Distributed Robust Optimization for Scalable Video Multirate Multicast Over Wireless Networks
Chenglin Li, Hongkai Xiong,Senior Member, IEEE, Junni Zou,Member, IEEE, and Chang Wen Chen, Fellow, IEEE
Abstract—This paper proposes a distributed robust optimiza- tion scheme to jointly optimize overall video quality and traffic performance for scalable video multirate multicast over prac- tical wireless networks. In order to guarantee layered utility maximization, the initial nominal joint source and network optimization is defined, where each scalable layer is tailored in an incremental order and finds jointly optimal multicast paths and associated rates with network coding. To enhance the robustness of the nominal convex optimization formulation with nonlinear constraints, we reserve partial bandwidth for backup paths disjoint from the primal paths. It considers the path- overlapping allocation of backup paths for different receivers to take advantage of network coding, and takes into account the robust multipath rate-control and bandwidth reservation problem for scalable video multicast streaming when possible link failures of primary paths exist. Specifically, an uncertainty set of the wireless medium capacity is introduced to represent the uncertain and time-varying property of parameters related to the wireless channel. The targeted uncertainty in the robust optimization problem is studied in a form of protection functions with nonlinear constraints, to analyze the tradeoff between robustness and distributedness. Using the dual decomposition and primal-dual update approach, we develop a fully decentralized algorithm with regard to communication overhead. Through extensive experimental results under critical performance factors, the proposed algorithm could converge to the optimal steady- state more quickly, and adapt the dynamic network changes in an optimal tradeoff between optimization performance and robustness than existing optimization schemes.
Index Terms—Multirate multicast, network coding, rate-distortion, robust optimization, scalable video coding.
I. Introduction
V
IDEO STREAMING over wireless networks has been compelling for a wide range of applications, fromManuscript received April 3, 2011; revised October 17, 2011; accepted November 17, 2011. Date of publication February 3, 2012; date of current ver- sion May 31, 2012. This work was supported in part by the National Natural Science Foundation of China, under Grants 60802019, 60928003, 60772099, and 60632040, in part by the Program for New Century Excellent Talents in University, under Grant NCET-09-0554, and in part by the Shanghai Rising- Star Program, under Grant 11QA1402600. This paper was recommended by Associate Editor S. Battiato.
C. Li and H. Xiong are with the Department of Electronic Engi- neering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail:
lcl1985@sjtu.edu.cn; xionghongkai@sjtu.edu.cn).
J. Zou is with the School of Communication and Information Engineering, Shanghai University, Shanghai 200072, China (e-mail: zoujn@shu.edu.cn).
C. W. Chen is with the Department of Computer Science and Engi- neering, State University of New York, Buffalo, NY 14260 USA (e-mail:
chencw@buffalo.edu).
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/TCSVT.2012.2186740
home entertainment and video surveillance to audiovisual communication [1]. To benefit the overall network utility, scalable video coding (SVC) is developed to adapt different user requirements and heterogeneous network conditions in a multirate multicast mode. SVC can allow rate adaptation not only at the encoder/decoder but also at the intermediate network nodes while achieving highly efficient rate-distortion (RD) performance [2]. An SVC stream comprising a base layer and one or multiple enhancement layers, with a flexible multidimension layer structure, can provide various operating points in spatial resolution, temporal frame rate, and video re- construction quality. For multirate multicasting, different SVC layers are transported in different Internet protocol (IP) mul- ticast groups and are subscribed by heterogeneous receivers with different computation and communication resources and capabilities. In communication networks, on the other hand, it has been demonstrated that coded network can achieve the capacity in single-source multiple-terminal multicast via algebraic operation at intermediate nodes [3]. Also, recent research has shown that network coding can significantly improve the network throughput and robustness to link or node failures [4] and packet losses [5], [6]. Distributed random linear coding schemes [7], [8] have been proposed for practical implementation of network coding. Chen et al. [9] developed adaptive rate-control algorithms for networks with and without coding subgraphs.
When both SVC and network coding are taken into account, layered multirate multicasting is equivalent to a generalized multisource problem where the progressive interlayer depen- dency is considered as fairness between different sources (layers) [10], and also network coding is implemented at relay nodes to enhance the communication efficiency of the network.
In practical wireless networks, dynamic network changes (e.g., fluctuations of links or link failures) might occur due to wire- less channels and mobility in wireless nodes [11]. Therefore, it is imperative to formulate the corresponding rate-control scheme in a robust way, to ensure service feasibility and availability against such uncertainties. Furthermore, the lack of centralized control in a wireless network requires that the scheme be performed in a distributed manner. In this paper, we aim to develop a robust and efficient optimization framework for scalable video streaming over wireless networks, where the layered SVC stream is generated at the source node, and distributed via network coding by relay nodes to different users through the wireless network. The objective is to maximize the overall video quality of all receivers with dynamic network changes.
