network in pushing the streaming rate to
the limits the total available bandwidth, rarely provides efficient
1520-9210/$25.00 © 2007 IEEE
1228 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 9, NO. 6, OCTOBER 2007
Fig. 1. Multipath network scenario.
strategies in the absence of complex and expensive transcoding
strategies for stored video streams.
The main contributions of this paper can be briefly summa-
rized as follows.
• We propose a general framework for streaming of pre-en-
coded media data in multipath networks, which encom-
passes network and media aware metrics.
• We perform the first theoretical media flow analysis on
the optimality of number, and choice of network paths, in
terms of end-to-end Quality of Service.
• We provide a linear time media aware routing algorithm
that outputs the optimal set of network paths to be used in
the streaming process, along with the corresponding flow
rate distribution.
The paper is organized as follows: Section II presents the
streaming framework and formulates our optimization problem.
The theoretical analysis of the streaming process is developed
in Section III. Section IV presents the routing algorithm and
Section V presents our main results. We present the related work
in Section VI, and conclude the paper in Section VII.
II. D
ISTORTION OPTIMIZED MULTIPATH MEDIA STREAMING
A. Multipath Network Model
We consider a framework where the media streaming appli-
cation uses a multipath network, which can be represented as
follows. The available network between a media server
and a
client
is modeled as a graph , where is the
set of nodes in the network, and
is the set of links or segments
(see Fig. 1). Each link
connecting nodes
and has two associated positive metrics
• available bandwidth
expressed in some appropriate
unit (e.g., kbps);
• average loss probability
, assumed to be inde-
pendent of the streaming rate.
Let
denote the set of available loop-free
paths between the server
and the client in , with the
total number of nonidentical end-to-end paths. A path
is defined as an ordered list of nodes and
their connecting links, such that no node appears more than
once, and that each link
between two consecutive nodes in
the path belongs to the set of segments
. Let further and
denote respectively the end-to-end bandwidth and loss proba-
bility of path
.Wedefine the bandwidth of an individual path
as the minimum of the bandwidths among all links on the
path (i.e., the “bottleneck bandwidth”). Hence, we have
(1)
Under the commonly accepted assumption that the loss
process is independent on two consecutive segments, the
end-to-end loss probability on path
becomes a multiplicative
function of the individual loss probabilities of all segments
composing the path. It can be written as
(2)
Finally, the media application sends data at rate
on path
, with a cost . The cost represents the price to be paid by
the streaming application, for using path
. As, in general, the
underlying transport medium should be transparent for the ap-
plication, we define the cost function as dependent only on the
total flow rate
sent by the application on path . A linear cost
relation is simply expressed as follows:
if is used, with
if is not used
(3)
where
is a constant (i.e., the cost factor is identical for any path
in ). In this network model, efficient streaming strategies
have to carefully allocate the rate between the different network
paths. The goal of the next sections is to get the best out of the
multipath network, both in terms of cost, and from a media-
driven quality of service perspective.
B. From Network Graph to Flow Tree
In order to study the flow rate allocation problem in multipath
networks, we use a flow tree representation of the network graph
. The media server becomes the root of the tree, and each flow
represents the share of the overall media stream, which is
sent on a network path
. The media stream is the composition
of individual media flows, and the client is represented as a set
of leaf nodes, with one leaf per flow. Note that several methods
in graph theory have been proposed for constructing such trees,
and we rather concentrate in this paper on the rate allocation
problem, among the branches of the tree. In this case, the rate
allocation becomes a flow assignment problem.
Considering that there is (at most) one flow for each network
path
, we can transform the original network graph into
a flow tree by duplicating any network edge and vertex that is
shared by more than one network path, as represented in Fig. 2.
Since the transformation from paths to flows is bijective, each
flow is characterized by a maximal end-to-end streaming rate,
and an end-to-end loss probability, as computed in Section II-A.
The flow
on path uses a streaming rate , with a loss
probability
, and a cost .
Due to the assumption of rate independent loss process, any
two flows in the tree are independent in terms of loss probability.
