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Joint Power Control and Rate Adaptation for Video Streaming in Wireless Networks with Time Varying Interference Ayaz Ahmad, Naveed Ul Hassan, Mohamad Assaad, and Hamidou Tembine

Abstract—This paper considers a cross-layer optimization framework for video streaming in multi-node wireless networks with time-varying interference environment. We develop a distributed joint power control and rate adaptation framework that exploits the time diversity of the wireless channels, satisfies the hard delay constraints associated to video applications, and respects a certain fairness criterion among the nodes. The proposed framework performs power allocation at the PHY/MAC layers to achieve a certain target SINR such that the difference between the arrival and the departure rates at the queues is very small, and performs video rate adaptation at the video coding layer (upper layer) according to the nodes’ demanded video quality, their channel conditions, and a given fairness criterion. A main challenge here is that the adaptation of the video rate and the power control are not performed at the same time scale. We deal with this issue and model the power and the rate variations of the nodes as linear stochastic dynamic equations, and formulate a risk-sensitive control problem that captures the hard delay constraints of the video services, and the fairness criterion for resources utilization. We provide optimal solution to this control problem, and illustrate the performance of our framework through simulations. Index Terms—Power Control, Rate Adaptation, Video Streaming.

I. I NTRODUCTION The recent advances in wireless communication technologies and their capabilities of providing the nodes/users with high data rates (e.g., 4G and future 5G) has motivated video streaming over wireless networks and its application is increasing very rapidly. In parallel, recent trends in wireless communications allow the terminals-devices to communicate directly between each other (device-to-device communications). The main reason behind this is to support proximity-based services such as social networking and media sharing applications. In future networks, the transmitters will be more and more Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported in part by the Lahore University of Management Sciences under the Research Startup Grant. A. Ahmad is with the Department of Electrical Engineering, COMSATS Institute of Information Technology, 47040 Wah Cantt. Pakistan, Email: [email protected]. N. U. Hassan is with the Department of Electrical Engineering, Lahore University of Management Sciences, 54000 Lahore, Pakistan, Email: [email protected]. M. Assaad is with Laboratoire des Signaux et Systemes (L2S, UMR CNRS 8506), CentraleSupelec, Gif-sur-Yvette, France, Email: [email protected]. H. Tembine is with the Department of Electrical & Computer Engineering, New York University, Abu Dhabi, Email: [email protected].

directly connected to their receivers. In this paper, we therefore consider a wireless network of multiple transmitter-receiver pairs that use the same bandwidth and thus cause interfere to each other. The network can be seen as an ad hoc or a deviceto-device network. In such contexts, the performance of the network can be improved by efficiently allocating the power to the nodes in different time slots. Adjusting the transmit power according to the node’s demanded bit rate and its channel quality is not only efficient in terms of node’s power consumption but can also help in reducing the interference caused to the neighboring nodes. Notice that in this case, the power allocation must be conducted independently by each transmitter without exchanging information with the other transmitters. On the other hand, video streaming over wireless networks with multiple nodes/receivers with different channel characteristics faces several challenges. These multiple nodes may demand the same video quality while the bit rate they can support and the packet loss they experience may be different due to their different channel conditions. In addition, the wireless radio resources are limited and therefore, they should be efficiently shared among multiple streaming users. In multi-node networks, in order that the video transmission is adapted to the individual channel condition of each node, one of the following three principle video streaming methods can be used. The first method consist in selecting one among several non-scalable bitstreams with different bit rates (and of course with different quality) associated to the same video [1], [2]. The choice of bitstream for each node is made according to its demanded/promised video quality and its bit rate support capability. The second video streaming method uses a single high quality bitstream which during streaming is converted into another bitstream with a transcoder to match the node’s requirements [3]. In the third method called the scalable video streaming, the video is encoded once in a single scalable bitstream that can be adapted to the node’s channel condition [4]. More specifically, this method uses scalable coding technique wherein a video is encoded into a single bitstream with a base layer and several enhancement layers. The base layer is non-scalable and is necessary for decoding the video stream, whereas the enhancement layers that improves its quality are scalable and can be truncated at any point to meet the quality-of-service (QoS) requirement and bit rate supporting constraint of each node. This method has small storage requirements, and provides more simplicity and flexibility in terms of bitstream truncation/switching. An