1051-8215/$31.00 c 2012 IEEE
The contribution of this paper is twofold. First, we are motivated to make a robust formulation of jointly optimizing overall video quality and traffic performance for scalable video multirate multicast over practical wireless networks. To guarantee layered utility maximization, we define a nomi- nal joint source and network flow optimization where each scalable layer is tailored in an incremental order and finds jointly optimal multicast paths and associated rates. At relay nodes, network coding is utilized for a decent multicast capacity. To enhance the robustness of the nominal convex optimization problem with nonlinear constraints, we reserve partial bandwidth for backup paths disjoint from the primal paths during the transmission of SVC multicast stream. It considers the path-overlapping allocation of backup paths for different receivers to take advantage of network coding, and takes into account the robust multipath rate-control and band- width reservation problem for SVC multicast streaming when possible link failures of primary paths exist. Impressively, an uncertainty set of the wireless medium capacity is introduced to represent the uncertain and time-varying property of param- eters related to the wireless channel. The targeted uncertainty in the robust optimization problem is studied in a form of protection functions with nonlinear constraints, to analyze the tradeoff between robustness and distributedness. Second, using dual decomposition and the primal-dual update approach, we develop a fully decentralized algorithm where the globally optimal solution is achievable via distributed computation, by iteratively updating optimization variables. Extensive numeri- cal and packet-level experimental results under critical factors demonstrate that the proposed algorithm (PA) can converge to the optimal steady-state more quickly, and adapt the dynamic network changes in an optimal tradeoff between optimization performance and robustness than existing schemes.
The remainder of this paper is organized as follows. Sec- tion II reviews related research on rate allocation schemes of scalable video streaming, as well as robust optimization frameworks. The notations, system models, and nominal flow control and resource allocation problem for SVC-based multi- rate multicasting with network coding-based routing over wire- less networks is formulated in Section III. In Section IV, the robust problem formulation is presented, with consideration of uncertain factors in practical wireless networks. The fully distributed algorithm for the robust optimization formulation is proposed in Section V, where we also provide an efficient implementation scheme. Extensive results of both numerical experiments and packet-level simulations are presented in Section VI. Section VII concludes this paper.
II. Related Work
Over the past years, a few of rate-control schemes have been developed in literature to address the scalable video streaming problem over networks [10], [12]–[14]. The dis- tributed rate allocation scheme in [10] addressed the problem of rate allocation for multiple SVC multicast sessions over wireless mesh networks, with the goal of minimizing total video distortion of all peers. To incorporate network coding as well as interlayer dependency into layered overlay multi-
cast over networks, Zhao et al. [12] proposed a distributed heuristic algorithm with suboptimal performance. Recently, Zou et al. [13] explored the path cost and price of each layer as the priority parameters to capture interlayer dependency and developed a rigorous distributed algorithm proven to be stable and convergent. As a further improvement, a joint source and flow optimization scheme with mathematically rigorous layer dependency constraints has been shown the best overall performance for heterogeneous receivers via a fully distributed implementation supporting partial layer reception in [14]. This kind of optimization problems, although can achieve optimal or near optimal performance, would make the unrealistic assumption that data defining the optimization formulation can be obtained precisely, which may lead to poor or even infeasible solution in practical dynamic networks.
Hereinafter, we call the corresponding optimization problems with deterministic parameters “nominal” problems.
As a framework of tackling optimization problems under data uncertainty, a couple of robust optimization schemes have emerged to seek a solution that remains feasible and near optimal under the fluctuation of parameters in the optimization formulation [15]– [18]. Each robust optimization problem is defined by three-tuple: a nominal formulation, a definition of robustness, and a representation of the uncertainty set. The process of making an optimization formulation robust can be viewed as a mapping from one optimization problem to another. In [15], it was demonstrated that optimal solutions of linear programming (LP) problems may become severely infeasible if the nominal data is slightly perturbed, also robust solutions of the above LPs which were in a sense immune against uncertainty were developed. A D-norm approach [16]
has been proposed to model the data uncertainty, and has advantages such as guarantee of feasibility independent of uncertainty distributions and flexibility in tradeoff between robustness and performance. Yang et al. [17], [18] discussed several efficient models for describing parameter uncertainty sets that can lead to decomposable problem structures, and applied these models in distributively solving a robust flow control problem in wireline networks. These work mainly fo- cus on the robust formulation of linear optimization problems and is thus not suitable for complicated video application.
III. System Models and the Nominal Optimization Problem Formulation
In this section, related system models are addressed. As a fundamental step for optimization formulation of robust SVC multirate multicast over wireless networks, in the sequel, we formulate the nominal optimization problem.
A. Wireless Network Model
Consider video content distribution over a wireless network G= (V, E) with a set of wireless links E and a set of wireless nodes V = {s} ∪ N ∪ D, where s denotes the source node, and N and D represent the sets of relay nodes and destination nodes, respectively. The SVC-coded stream is encoded at the source s and then multicasted to destination nodes through a wireless network with network coding-based routing.
In wireless networks, the capacity of a wireless link is interrelated with other adjacent wireless links. Consequently, we should consider the wireless link contention in a shared transmission medium by introducing constraints of the location-dependent contention among the competing wireless data flows [19]. In the target problem formulation, the assumption is that the wireless medium capacity is shared among a wireless link l and the cluster of its competing links.
A typical protocol model considers the spatial locations of the nodes and determines that transmission can be successfully received by its intended recipient [20]. It hypothesizes that any link originating from node k will interfere with link (i, j) if the link distance d(k,j) < (1 + γ)d(i,j), γ ≥ 0 and define
(i,j)for each link (i, j)∈ E as the cluster of links that cannot transmit when link (i, j) is active. As compared to individual links in traditional wired network, the notation of cluster can be treated as a basic resource unit. Wireless data flows compete for the capacity of individual cluster that is equivalent to the capacity of the wireless shared medium. Hence, the wireless network channel capacity constraint [21] is
0≤ f(i,j) 1− ρ(i,j)
+
(p,q)∈(i,j)
f(p,q) 1− ρ(p,q)
≤ C(i,j),∀(i, j) ∈ E (1)
where f(i,j) denotes the physical flow rate that is required to transmit through link (i, j), C(i,j) is defined as the maximum rate of link (i, j) and its corresponding cluster (i,j)supported by the wireless shared medium, and ρ(i,j)is assumed to be the packet loss probability at wireless link (i, j). Theoretically, this packet loss rate (PLR) can be derived from the Gilbert–Elliott model [22], [23].
