An Optimization Theoretic Framework for Video Transmission with Minimal Total Distortion over Wireless Networks
© Pejman Goudarzi et al. 2009
Received: 17 February 2008
Accepted: 10 January 2009
Published: 24 February 2009
Optimization theoretic-based rate allocation strategies can be used for the aim of allocating some optimal rates to the competing users in wireless ad hoc networks. By considering different objective functions (such as congestion level, total packet loss, etc.), the researchers propose some optimization framework by which the problem can be solved. Due to the rapid increase in the development of different video applications in such environments and the existence of difficulties in satisfying the prespecified QoS limits, increasing the perceived video quality can be considered as an important and challenging issue. The quality of the received video stream is inversely proportional to the amount of distortion which is being imposed on the video stream by the network packet loss and the video encoder. The main contribution of the current paper is to introduce an optimization theoretic framework in which by optimal rate allocation to some competing video sources, the aggregate distortion associated with all of the sources can be minimized. The numerical results verify the claims.
Convex optimization theory is an important tool for many rate allocation algorithms in wireline or wireless networks. Wireless ad hoc networks are computer networks in which the communication links are wireless. The network is ad hoc because each node is willing to forward data for other nodes, and so the determination of which nodes forward data is made dynamically based on the network connectivity. This is in contrast to wired network technologies in which some designated nodes, usually with custom hardware (variously known as routers, switches, hubs, and firewalls), perform the task of switching and forwarding the data. Ad hoc networks are also in contrast to managed wireless networks, in which a special node known as an access point manages communication among other nodes. Ad hoc networks can form a network without the aid of any pre-established infrastructure .
The requirements of a specific set of QoS parameters (delay, jitter, packet loss, etc.) must be guaranteed for each real-time application. However, for most real-time applications of wireless ad hoc networks, intrinsic time-varying topological changes provide challenging issues in guaranteeing these stringent QoS requirements.
Due to dynamic nature of these networks, traditional routing protocols are useless. So, special proactive/reactive multihop routing protocols such as DSDV/AODV are developed. Some of these routing protocols introduce more than one feasible path for a source-destination pair. These categories of routing algorithms are called multipath routing algorithms . Multipath routing schemes can reduce interference, improve connectivity, and allow distant nodes to communicate efficiently . In multipath routing, multiple multihop routes or paths are used to send data to a given destination. This allows a higher spatial diversity gain and throughput between source and destination nodes. On the other hand, it is obvious that inherent load balancing feature of the multipath routing algorithms has the capability of reducing the congestion as well as increasing the throughput of the user traffic in multihop wireless ad hoc networks. Moreover, using multiple paths between any source-destination pair can improve the important reliability and availability features of the routing strategy.
Multipath routing can provide both diversity and multiplexing gain between source and destination. However, multihop and multipath routing can also increase the total packet loss between the source and destination, especially if there is congestion in the paths or if the bit error rate of the paths is high due to the bad wireless link conditions (existence of high noise or interference levels). Therefore, supporting multimedia data with stringent maximum loss requirement over multihop ad hoc networks with multipath routing can be considered as an important and challenging research area.
Sending multimedia traffic over wireless ad hoc networks is a challenging issue, and many active research areas exist that all try to propose a solution to the problem from different points of view.
Some researchers such as those in [3, 4] try to use adaptive link layer techniques for throughput optimization. The authors in  propose a mathematical framework in which they vary adaptively the constellation size of an MQAM modulator in order to maximize the single user throughput.
In [5, 6] a congestion-minimized stream routing approach is adopted. In  the authors analyze the benefits of an optimal multipath routing strategy which seeks to minimize the congestion, on the video streaming, in a bandwidth limited ad hoc wireless network. They also predict the performance in terms of rate and distortion, using a model which captures the impact of quantization and packet loss on the overall video quality.