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overview of this technique can be found in [5]. Several papers have shown the advantage of using scalable video coding and dynamic adaptive streaming over HTTP (e.g. [6]–[8]). In these works, the rate adaption and thus the video quality selection can be performed at the client side, based on the current network condition. One among the above three methods which is more suitable according to some given preferences, and limitations, can be chosen for video streaming in multi-node wireless networks. In summary, the transmission of video streaming can be improved by adapting the rate of the bitstream at the video coding layer and allocating efficiently the power at the PHY/MAC layer. These two (rate adaptation and power allocation) are performed at two different time scales which makes the problem challenging. In addition, the rate adaptation must be done such that a fairness among nodes is achieved. This paper deals with this problem. A. Related Work Resource allocation for wireless video streaming has attracted worldwide attention in recent years, and a rich literature exists in this area. A comprehensive survey on cross-layer perspective of video transmission over lossy wireless networks is presented in [9] which discusses the challenges related to all layers of the protocol stack and state-of-the-art solutions for these challenges. Most of the existing work on resource allocation for multiuser video streaming do not consider joint video bitstream adaptation and distributed power control (at different time-scale) in an interfering wireless networks but only focus on one or a combination of the following: i) users’ video quality improvement, ii) dynamic rate adaptation to match the wireless channel capacity, iii) packet scheduling at the MAC layer (mainly for cellular networks), and buffer management. In [10], [11], channel-aware schedulers for non-scalable video streaming have been proposed that dynamically adjust the data rates of the users and perform packet scheduling while accounting for video distortion and delay deadlines of the packets. A cross-layer framework is proposed in [12] that aims at meeting the instantaneous rate, and the deadline requirements of video traffic. In [13], [14], the problem of joint scheduling and buffer management for scalable video transmission has been addressed. The problem of rate control, buffer management and video quality is also addressed in [15], [16] which jointly controls the encoder rate and the network congestion. In [17], an adaptive layer switching algorithm is proposed that adapts the video rate to match the achievable network throughput and adjusts the video layers at the video source to guarantee a QoS constraint. Other studies on cross layer rate adaptation for video streaming can also be found in [18], [19]. In [20], scheduling for multi-quality video streaming in cellular wireless networks is studied that improves the users’ Quality of Experience (QoE) by reducing the playback interruptions and by providing the best possible video quality to the users. In [21], a novel MAC-level multi-cast protocol is developed to enhance the reliability and efficiency of multicast services in IEEE 802.11n WLANs. In [22], the distortion