B. SVC Coding Model
The layered scalability is adopted here, and assume that the SVC video stream is encoded into a set of M layers {L1, L2, . . . , LM} with a predefined encoding rate based on the network condition. According to the encoding rates of layers, we can make the optimal adaptation decision in the scalability cube model illustrated in Fig. 1 by mapping from an SVC elementary stream with fully scalable representation into the layered representation. Correspondingly, the multicast of SVC video stream is divided into M multicast sessions.
Each multicast session m has one source node s, a set of destination nodes D, and a set of relay nodes N. In order to successfully decode received SVC video streams, we should make sure that all destination nodes are able to subscribe to SVC multicast layers in an incremental order, since layer m+ 1 is not decodable without its previous layers 1 to m.
In accordance with the SVC layer dependency constraint, lower layers need be received before higher layers for each destination node to promise the most efficient SVC decoding.
Considering both fluctuating network adaptation and optimization condition [27], [28], each layer is distributed over a multicast session at a variable transmission rate within a tolerable rate region [rminm , rmmax]. Mathematically, the upper bound rmmax(e.g., the encoding rate with a resilient margin) and the lower bound rmmin (e.g., the minimum partial margin for layer m) are specified for a confidence interval of the layered transmission rate in layer m. It differentiates the layers with
Fig. 1. Typical structure of scalable video bitstream with multiple dimen- sions.
the piecewise confidence intervals along the layer-dependent direction, namely, the achievable transmission rate for each layer is mathematically extended from an encoding rate point to a tolerable rate region. From the layered optimization, the fine-granular continuity of the targeted variables (rates) could specifically urge the convexity of optimization problem for developing a distributed solution, and support a strong notion of fairness.
C. Network Coding Model
To transmit multiple multicast sessions over a shared net- work, we might perform network coding across session to achieve the optimal throughput. However, combining data belonging to different layers makes it difficult to recover all original data for destination nodes that only receive partial layers. Thus, network coding is limited within each session in this paper. This approach is often referred to as intra-session coding or superposition coding [26].
For each multicast session, we find multiple paths from the source node to destination nodes using existing multipath routing schemes [29], [30], which are chosen based on general cost criteria that are independent of flow rates. Since each multicast session uses only a limited set of paths, it is shown that such approach may give lower rates compared with optimizing over the entire network, but it is much less complex [9]. For each destination node d ∈ D, we use a matrix Hd = {hldj} to represent the relationship between its transmission paths and corresponding links. More specifically, suppose destination node d has J (d) alternative paths from source node s, then hldj= 1 if the path j of node d uses link l, and hldj= 0 otherwise. In network coding-based routing, let Rmdj denote the information flow rate of destination node ds jth path in multicast session m, and flmrepresent the physical flow rate for link l in multicast session m.
With intra-session network coding, flows to different des- tinations of a multicast session are allowed to share network capacity by being coded together. Therefore, it is only required to set the actual physical flow on each link to be the maximum of the individual destinations’ information flows, and accord- ingly the network coding constraint is
J(d) j=1
hldjRmdj≤ flm,∀m ∈ M, ∀l ∈ E, ∀d ∈ D. (2)
D. RD Model
From the perspective of application-layer quality of service, RD-related models [31] could be picked as the optimized
targeted utility for video applications as follows:
De(Re) = θ Re− R0
+ D0 (3)
where De is the distortion of the encoded video sequence and Re is the encoded rate. The remaining variables θ, R0, and D0 are the parameters of the RD model, which can be fitted to empirical data from trial encodings using nonlinear regression techniques.
To characterize the video streaming performance of each layer m, we introduce a utility function Um(·), which is continuously differentiable, increasing, and strictly concave with respect to the receiving rate. In this paper, we multicast the video streams to all destination nodes and attempt to maximize the total utility of all recipients, and accordingly the objective function is
max
d∈D
m∈M
Um(Rmd) = max
d∈D
m∈M
Um
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠ (4)
where Rmd denotes the received rate at destination node d in multicast session m.
Within the context of SVC, the utility function Um(Rmd) is defined for destination node d to represent the corresponding distortion decrement when a new layer m is successfully received and decoded as follows:
Um
Rmd
=−
De
m−1
i=0
Rid+ Rmd
− De
m−1
i=0
Rid
. (5)
E. Nominal Optimization Problem Formulation
In SVC streaming problems, layered multirate multicasting is equivalent to a generalized multisource problem where the interlayer dependency is considered. The proposed nominal optimization problem integrates the prior context of source de- composition into the layered multirate multicast optimization, which can be formulated as follows:
P1 : max
R
d∈D
m∈M
Um
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠ (6)
s.t.