Some researchers such as Agarwal , Adlakha , and Zhu  follow some congestion-aware and delay-constrained rate allocation strategies. Agarwal and Goldsmith  introduce a mathematical constrained convex optimization framework by which they can jointly perform both rate allocation and routing in a delay-constrained wireless ad hoc environment. Adlakha et al. extend the conventional-layered resource allocation approaches by introducing a novel cross-layer optimization strategy in order to more efficiently perform the resource allocation across the protocol stack and among multiple users. They showed that their proposed method can support simultaneous multiple delay-critical application sessions such as multiuser video streaming .
For multipath video streaming over ad hoc wireless networks, received video quality is influenced by both the encoder performance and the delayed packet arrivals due to limited bandwidth. Hence, Zhu et al. propose a rate allocation scheme to optimize the expected received video quality based on simple models of encoder rate-distortion performance and network rate-congestion trade-offs . As the quality of wireless link varies, video transmission rate needs to be adapted accordingly.
In , Setton et al. analyzed the benefits of optimal multipath routing on video streaming in a bandwidth-limited ad hoc network. They show that in such environments the optimal routing solutions which seek to minimize the congestion are attractive as they make use of the resources efficiently. For low-latency video streaming, they propose to limit the number of routes to overcome the limitations of such solutions. To predict the performance in terms of rate and distortion, they develop a model which captures the impact of quantization and packet loss on the overall video quality.
In , measurements of packet transmission delays at the MAC layer are used to select the optimal bit rate for video, subsequently enforced by a transcoder. The benefit of cross-layer signaling in rate allocation has also been demonstrated in , where adaptive rate control at the MAC layer is applied in conjunction with adaptive rate control during live video encoding. The authors in  propose a media-aware multiuser rate allocation algorithm in multihop wireless mesh networks that can adjust the video rate adaptively based on both video content and network congestion and show the benefits of their work with respect to the well-known TCP friendly rate control (TFRC) .
In the current work, a similar approach such as  is being adopted by which a constrained optimization framework is introduced for optimal rate allocation to the real-time video applications. In , the authors do a similar optimization but they take the average congestion of the overall network as the QoS criterion and minimize it to find the optimal solution for rate allocation on the available paths using simulations. In , the authors propose a distributed rate allocation algorithm which minimizes the total distortion of all video streams. Based on the subgradient method, their proposed scheme only requires link price updates at each relay node based on local observations and rate adaptations at each source node derived from rate-distortion (RD) models of the video. They show by simulation that their proposed scheme can achieve the same optimal rate allocation as that obtained from exhaustive search.
The presented work in this paper differs from that of  in that, in our work, we have assumed that each video source may use multipath routing for partitioning and transmission of the total video traffic. On the other hand, we have included the effect of the packet loss in the perceived video distortion. Our work differs from [6, 7] in that we have used the total distortion as an objective Quality of Experience (QoE) measure in place of the QoS criterion used in [6, 7]. In order to compute the total distortion, we have assumed that multiple video sources use the same wireless ad hoc medium for transmission, and their associated distortions are additive . On the other hand, the presented work differs from  in considering more than one (and possibly interfering) multipath-routed video sources which compete for the available bandwidth in a bandwidth-limited wireless ad hoc network.
The paper's objective is to develop an optimal rate allocation framework bases on which the overall distortion of all the video sources is minimized. We also have used a penalty function approach for finding an iterative solution algorithm for the proposed constrained optimization problem such as those introduced in [15, 16].
The rest of the paper is organized as follows. In Section 2 the relationship between the allocated rate and the resulting distortion is introduced. In Section 3 the proposed optimization framework has been developed in detail. Section 4 is devoted to the numerical analysis, and finally in Section 5 some concluding remarks are presented.
2. Video Distortion Model
For smooth playback of a live video, transmitted packets must meet maximum allowed delay constraint. Therefore, the packets with a greater delay are useless so that they are supposed as packet loss. On the other hand, the total distortion of decoded video is the superposition of the distortion caused by video encoder ( ) and the distortion caused by packet loss or late arrivals during the transmission ( ) .
where is the number of paths in a multipath-routed source-destination pair, and is the rate allocated to the th path. The above assumption is achievable by using adaptive source rate control algorithms such as those mentioned in [18–20].