of the received video is minimized by creating multiple subflows from one video and assigning them different priorities according to their importance In [23], a rate scheduling scheme is developed that exploits the dependency among video blocks to enable the delivery of important blocks and reduce the number of undecodable blocks that might have lost their reference blocks. A cross-layer design for QoS support in wireless sensor networks with multimedia application that maximizes the number of video sources under constraint on QoS of each individual source is investigate in [24]. Improving the coding efficiency, matching the coded video streams with the channel capabilities, and adaptive coding and modulation schemes for video streaming application are also investigated. In [25], network coding techniques are used to performs joint multi-cast paths selection, associated rates control and bandwidth reservation for scalable video multi-cast streaming applications. In [26], scalable video broadcasting schemes for mobile TV broadcast networks are proposed which improve the coding efficiency and allow the coded video streams to be better matched with the transmission capability of the mobile devices. In [27], an approach for scalable video coding combined with adaptive modulation and coding schemes (MCS) and wireless multi-cast has been proposed. This approach improves the users’ video quality by adaptive MCS selection for each video layer. In [28], a crosslayer resource allocation scheme for assigning subcarriers to users is proposed that is based on video demands of the users and the channel quality. In this scheme, after performing the subcarriers allocation, the users adapt their modulation schemes according to the channel quality. They design a cross-layer resource allocation algorithm to assign subcarriers to users based on both the demand of the video and the quality of the channel. Once the resource allocation decision is made, the users then periodically adapt the modulation format of the subcarriers allocated according to the evolution of the CSI for the duration of the GOP. Packet scheduling in the context of video streaming over heterogeneous access networks has also been studied. The work in [29] aims at determining that which video packets should be transmitted over each access network in heterogeneous access networks in order to minimize the distortion of the received videos. In [30], a cross-layer framework for improving the user’s QoE and energy efficiency of the heterogeneous wireless multimedia broadcast receivers is investigated. The study of distributed power control and the effect of interference in the context of video streaming has not been given enough attention by the researchers and only few works have considered these important issues (e.g., [31], [32]). The authors in [31] proposed an algorithm for video streaming over OFDMA downlink which maximizes the users’ average PeakSignal-to-Noise-Ratio (PSNR) subject to transmission power constraint. However, this work deals a single cell network scenario with orthogonal users under the assumption of no inter-cell interference. Joint source adaptation and resource allocation for video streaming in CDMA networks is studied in [32] where for the uplink transmission, the interference and the maximum received power values are assumed to be not



changing frequently. A cross-layer optimization approach for joint channel allocation, relay selection, rate and power control for maximizing the sum PSNR of multiple video sessions in the context of cognitive cooperative ad hoc networks is proposed in [33]. This work concludes that cooperative relaying can improve the quality of service of secondary users. In [34], another cross-layer framework for joint power control, rate control and relay selection in cooperative networks for video streaming is considered which demonstrates that cooperative relaying results in nodes’ energy saving without a perceivable degradation in video quality. In [35], the authors investigate the properties of energyefficiency and spectrum-efficiency for video streaming over mobile ad hoc networks by developing an energy-spectrumaware scheduling scheme. The problem considered in this paper is different from the existing work described as follows. We consider a network consisting of transmitter-receiver pairs that interfere with each other where the resource allocation is performed separately by each transmitter at each time slot. Since no central entity exists in our framework, multi-user scheduling is not considered. To the best of our knowledge, dealing with power control, interference management, and video control in a distributed way and at different time-scales has not been considered before in the literature. In our work, the state of each node (channel state, SINR) lies in a continuous set which is different from the existing work on MDP (Markov Decision Process) that assumes a finite state space set. To improve the performance of video streaming, joint video rate adaptation (at the coding layer) and power control (at the PHY layer) is necessary. In this paper, we develop a distributed joint power and video rate adaptation framework as well as explicitly consider the target fairness constraints in multi-node wireless network with interference.

the problem difficulty. If during a given period of time, a high interference is exerted on a given receiver node (due to the fact that the other nodes have high bit rates to transmit), the node will not have a high transmission rate the high video quality requirements will not satisfied. An immediate solution is to decrease the video quality required for this node by decreasing the video bitstream rate. In this case, in the next slots, the node will transmit at low power while the other nodes keep transmitting at high power (and then causing high interference on the node). This will imply that the node will keep having a lower video quality as compared to the other nodes. Therefore, a more clever adaptation of the video rate must be performed in order to ensure fairness among nodes. The solution must depend on the link quality, interference, fairness between nodes in terms of video quality, etc. Therefore, the problem is a stochastic control problem with two different timescales. The joint adaptation of video bitstream and power (allocated separately by each node) taking into account the QoS fairness between nodes, the channel variations and the time-varying interference is not at all an easy task. The novelty of our problem, and its solution lies in the following. •