1)
J(d)
j=1
hldjRmdj≤ flm;∀m ∈ M, ∀l ∈ E, ∀d ∈ D
2)
m∈M
flm
(1− ρl)+
k∈(l)
m∈M
fkm
(1− ρk) ≤ Cl;∀l ∈ E
3) rminm ≤
J(d)
j=1
Rmdj≤ rmmax, or
J(d) j=1
Rmdj= 0;∀m ∈ M, ∀d ∈ D
4)
J(d)
j=1
Rmdj
rmmin ≥
J(d) j=1
R(m+1)dj
r(m+1)max
;∀m ∈ {1, 2, . . . , M − 1}, ∀d ∈ D
5) Rmdj≥ 0; ∀j ∈ J(d), ∀m ∈ M, ∀d ∈ D 6) flm≥ 0; ∀l ∈ E, ∀m ∈ M.
Constraint 1) specifies the required physical flow rate on each link for each layer under the network coding condition.
With network coding, different destinations will not compete for link bandwidth within the same layer, therefore the re- quired physical flow rate on link l for layer m is the largest information flow rate on link l consumed among all destination nodes. Note that the impact of network coding is embedded in this constraint. Constraint 2) characterizes the wireless link contention in a shared medium. For each wireless link l, the sum of ls actual physical flow rate and the actual physical flow rates of links in (l) cannot exceed the maximum rate Cl. Constraint 3) gives the lower bound and upper bound of the transmission rates allocated for layer m, denoted by rminm , and rmmax, respectively. The SVC layer dependency constraint is taken into account and promised by Constraints 3) and 4), according to [14, Prop. 1]. Constraints 5) and 6) specify that the allocated rates and required physical flow rates are nonnegative.
To make sure the convexity of the proposed optimization problem P1, Constraint 3) needs to be redefined to meet the convexity requirement. Based on the nonnegativity Constraint 5), it can directly imply J(d)
j=1 Rmdj≥ 0 from Rmdj≥ 0. Hereby, we can simply extend the second equality term J(d)
j=1 Rmdj = 0 in Constraint 3) to J(d)
j=1 Rmdj ≤ 0 because J(d)
j=1 Rmdj = 0 can be promised along with the nonnegativity Constraint 5).
Therefore, Constraint 3) is formalized as rminm ≤ J(d)
j=1 Rmdj ≤ rmaxm , or J(d)
j=1 Rmdj ≤ 0, and further simplified as a cubic inequality ( J(d)
j=1 Rmdj)( J(d)
j=1 Rmdj−rmmin)( J(d)
j=1 Rmdj−rmaxm )≤ 0.
We can see that the nominal optimization problem P1 has a unique optimal solution since its objective function is strictly concave and the solution space defined by the constraints is convex. In other words, this is a convex optimization problem with either centralized or distributed feasible solutions under the assumption that the input parameters are precisely known and equal to some nominal values. This approach, however, does not take into account the influence of data uncertainties on the quality and feasibility of the practical problem. It is therefore noticeable that as the parameters take values different from the nominal ones, several constraints may be violated, which may further lead the optimal solutions obtained by the nominal problem to no longer optimal or even infeasible ones.
To tackle the optimization problems under data uncertainty in practice, in the following sections, we will extend the nominal problem with deterministic parameters into a robust optimization problem that is still feasible in practical wireless networks, and accordingly develop a distributed solution to the proposed robust optimization problem.
IV. Robust Multipath Rate Control for SVC Multicast Streaming
In this section, we consider a practical wireless network where some link failures might occur due to wireless channels and mobility in wireless nodes [11]. To enhance the robustness
of the wireless communication network, an efficient way is to reserve partial bandwidth for backup paths disjoint from the primal paths during the transmission of SVC multicast stream, such that a partial or all of the traffic would be rerouted to the corresponding disjoint backup paths when the primary paths fail. Here we consider the robust multipath rate-control and bandwidth reservation problem for SVC multicast streaming when possible link failures of primary paths exist, to ensure service feasibility and availability in the presence of link failures. In the meanwhile, an uncertainty set of the wireless medium capacity is introduced to represent the uncertain and time-varying property of parameters related to the wireless channel, in accordance with the fluctuation and perturbation in practical wireless networks.
A. Preliminaries to Distributed Robust Optimization
According to the form of the nominal optimization problem P1, we focus on a class of optimization problems with the following nominal form: maximization of a concave objective function over a given data set characterized by linear con- straints as follows:
maxx U(x) (7)
s.t. Ax c
where A is an M× N matrix, x is an N × 1 vector, c is an M×1 vector, and Ax c denotes that c−Ax is componentwise nonnegative.
The uncertainty of (7) may exist in the objective function U(x), the matrix parameter A, and the vector parameter c. It is demonstrated by [16] that the uncertainty in the objective func- tion U(x) can be converted into uncertainty of the parameters defining the constraints. In the next section, we show that it is also possible to convert the uncertainty in c into uncertainty in Ain certain cases. In the remainder of this section, therefore, we focus on studying the uncertainty in the matrix parameter A, the structures and physical meaning of which can readily lead to distributed algorithms in many networking problems.
Furthermore, quantifying the tradeoff between robustness and distributedness is also studied in this paper.
In the robust counterpart of (7), the constraint Ax c is required to be valid for any A ∈ A, where A denotes the uncertainty set of A. In this case, the definition of robustness is the worst-case robustness [34], since the solution of the robust optimization problem is always feasible. However, this uncer- tainty definition may be too conservative. A more meaningful choice of robustness is the chance-constrained robustness, where the probability of infeasibility is upper bounded. By solving the worst-case robust optimization problem over a properly selected subset of the exact uncertainty set, the chance-constrained robustness of the robust solution can be flexibly adjusted.