However, flows may be dependent in terms of aggregated band-
width, since they may share joint bottleneck links. The flow tree
representation allows us to explicit the constraints imposed on
a valid rate allocation. These constraints are imposed by band-
width limitation on the network links, and flow conservation in
the network nodes. The necessary and sufficient conditions for
the flow tree model to be a valid representation of the original
network graph can finally be grouped into single flow, and mul-
tiple flow constraints, and expressed as follows.
JURCA AND FROSSARD: MEDIA FLOW RATE ALLOCATION IN MULTIPATH NETWORKS 1229
Fig. 2. Equivalent transformation between a network graph and a tree of paths between the server and the client.
1) Single Flow Constraints:
• path bandwidth limitations:
, ;
•flow conservation at intermediate nodes: for every node
, , where and are the
incoming and respectively outgoing rates of
passing
through node
.
2) Multiple Flow Constraints:
• link bandwidth limitations:
•flow conservation at intermediate nodes: for every node
:
C. Media-Driven Quality of Service
The end-to-end distortion, as perceived by the media client,
can generally be computed as the sum of the source distortion,
and the channel distortion. In other words, the quality depends
on both the distortion due to a lossy encoding of the media infor-
mation, and the distortion due to losses experienced in the net-
work. The source distortion
is mostly driven by the source or
streaming rate
, and the media sequence content, whose char-
acteristics influence the performance of the encoder (e.g., for
the same bit rate, the more complex the sequence, the lower the
quality). The source distortion decays with increasing encoding
rate; the decay is quite steep for low bit rate values, but it be-
comes very slow at high bit rate. The channel distortion
is
dependent on the average loss probability
, and the sequence
characteristics. It is roughly proportional to the number of video
entities (e.g., frames) that cannot be decoded correctly, and an
increase in loss probability augments the channel distortion
.
Overall, the end-to-end distortion can thus be written as
(4)
where
represents the set of parameters that describe the
media sequence. This generic distortion model is quite com-
monly accepted, as it can accommodate a variety of streaming
scenarios. For example, when error correction is available, the
total streaming rate has to be split between the video source
rate that drives the source distortion
and the channel rate,
which directly influence the video loss rate
[9].
The total streaming rate
, and the end-to-end loss proba-
bility
directly depend on the path selection, and the flow rate
allocation. In the multipath scenario described before, the media
application uses rate allocation
, where the flow
rate
, with , represents the streaming rate on path
. The total media streaming rate is expressed as
(5)
The overall loss probability
experienced by the media appli-
cation can be computed as the average of the loss probabilities
of the
paths
(6)
It is important to note that increasing
with the addition of
a path reduces the source distortion. However, the addition of a
path generally impacts the loss probability
, and may augment
the channel distortion. The optimal flow rate allocation therefore
results from a trade-off between increase the streaming rate, and
controlling the end-to-end loss probability. Finally, since paths
may not be completely disjoint,
is a valid rate allocation on the
network graph
, if and only if can simultaneously accom-
modate the flow rates on all paths in
. A necessary condition
for the equality in the right side of (5) to be verified requires that
all bottleneck links of the
streaming paths are disjoint. Suffi-
cient conditions for valid rate allocation are analyzed in the next
section.
D. Multipath Rate Allocation: Problem Formulation
We consider the problem of the optimal routing and rate allo-
cation strategy, for a given video stream that can be split into
flows sent on different network paths between the streaming
server, and the media client. The rate constraints are directly
given by the network status, as shown before, and the overall
streaming rate can be adapted by simple operations at the server
(e.g., packet filtering). We can formulate the optimal multipath
rate allocation problem as follows.
Given a network graph
, the optimization problem consists
in jointly finding the optimal sending rate for a video packet
stream, along with the optimal subset of network paths to be
used for transmission, such that the end-to-end distortion is
minimized. Equivalently, using the flow tree representation of
the network graph proposed in Section II-B, the optimization
problem translates into finding the optimal rate allocation for
each of the flows in the tree, such that the video distortion is
minimized. It can be formulated as follows.