•

B. Motivation and Contributions The temporal variations of wireless fading channels can be exploited by optimally controlling the power, and adapting the video bitstream/rate in different time slots. However, in order to develop an optimal power control and source rate adaptation algorithm, channel gain values and packet arrival rates for current and future time slots are required. Unfortunately, the information about the future channel conditions and arrival processes is not available which makes this problem very challenging. It is worth mentioning that the power control must be performed separately by each transmitter taking into account the channel and interference variations. In this paper, we consider this challenging problem of distributed joint dynamic power control and bitstream adaptation for video streaming in a multi-node wireless networks with time-varying interference. The main objective is to jointly control the power at the PHY/MAC layer and the video rate at the video coding layer (upper layer) of all nodes such that they are provided with good video quality for minimum power consumption while respecting a certain fairness/satisfaction criterion among the nodes. The control of power and video rate are performed at different time-scales. The following provides an idea on

•

•

•

•

We consider video streaming in a multi-node wireless network with interference. Unlike the underlying approach of assuming no interference at all or assuming constant/fixed interference in many of the available solutions for resource allocation in wireless networks, we assume that the wireless channel and the interference gains of the nodes are both time varying. We propose a joint dynamic power control and video rate adaptation framework whose formulation is even a difficult task. The difficulty arises due to the fact that the power control at the PHY/MAC layer should be performed at each time slot (usually ≤ 1ms) whereas the video rate, at the video coding layer (upper layer), should be adapted after a longer time (i.e., two different time scales should be jointly dealt with). In this work, we address these issues, and formulate a framework that allows instantaneous power control and average video rate adaptation jointly. In addition to the exploitation of the time varying nature of the channels, we also introduce a fairness/satisfaction criterion among nodes so that irrespective of its channel condition, each node can get a promised share in system’s total resources/capacity which is a challenging goal. As this paper considers video streaming, we guarantee the associated stringent delay constraints. In other words, the packets in the buffer of the transmitting node are delivered to the receiving within a very small period of time and at small transmit power. In order to solve the joint power control and rate adaptation problem, we analyze the Channel to Interference and Noise Ratio (CINR) distribution, and model the power control and rate adaptation for each node as linear stochastic dynamic equations and write the problem as a dynamic stochastic control problem. We then formulate a risk-sensitive control problem that



jointly performs power control and rate adaptation for all nodes while satisfying the stringent delay constraints of the streaming applications, and respecting the nodes’ fairness/satisfaction criterion. There is a design parameter µ, which could be used to give less or more weight to the rate deviation term in the cost function. This parameter may be used to control rate deviation so that the video quality changes smoothly with almost zero jitter. C. Paper Organization The rest of the paper is organized as follows. In Section II, we describe the system model and problem statement. In Section III, we develop a stochastic framework for joint power control and rate adaptation. Risk sensitive control problem and its optimal solution are presented in Section IV. Simulation results are presented in Section V while the paper is concluded in Section VI. In this paper, the lowercase letters denote scalars while the uppercase and lowercase boldface letters denote matrices, and vectors respectively. The key notations and symbols used in this paper are summarized in Table I. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT A. System Model We consider video streaming in a multi-node wireless network that consists of a geographical area with pairs of transmitter and receiver nodes. It can be an ad hoc network or a device-to-device communication model. Each transmitter is connected to one receiver. The network conditions are assumed to be dynamic. In this network, some of the nodes act as video transmitters while the others act as receivers. We assume a Code Division Multiple Access (CDMA) system where the transmitters share the same bandwidth for video transmission. Owing to this fact, interference is caused at the receivers, and consequently, the achieved rate at each receiver node not only depends on its channel conditions and allocated resources but also depends on the interference caused to it by the transmissions of the other nodes. The network model considered in this paper is distributed and does not contain any central entity to handle the resource allocation. Therefore, multi-user scheduling (that requires a central entity) is not considered and only power control is considered. The interference caused to any receiver node is equal to the sum of interferences exerted by all the other transmitters in the area. The CINR at receiver node k for the signal transmitted to it by node j can be defined as, Gk,j (t)

gk,j (t) = η+

K ∑

(1)

pl (t)Gk,l (t)