IfA is assumed to be an arbitrary uncertainty set, the robust optimization problem is difficult to solve even in a centralized manner [15]. Therefore, we focus on the study of constraint- wise (row-wise) uncertainty set, where the uncertainties be- tween different rows in matrix parameter A are decoupled.
Though restricted, this type of uncertainty set characterizes the data uncertainty in many practical problems, and also leads the robust optimization problem to a distributively solvable formulation.
To introduce uncertainty in the matrix parameter A, denote the jth row of A by aTj, which lies in a compact uncertainty setAjT. Then the robust optimization formulation of (7) can be written as follows:
maxx U(x) (8)
s.t. aTjx≤ cj; ∀aj ∈ Aj, ∀1 ≤ j ≤ M.
It is shown by [17] that the uncertainty sets in robust optimization problem (8) can be equivalently written in the form of protection functions. Denote the matrix parameter in the nominal counterpart of (7) as a coefficient matrix ¯Awith the jth row’s coefficient ¯aj ∈ Aj when there is no uncertainty, we have the following proposition as proven in [17].
Proposition 1: Problem (8) is equivalent to the following convex optimization problem:
maxx U(x) (9)
s.t. ¯aTjx+ gj(x)≤ cj; ∀1 ≤ j ≤ M where
gj(x) = sup
aj∈Aj
(aj− ¯aj)Tx (10) is the protection function for the jth constraint depending on both the uncertainty set Aj and the nominal row coefficient
¯aj. Each gj is a convex function.
B. Robust Formulation
As the extension of the nominal problem with deterministic parameters as proposed in Section III, the robust formulation of multipath rate-control and bandwidth reservation problem for SVC multicast streaming is taken into account. First, to ensure robust data transmission against the wireless link failures and thus enhance the robustness of the wireless com- munication network, each destination node d also determines a backup path when it joins the network, in addition to the set of J (d) primary paths. To take advantage of network coding, the path-overlapping allocation of backup paths for different receivers is considered. The nonnegative backup path choice matrix is denoted as
Bd={bdl},
bld∈ [0, Bd], link l is on ds backup path bld= 0, otherwise
where Bd >0 indicates the maximum percentage that desti- nation node d allocates its rate to the backup path. The actual rate percentage of backup link bld could be a random variable between 0 and Bd, depending on whether the primary paths fail. Furthermore, we assume that a path can only be selected as either a primary path or a backup path but not both for the same destination node.
Second, due to the fluctuation and perturbation in practical wireless networks, the rate of each wireless link and its corre- sponding cluster supported by the wireless shared medium are
time-varying and cannot achieve the maximum rate Cl most of the time. To represent the uncertain property of parameters related to the wireless channel, we introduce a perturbation factor δ ∈ [, 1], which indicates the instant percentage of maximum rate Cl that can be practically achieved by the wireless channel. Therefore, the actual rate of wireless link l and its corresponding cluster (l) fluctuates between the lower bound Cl and the upper bound Cl. Accordingly, the robust multipath routing rate allocation problem for SVC multicast streaming is given by
P2 : max
R
d∈D
m∈M
Um
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠
(11) s.t.
1)
J(d) j=1
hldjRmdj+ gml (Bv, R)≤ flm;∀m ∈ M, ∀l ∈ E, ∀d ∈ D
2)
m∈M
flm
(1− ρl) +
k∈(l)
m∈M
fkm
(1− ρk)≤ δCl;∀δ ∈ [, 1],
∀l ∈ Eand 3), 4), 5), 6) in P1
where gml (Bv, R) is the protection function for the traffic from a set of destination nodes {v} ⊆ D whose backup paths use link l.
Stipulating that nature will be restricted in such behavior that only a subset of destination nodes will suffer from link or path failures in order to adversely affect the solution, we propose a method similar to the D-norm approach [16], [35] to model the protection function gml (Bv, R). Denote the notation D as the subset of destination nodes that might experience link or path failures, and represents the number of destination nodes within this subset, i.e., D ⊆ D and
|D| = . It is noted that the definition of robustness associated with the D-norm formulation is to maintain feasibility if at most out of all possible |D| destination nodes experience link or path failures. Considering network coding condition that different destination nodes will not compete for link bandwidth within the same layer, the protection function can be written as
gml (Bv, R) = max
D⊆D
⎧⎨
⎩max
v∈D
blv(
J(v) j=1
Rmvj)
⎫⎬
⎭. (12)
In (12), is a parameter instead of a optimization variable. If
= 0, then gml (Bv, R) = 0, and the corresponding constraint is reduced to the nominal constraint. On the contrary, if =|D|, then gml (Bv, R) = maxv∈Dblv( J(v)
j=1 Rmvj) and the corresponding constraint becomes the worst-case formulation [16]. Therefore, the value of theoretically takes control of the tradeoff between robustness and performance.
Centralized algorithms can be used to solve the convex optimization problem P2. In practice, however, such solutions require global information and coordination between all nodes and links, which is very costly and sometimes infeasible [34],
[36]. In comparison, distributed algorithms can be used to dynamically adjust the rates in accordance with changes in the network. In the following section, we will develop a distributed solution of the proposed multipath rate-control problem. Since robustness, i.e., service reliability, is taken into account in this application, it is most likely that such a change and corresponding update are only required infrequently.
V. Distributed Algorithm
In this section, we develop a distributed solution to the pro- posed robust optimization problem P2 that allows each node and link to control and update the transmission parameters by itself.