Multimedia Rate Allocation Problem (MMR): Given the
network graph
, the number of different paths or flows ,
the video sequence characteristics
, and the total streaming
1230 IEEE TRANSACTIONS ON MULTIMEDIA, VOL. 9, NO. 6, OCTOBER 2007
budget , find the optimal rate allocation
that minimizes the distortion metric :
(7)
where
and , under the
following constraints.
1) Budget Constraints:
.
2) Single Flow Constraints.
3) Multiple Flow Constraints.
In the next section, we present a detailed analysis of a typical
distortion model for video sequences. While the nonconvexity
of the optimization metric will not allow for an easy solution by
integration of the constraints into a Lagrangian formulation, our
analysis eventually allows us to define a simple algorithm, able
to find the optimal rate allocation with linear time complexity.
III. F
LOW RAT E
ALLOCATION ANALYSIS
A. End-to-End Distortion Model
We introduce in this section a quite generic distortion model,
which is able to capture the influence of the average encoding
rate on the source distortion, as well as the impact of losses on
the channel distortion. Recall that our objective is to find the
best flow rate allocation, on a multipath networks with known
average statistics. Hence, we are looking for an average distor-
tion model, which is able to estimate the video quality of service
in a stationary regime.
In low to medium bit rate video streaming, it is commonly
accepted that the source distortion is a decaying exponential
function on the encoding rate, while the channel distortion is
proportional to the number of lost packets (i.e., the packet loss
probability, when the number of packet per frame is indepen-
dent of the bit rate) [10]. Hence, we can explicitly formulate the
Mean-Square Error distortion metric as:
(8)
where
and are parameters that de-
pend on the video sequence. This distortion model is a simple
and general approximation that follows closely the behavior of
more sophisticated distortion measures, such as those proposed
in [11]–[13]. Since it is suitable for most common streaming
strategies where the number of packets per frame is independent
of the encoding rate, we use the model of (8) in the remainder of
that paper. It can be noted that our simple model does not take
into account the exact characteristics of the loss process, and that
it mostly captures the effect of independent losses. We assume
that bursts of losses on the video packet stream are quite unlikely
due to the partitioning in multiple flows. Simple interleaving can
also be applied to reduce the effects of bursts, if delay permits it.
Finally, we should stress out that bursts of video packets losses
are in general less penalizing for the channel distortion [14], so
that our model has the advantage to provide a worst case esti-
mate of the end-to-end distortion.
Fig. 3. Overall distortion measure for two network paths in function of avail-
able rates,
=1
:
76
1
10
,
=
0
0
:
658
,
= 1750
,
p
=0
:
02
, and
p
=0
:
04
.
Before going deeper in the analysis of flow rate allocation,
we propose a simple example to illustrate the behavior of the
end-to-end video distortion in a multipath scenario. We con-
sider a basic network scenario consisting of two disjoint network
paths,
and , with bandwidth , and
loss probabilities
and , respectively. Consider
two independent flows
and composing the same video
stream, and traversing the two network paths with streaming
rates
, and . The evolution of the distortion
function given in (8) is presented in Fig. 3, for a test video se-
quence (i.e., Foreman CIF).
As expected, we observe that the decrease in distortion is
larger if we increase the rate of flow
, than if we equivalently
increase the rate of flow
. This behavior is due to the lower
loss probability that affects the path followed by the flow
.In
the same time, we observe that the distortion metric is always
decreasing with the increase of
, hence it is optimal to fully
utilize the bandwidth of the path with the smallest loss prob-
ability. In this case, for a given packet loss rate, it is better to
increase the quality of each video frame by augmenting the rate
, as expected.
More interestingly, Fig. 4 shows that the behavior of the dis-
tortion as a function of the rate
, depends on the value of the
rate
. For high values of , the distortion can even increase
with growing rate
. Beyond a given value of the streaming rate
on the most reliable network path, adding an extra flow can de-
grade the end-to-end quality of the media application since the
packet loss rate increases. In this case, the negative influence of
the error process on the second network path is greater than the
improvement brought by additional streaming rate. Such a be-
havior is the key to explain why using all the paths to their full
bandwidth does not necessarily result in an efficient streaming
strategy. Finally, the same type of behavior can be observed for
stored video packet streams that are built on video packets, and
error control packets (e.g., Forward Error Correction). In this
case, the sensitivity of the channel distortion is obviously lower
for low error rates, but rapidly increases when the channel pro-
tection becomes insufficient.