l=1,l̸=j

where pl is the power transmitted by the node l, Gk,j is the path gain including shadowing effect between the nodes j and k, η is the power of white noise at the receiver node k and K is the total number of interfering nodes in the area. We assume that the video source/transmitter node can provide several bitstreams of the same video with different

rates. Each particular rate corresponds to a given QoS level. The higher the quality of the video, the higher the bit rate is needed for its transmission (in video streaming applications, PSNR is normally regarded as a measure of video quality. PSNR is a non-decreasing function of the video bit rate). Let ∗ rk,j (t) denote the video bitstream rate intended to be sent to node k by another node j at the tth time slot. We will call this bitstream rate as the “arrival rate” in the remainder of this paper. Notice that we ignore the signaling overhead (control information, CRC, etc.) required in each of the OSI layers. The data rate of each bitstream perceived at the video coding layer is the same as that at the physical layer. The actual transmitted/achieved data rate at the PHY layer at time t from node j to node k depends on the actual SINR γk,j (t) which we denote by rk,j (t). It is obvious that video streaming delay is a function of the deviation between the arrival rate and ∗ the actual data rate, i.e., rk,j (t) − rk,j (t). In order to satisfy the stringent delay requirements of the streaming application, ∗ each rk,j (t) should approach to the corresponding rk,j (t). At the PHY layer, we can adapt each rk,j (t) by performing power control according to the instantaneous channel conditions. As we consider video streaming in multi-node network, in addition to satisfying the stringent delay constraint, we also opt ∗ to adapt each node’s video bitstream (i.e., rk,j (t)) according to its time varying channel and interference gains, and the ∗ available communication resources. Since rk,j (t) is the video coding layer (upper layer) parameter while rk,j (t) is the PHY layer parameter and both should be optimized dynamically, we shall propose a cross-layer optimization framework which will enable to perform joint power control and rate adaptation.

B. Fairness Considerations In this paper, we assume that each node can have a different quality-of-service (QoS) requirement with a different level of best quality video. Since each node is promised to be provided with a certain QoS (bitstream/rate) despite having very bad channel conditions, we also introduce a satisfaction/fairness criterion among the nodes such that each node is provided with a certain portion of system resources and a certain data rate which are both functions of its promised maximum arrival rate and the system’s total capacity/resources. To this end, we introduce a satisfaction parameter sk,j (t), a fairness parameter T fk,j (t), and a target fairness parameter fk,j for each node. In practice, it is more likely that these satisfaction and fairness parameters are updated once during each time window Tw = N ∗ T where N is a given integer and T is the time slot duration. The reason is that the fairness and satisfaction are in general perceived by the users after some transmission time especially in dynamic environment where the bit rates received by the users are time-varying. The aforementioned parameters are therefore defined as follows. max be the Definition II.1. (Satisfaction Parameter) Let rk,j maximum possible arrival rate of node k that corresponds to the best video quality the node can be provided with, and ∗ rk,j (t) be its arrival rate at time t, then the user satisfaction



TABLE I N OTATIONS Notation

Symbol(s)

CINR at receiver k for signal transmitted by transmitter j at time t

gk,j (t)

Video (Arrival) rate; Actual physical layer achieved data rate

r ∗k,j (t); rk,j (t)

Transmit power and the corresponding SINR

pk,j (t); γk,j (t)

Feasible power and the corresponding SINR

f pfk,j (t), γk,j (t)

Target fairness; Achieved fairness

T ; f fk,j k,j (t)

10 log(.) (.) Gaussian noise that models variation of gk,j (t) from time instant to another

ng (t)

State vector; Control vector; and Noise vector, respectively

xk,j (t); uk,j (t); nk,j (t)

Quadratic cost; Risk sensitive cost

Jk,j ; Jk,j

parameter is given by 1 Tw

s˜k,j (pTw ) =

t=pT ∑w

∗ (t) rk,j

t=(p−1)Tw +1 max rk,j

(2)

where p is an integer. Definition II.2. (Fairness Parameter) The fairness parameter of user k is defined in terms of satisfaction parameters as s˜k,j (pTw ) fk,j (pTw ) = ∑K ˜l,j (pTw ) l=1 s