A. Equivalent Formulation
Observing P2, it is uncertain that Constraints 1) and 2) prevent the robust optimization problem from a fully dis- tributed solution. We first show that the perturbation factor δin Constraint 2) can be moved into the left part and thus the uncertain Constraint 2) is converted to a linear constraint with certain parameters.
Proposition 2: For any link l∈ E, the constraint
m∈M
flm
(1− ρl)+
k∈(l)
m∈M
fkm
(1− ρk) ≤ δCl,∀δ ∈ [, 1] (13) is equivalent to the following constraint:
m∈M
flm
(1− ρl)+
k∈(l)
m∈M
fkm
(1− ρk)≤ Cl. (14) Proof: See Appendix A.
As for Constraint 1), it is shown by Proposition 3 that this nonlinear constraint with the protection function gml(Bv, R) can be replaced by a set of linear constraints.
Proposition 3: For any link l ∈ E, any destination node d ∈ D, and any layer m ∈ M, the single constraint
J(d)
j=1
hldjRmdj+ gml (Bv, R)≤ flm (15)
is equivalent to the following set of constraints:
J(d)
j=1
hldjRmdj+ blv
⎛
⎝J(v)
j=1
Rmvj
⎞
⎠ ≤ flm,∀v ∈ D,∀D⊆ D. (16)
Proof: See Appendix B.
Note that the number of constraints in (16) is |D|
· , and increases quickly with |D| and . However, due to the spe- cial structure of the protection function that utilizes network coding, most of the constraints in (16) should be inactive and thus redundant. This motivates us to formulate the protection function gml (Bv, R) into another set of linear constraints, the number of which is much less.
Proposition 4: For any link l ∈ E, any destination node d ∈ D, and any layer m ∈ M, the single constraint (15) is equivalent to the following set of constraints:
J(d)
j=1
hldjRmdj+ blv
⎛
⎝J(v)
j=1
Rmvj
⎞
⎠ ≤ flm,∀v ∈ D. (17)
Proof: See Appendix C.
It can be observed that the set of constraints (17) no longer relies onDand . Propositions 2 and 4 together transform the robust optimization problem P2 into a simplified formulation P3 independent of uncertainty δ, , andD as follows:
P3 : max
R
d∈D
m∈M
Um
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠ (18)
s.t.
1)
J(d)
j=1
hldjRmdj+blv
⎛
⎝J(v)
j=1
Rmvj
⎞
⎠ ≤flm;∀v, d∈D, ∀m∈M, ∀l ∈ E
2)
m∈M
flm
(1− ρl)+
k∈(l)
m∈M
fkm
(1− ρk) ≤ Cl;∀l ∈ E and 3), 4), 5), 6) in P1.
To analyze the impact of the protection parameters, Bd
denotes the maximum backup percentage of destination node ds total rate on the backup path. If Bd = 0, then there is no rate allocated for the backup paths, and Constraint 1) in P3 is reduced to the nominal constraint. As Bd increases from 0 to 1, the protection rate of backup path becomes larger, which may cause the overall allocated rates for destination nodes to decrease. Similarly, reflects the extent of fluctuation of wireless links’ maximum transmission rate. When = 1, the wireless channel condition is determined with known maximum transmission rate, and Constraint 2) in P3 is reduced to the nominal constraint. As decreases, the protection level for wireless channel’s fluctuation becomes greater, and accordingly the overall allocated rates for destination nodes may decrease. Therefore, Bd and can control the tradeoff between robustness and performance in P3.
B. Dual Decomposition
Decomposition theories provide a mathematical foundation for the design of modularized and distributed control of net- works [36]. The decomposition procedure aims to decompose a large and complex optimization problem into a set of small subproblems, which can be then solved by distributed and often iterative algorithms that converge to the global optimum.
It is observed that problem P3 can be decoupled using dual decomposition. More specifically, by relaxing the coupling Constraints 1)–4) in P3 with Lagrange multipliers λ, μ, η, and
Fig. 2. Schematic diagram of dual decomposition.
θ, respectively, the Lagrangian of P3 is obtained as follows:
L(R, f, λ, μ, η, θ) =
d∈D
m∈M
Um
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠
−
l∈E
d∈D
m∈M
v∈D
λmld(v)
⎡
⎣J(d)
j=1
hldjRmdj+ blv
⎛
⎝J(v)
j=1
Rmvj
⎞
⎠ − flm
⎤
⎦
−
l∈E
μl
m∈M
flm
(1− ρl) +
k∈(l)
m∈M
fkm
(1− ρk) − Cl
−
d∈D
m∈M
ηmd
⎡
⎣
⎛
⎝J(d)
j=1
Rmdj
⎞
⎠
⎛
⎝J(d)
j=1
Rmdj− rminm
⎞
⎠
⎛
⎝J(d)
j=1
Rmdj− rmmax
⎞
⎠
⎤
⎦
−
d∈D M−1
m=1
θdm
⎡
⎢⎢
⎢⎢
⎢⎣
J(d) j=1
R(m+1)dj r(m+1)max
−
J(d) j=1
Rmdj rminm
⎤
⎥⎥
⎥⎥
⎥⎦
(19)
and the corresponding Lagrange dual function is g(λ, μ, η, θ) = sup
R 0, f 0L(R, f, λ, μ, η, θ). (20) Then the Lagrange dual problem of P3 can be formulated as
λ 0, μ 0, η 0, θ 0min g(λ, μ, η, θ). (21) According to convex optimization theories [36], [37], if the original problem P3 is convex, it is equivalent to its Lagrange dual problem in (21). Then, P3 can be decomposed into a master dual problem P3a and a set of subproblems P3b–P3c that can be solved in a distributed manner as follows:
P3a : min
λ,μ,η,θ g(λ, μ, η, θ) (22)
s.t. λ 0, μ 0, η 0, θ 0 P3b : max
R Ub(R) (23)
s.t. Rmdj≥ 0; ∀j ∈ J(d), ∀m ∈ M, ∀d ∈ D P3c : max
f Uc(f) (24)
s.t. flm≥ 0; ∀l ∈ E, ∀m ∈ M
where L(R, f, λ, μ, η, θ) = Ub(R) + Uc(f), Ub(R) and Uc(f) denote the summation of terms relating to R and f in (19), respectively.