JURCA AND FROSSARD: MEDIA FLOW RATE ALLOCATION IN MULTIPATH NETWORKS 1231
Fig. 4. Overall distortion behavior as a function of
r
, for various fixed values
of
r
.
B. Maximum or Null Flows
We now generalize the previous observations, and derive the-
orems that guide the design of an optimal rate allocation strategy
for a given video packet stream. This section shows that, in the
optimal rate allocation, a flow is either used at its full bandwidth,
or not used at all. Furthermore, the optimal rate allocation al-
ways chooses the lowest loss probability paths, i.e., a path shall
not be selected, unless all other paths with a lower loss proba-
bility have been picked before. We start from an ideal streaming
scenario with unlimited budget and disjoint network paths, and
eventually add budget and flow constraints, which are however
shown not to affect the initial findings.
Assume that the
disjoint network paths are represented into
a tree of flows as explained in Section II-B. Without loss of gen-
erality, we further assume that flows
with , are
arranged in increasing order of the loss probability, i.e.,
. We note that, from the distortion metric point of
view, any two flows
and , with rates and and traversing
paths
and with the same loss probability , can be
observed as a single flow affected by the same loss probability
, and having an aggregated rate . Under these generic
settings, we first claim that the optimal rate allocation either uses
a network path to its full bandwidth, or does not use it at all.
Theorem 1 (On-Off Flows): Given a flow tree with indepen-
dent flows
having rates and a distortion metric as
defined in (8), the optimal solution of the MMR problem when
all the paths are disjoint, lies at the margins of the value in-
tervals for all
, i.e., the optimal value of is either 0 or ,
.
Proof: Deriving the distortion
given in (8), with respect
to the rate
, , we obtain
Observe that the condition for an extremum,
for any , implies:
where and are constants independent on . Since
, the equation has a single finite solution
In the same time, the derivative in any point is posi-
tive, while to the right of the optimal value, it is negative (since
, and all other terms are positive). Hence, is a
point of local maximum for the distortion function
, which
means that only values at the margins of the value interval for
can minimize the objective function.
1
It can be further observed that, in the case of , ,
for any positive value of
(since , and
, ). Hence the value always minimizes
the objective function, and is part of the optimal solution.
Corollary 1: Given a flow tree with independent flows
having rates and a distortion metric as defined in (8),
the optimal solution of the MMR problem when all paths are
disjoint, allocates
, where the path is the path with
the lowest loss probability.
Theorem 1 greatly reduces the search space for an optimal so-
lution to the MMR optimization problem. Hence we can rewrite
the optimal streaming solution as a vector
of boolean values
for each flow , where means that path is used
with full rate
, and denotes the fact that the path
is not used by the streaming application. The previous corol-
lary further says that
is part of the
optimal solution.
For bounded intervals for all rates
, computations are
sufficient for finding the optimal solution vector. For practical
scenarios, with a limited number of available network paths be-
tween a server and a client, this number of computations is in
general quite low. We can however further constrain the search
space by considering that the optimal rate allocation always uses
first the network paths with the smallest loss probabilities.
Theorem 2 (Parameter Decoupling): Given a flow tree with
independent flows
having rates and a distortion
metric as defined in (8), the structure of the optimal rate alloca-
tion is
.
Proof: We prove the result by induction. Recall that the
network paths/flows are arranged in increasing order of their
loss probabilities
. We have already seen that
is part of the optimal solution. Next we show that,
for
, cannot
be part of the optimal solution.
For the sake of clarity, let us remove
’s with from
the notation, since they stay constant in our proof. By contra-
diction, assume that
is part of the optimal solution. It means
that
. Since the paths are ordered
with increasing values of the loss probabilities and considered
to be disjoint, we can always transfer part of the rate from
1
Since
r
is the only finite solution, this statement is valid even if
r
is not
contained in
[0
;b
]
.