3. Proposed Optimization Framework
In the above calculations it is assumed that by a properly designed MAC protocol, there exist no collisions so that the inter node interference (INI) can be neglected. is the distance between node and node , denotes the path loss exponent, represents the power spectral density of the noise, is the system bandwidth, and is the coding gain. It is also assumed that the transmission power is equal at all nodes. In the following, we will assume that the nodes are static with . While this is a simplified model for the wireless ad hoc network, the analysis we present can easily be extended to the more sophisticated link capacity calculations.
We assume a simple strong line of sight (LOS) with BPSK signaling for node's wireless transmissions and also neglect the interfering effect of wireless transmissions between different paths . We have used the Independent Basic Service Set (IBSS) setup (DCF mode) for implementing the MAC layer of the 802.11 WLAN standard which enforces the WLAN network in the ad hoc mode. It is assumed that BPSK DSSS is used in the physical layer. As the Bit Error Rate (BER) performance of the BPSK spread spectrum system in an AWGN environment is identical to that of conventional coherent BPSK system , it is sufficient to calculate the latter performance for evaluating the BER of the proposed system.
In the current paper, our objective is to minimize the total distortion associated with multiple video sources. Thus, a mathematical formulation must be presented to express the distortion of each video source in terms of its allocated rate. According to (3), this distortion is a function of the PER associated with each video source. In the following paragraphs, the packet error rate computation method is presented.
where is a physical constant, is the (nonempty) set of wireless links associated with the th flow of the th video source, and and are the transmitted power and the total transmission rate associated with the th link in the th flow of the th source, respectively. As it is said before, we assume that is fixed during transmission and therefore does not depend on the transmission data rate .
Assume that for each common link there exists an associated set which represents the set of all ordered pairs (path, source) that use the common link in the path of the source (e.g., in Figure 2, the path 1 of source 2 shares the common link with the path 2 of source 1). So the ingress and egress nodes associated with this common link are common between more than one flow.
Now we are in a position that must compute the congestion-related part of the PER.
As in  simple M/M/1 queueing model and FIFO service discipline are adopted for the nodes. With the assumption of M/M/1 queueing model, the service time of each queue is an exponentially distributed random variable . We also assume that this service times are independent. On the other hand, the end-to-end delay of each path belonging to the video source is equal to the sum of these independent random variables. Ignoring the source and destination nodes (hops), the total number of nodes in and the number of noncommon nodes in and common nodes in would be , , and , respectively, for each .
Now, we propose another theorem based on which the existence and uniqueness of the solution vector of the system (23)–(25) that can be proved.
From (92) and the convexity of the constraint set (24)-(25) and (51), it can be deduced that the constrained optimization problem (23) has a unique and optimal solution vector .
The rest of the proof is the same as that of Theorem 2.
3.2. Corollary A
3.3. Corollary B
where is some positive and sufficiently small constant that guarantees the convergence .
The stability of the discrete-time iteration (112) can be proved in the same way as that proposed in .
In reality, due to the nodes mobility, there may exist estimation errors or uncertainties in some of the parameters (e.g., link capacities) associated with constrained optimization problem (23)–(25). This may cause an optimal and unique solution that can hardly be derived or cannot be reached at all by the proposed iterative algorithm in (112). Hence, some modifications must be applied in the proposed method. In general, if it can be assumed that the estimation error in the link capacities is such that the resulting uncertain constraint set in (112) can be a subset of a given uncertainty set , then it can be shown that by adopting the robust convex optimization theory [24, 25], an optimal solution can still be found.
4. Numerical Analysis
In the current work, an optimization framework is introduced by which the rate allocation to each path of a multipath wireless ad hoc network can be performed in such a way that the total distortion of multiple video sources resulting from the network congestion and wireless environment can be minimized.
Main application of such algorithms is in rate allocation to those subsets of real-time traffics which require a minimum level of total distortion. As we have used a simple LOS propagation model for the mobile nodes and ignored the mobility, a more powerful algorithm which can support more general multipath fading propagation models and the mobility can be considered for future research.
This work was supported by Iran Telecommunication Research Center (ITRC).
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