As illustrated by the relationship in Fig. 2, subproblems P3b and P3c are controlled by the master dual problem
P3a through Lagrange prices. In the meanwhile, they can conversely affect the master dual problem through primal variables. At the lower level, subproblems P3b for each d, j, and m, and P3c for each l and m can be solved separately.
Accordingly, the primal variables R and f are updated, re- spectively, and then sent to the master dual problem P3a. At the higher level, the master dual problem P3a aims to update Lagrange prices (dual variables) λ, μ, η, and θ, which are further used for the solution of P3b and P3c.
Since the objective functions of the master dual problem P3a and subproblems P3b–P3c are differentiable with respect to the dual variables λ, μ, η, and θ, and primal variables R and f, all problems can be solved by the gradient algorithm [38], [39]. Based on this observation, we propose the following primal-dual algorithm that updates the primal and dual vari- ables simultaneously to solve the optimization problem P3:
Rmdj(t + 1) =
Rmdj(t) + α(t)∂L(R, f, λ, μ, η, θ)
∂Rmdj
+
(25)
flm(t + 1) =
flm(t) + α(t)∂L(R, f, λ, μ, η, θ)
∂flm
+
(26)
λmld(v)(t + 1) =
λmld(v)(t)− α(t)∂L(R, f, λ, μ, η, θ)
∂λmld(v)
+
(27)
μl(t + 1) =
μl(t)− α(t)∂L(R, f, λ, μ, η, θ)
∂μl
+
(28)
ηmd(t + 1) =
ηmd(t)− α(t)∂L(R, f, λ, μ, η, θ)
∂ηmd
+
(29)
θdm(t + 1) = [θmd(t)− α(t)∂L(R, f, λ, μ, η, θ)
∂θdm ]+ (30) where t denotes the iteration index, α(t) are positive step sizes, and [·]+denotes the projection onto the set of nonnegative real numbers.
In terms of physical meanings of Lagrange multipliers, λ represents the “congestion prices” of information flow at wireless links, i.e., λmld(v)can be considered as the “congestion price” of information flow at link l for destination node d (l is on ds primary paths) and v (l is on vs backup path)s bandwidth requirement in layer m. At link l, if the total infor- mation flow bandwidth demand J(d)
j=1 hldjRmdj+ blv( J(v)
j=1Rmvj) in layer m exceeds the supply flm, then the “congestion price”
λmld(v) will increase. Accordingly, in problem P3b, Rmdj will decrease in order to meet the link’s bandwidth requirement of information flow, flm, and vice versa, and μ specifies the
“aggregated congestion prices” of physical flows at wireless links. At link l, if the total physical flow bandwidth demand
m∈M flm
(1−ρl) +
k∈(l)
m∈M fkm
(1−ρk) exceeds the supply Cl, then the “aggregate congestion price” μl will increase.
Consequently, flm in problem P3c will decrease in order to meet the link’s bandwidth supply of physical flow, Cl, and
Algorithm 1 Distributed optimization algorithm Initialization
Set t = 0, and set Rmdj(0), flm(0), λmld(v)(0), μl(0), ηmd(0), and θdm(0) to some nonnegative value for all d, v, m, l, and j.
repeat
Update at wireless link l∈ E:
1) Receives fkm(t) from the cluster{k|k ∈ (l)};
2) Receives μk(t) from the cluster{k|k ∈ (l)};
3) Receives Rmdj(t) from{d ∈ D|hldj= 1 or bld >0};
4) Fetches λmld(v)(t), μl(t), flm(t) stored in local proces- sors;
5) Updates flm(t), λmld(v)(t), and μl(t) by (26)–(28), respectively;
6) Transmits the control packet (CP) comprising the updated physical flow rate flm(t + 1) to the cluster{k|k ∈ (l)};
7) Transmits CP that comprises the updated price μl(t+
1) to the cluster{k|k ∈ (l)};
8) Sends the CP that comprises updated price λmld(v)(t+1) in the downstream direction to the subset {d ∈ D|hldj = 1} ∪ {v ∈ D|blv>0}.
Update at destination node d ∈ D:
1) Receives λmld(v)(t) from {l ∈ E|hldj= 1 or blv>0};
2) Fetches Rmdj(t), ηmd(t), θmd(t) stored in local proces- sors;
3) Updates Rmdj(t), ηmd(t), and θdm(t) by (25)–(30), re- spectively;
4) Sends the rate packet (RP) comprising Rmdj(t + 1) in the upstream direction to {l ∈ E|hldj= 1 or bld>0}.
until All variables converge to the optimums or the maxi- mum iteration number is achieved.
vice versa. Similarly, the other two Lagrange multipliers, η and θ, can be considered as the “SVC encoding prices” for each destination node in a multicast session. Furthermore, all updating operations are distributed and can be implemented at individual links and nodes using only local information.
The convergence behavior of the PA can be analyzed by applying Lyapunov stability theorem [14], [38], [40]. Regard- ing the proposed primal-dual algorithm outlined in (25)–(30) as a nonlinear autonomous system, it can be shown in a similar way as in [14] that the equilibrium point of this dynamic sys- tem is asymptotically stable. Therefore, the global asymptotic stability of the primal and dual variables in (25)–(30) can be guaranteed, which leads to the convergence behavior of the distributed solution to the dual problem of P3. Since P3 is convex, we can solve it through the equivalent dual problem using the proposed distributed algorithms [36].
C. Implementation Issue
To implement the proposed distributed algorithm, each link l or destination node d is treated as an entity capable of processing, storing, and communicating information. In practice, each link l = (i, j) is delegated to its sender node i, and all computations related to link l = (i, j) will be executed on node i. Here, we assume that the processor for link l keeps track of variables flm, λmld(v), and μl, while the processor of
Fig. 3. Network topologies associated with maximum transmission rates, where (a) is a typical butterfly topology for numerical experiment and (b) is with additional backup path.
destination node d keeps track of variables Rmdj, ηmd, and θmd. A distributed implementation of the proposed iterative algorithm is summarized in Algorithm 1.
In summary, the centralized approach requires all of the above primal and dual variables shared in the entire network and thus causes a great amount of communication overhead.
Utilizing the proposed distributed algorithm, however, the communication overhead only comprises the sending overhead of λmld(v)(t + 1), Rmdj(t + 1) and the transmitting overhead of flm(t + 1), μl(t + 1) at each iteration.
In terms of amount of message passing, the overhead of the proposed distributed algorithm consists of two parts: the network coding overhead and the communication overhead.
According to [8], the side information required by network coding is the h-dimensional global encoding vector in the header of each packet, where h is the number of source packets that need to be transmitted. Therefore, the cost of the practical network coding scheme is the overhead of transmitting h extra symbols in each packet. Considering a large size packet with the number of both header and payload symbols far more than h, such overhead is approximately negligible. For example, in the Internet, a typical maximum packet size excluding headers is somewhat larger than 1400 bytes. Thus, each IP packet can carry about 1400 symbols if each symbol takes up 1 byte. If his 50, then the overhead introduced by the side information of network coding is approximately 50/1400≈ 3%.
On the other hand, the communication overhead is the CP and RP information at each iteration. Consider the imple- mentation issues [27] and take Fig. 3(a), e.g., at the end of each iteration, wireless link (n1, n3) needs only to send its CP downward to destination nodes d1, d2and to wireless link (n2, n3), while destination node d1 needs only to send its RP upward to the links belonging to its two primal paths and one backup path. Supposing each updated primal or dual variable is float type that takes up 4 byte, and let M = 3, then the CP of link (n1, n3) requires 3×2×2×4+1×4+1×3×4 = 64 bytes and the RP of node d1 requires 3× 2 × 4 = 24 bytes. Thus, the total communication overhead sums up to 88 bytes. For the same configuration of IP packet with a packet size of 1400 bytes, the communication overhead introduced by the PA is 88/1400 = 6%. Furthermore, it can be noted that these packets (CPs and RPs) in practical implementation need not be communicated as separate packets; the CPs can be conveyed
Fig. 4. Convergence performance of the PA. (a) Allocated rate for d1and d2. (b) Physical flow rate for wireless links.
through a field in the video data packets, while the RPs can be conveyed through a field in the acknowledgement (ACK) packets. The maximal additional delay introduced by sending these packets is the one way propagation delay of the particular multicast destination node.
When a wireless network initializes multicast sessions, the proposed distributed algorithm would run and determine the optimal resource allocation rates and routes for all destination nodes through the convergence time. Within this duration, the total overhead comprises both network coding overhead and the communication overhead. Once it converges to the optimal values, the entire network will work with certain tolerance to dynamic network changes with only network coding overhead.
VI. Experimental Results
In this section, we will evaluate the overall performance of the PA. Both numerical and packet-level simulations are conducted over a typical butterfly network and a large-scale wireless network, respectively. The corresponding simulation results evaluate the convergence behavior of the proposed distributed algorithm, and further demonstrate that the PA can promise the optimal overall video quality for all destination nodes with appropriate protection against dynamic network changes.
A. Algorithm Behavior and Performance Evaluations First, we implement numerical experiments and evaluate the PA in the wireless network with a simple but classical butterfly topology, as illustrated in Fig. 3(a). Here, s, ni, and direpresent source node, relay node, and destination node, respectively.
The maximum transmission rates of wireless links are shown in Fig. 3(a). Suppose that the packet loss probability of each wireless link is ρl = 0.1, and set the protection parameters Bd = 0.5, = 0.9.
In the numerical experiments, we assume that the video bit stream has three layers, with the base layer at a rate of 3 (data units/s), the first enhancement layer at a rate of 2 (data units/s), and the second enhancement layer at a rate of 1 (data unit/s).
In Fig. 3(a), it is obvious that each destination node has three alternative paths from the source node. We specify two primal paths for each destination node, which are s → n1 → d1, s→ n2 → n3 → n4 → d1, and s→ n2 → d2, s → n1 → n3→ n4 → d2, respectively. Accordingly, the rest two paths,