(3)

duration. This means that the source rate stays constant during m time slots and updates its value only at time slots pW (where p ∈ N). It is worth noting that W < Tw (Tw = qW where q ∈ N). In practice, Tw is at least 20x bigger than W so that the source rate controller can have enough long time to adapt the source rate for a given fairness value fk,j (pTw ). Different time windows for power, rate and fairness update are depicted in Fig. 1. In the next section, and in order to tackle the above described problem with different time scales, we model the power, actual transmit rate, and arrival rates of the nodes as linear stochastic dynamic equations. Fairness Update Window

which denotes the actual achieved fairness of the user k in the current time window pTw . Definition II.3. (Target Fairness) Let wk,j be a weight for user k that corresponds to its priority among the nodes, then the target fairness parameter of user k is given as follows max wk,j rk,j T fk,j = ∑K max l=1 wl,j rl,j

Arrival Rate Update Window

(4)

The aim of the above described fairness parameters is to ensure a fair video quality among users where the video quality is defined in terms of the bit rate requirements. It is worth noting that our developed solution in the sequel requires only a numerical value of a given fairness parameter fk,j (pTw ) regardless the definition itself of this fairness parameter. The readers can see later in this paper that our framework in general and equally holds for other fairness metrics/criteria (e.g., one can define a fairness parameter based on PSNR measurements, etc.). The main objective now is to jointly control the power, and adapt the arrival rate such that the stringent delay constraints associated to nodes streaming applications are satisfied as well as the fairness criterion is met. One of the main challenges as mentioned earlier is that the power control and video rate control do not have the same time scale. The transmit power is controlled at each time instant t whereas the time scale of the source/video rate control is slower and is denoted by W = m × T where m is an integer and T is the time slot

T

Time scale (t)

Power Update

Fig. 1. Time Windows for Power, Rate and Fairness update

III. S TOCHASTIC F RAMEWORK FOR J OINT P OWER C ONTROL AND R ATE A DAPTATION In this section, we develop a stochastic linear dynamic framework for source rate adaptation and power control. We start the analysis by understanding the dynamics of the CINR, gk,j (t). In the next step, we use these results to formulate the dynamics of the node’s power and data rate as stochastic linear equations. A. CINR Dynamics There are different approaches in the literature to approximate the distribution of signal-to-interference ratio (SIR) (e.g.,



[36]–[38]). The approach we use in this paper is described as follows. We approximate under given power allocation the product of lognormal and Chi-Square variables by a lognormal variable and then approximate the sum of lognormal variables by another log-normal variable using Fenton-Wilkinson approximation [37], [39], [40], [41]. Using this approach, we find that the probability distribution function of CINR gk,j (t) (equation (1)) can be approximated by a log-normal distribution. This approximation of the CINR dynamics (even it is an approximation of the distribution function) provides an interpretation/motivation to model the adaptation of the power and video stream rate as stochastic linear dynamic equations. The obtained log-normal approximation may not be accurate in practice especially that it was obtained for given powers of the interferers. However, the linear dynamic with additive disturbance considered here still hold (but maybe with other distribution) and the developed framework is useful and makes sense. In the numerical results, the simulation is conducted without using the lognormal approximation and one can see that our developed framework provides good results and outperforms existing work. Proposition III.1. The probability distribution function of the CINR, gk,j (t) can be approximated by a log-normal distribution. Proof. See Appendix A for the proof. Let g k,j (t) = 10 log(gk,j (t)) which will be referred to as“LogCINR” in the rest of the paper. The operation log(.) denotes the base-ten logarithm. The Log-CINR variations from one time slot to another for given transmit powers of the interferes can be modeled according to the following proposition. Proposition III.2. Log-CINR variations from one time slot to another can be modeled as, g k,j (t + 1) − g k,j (t) = ng (t)

(5)

where ng (t) is a Gaussian random process. Proof. Owing to the fact that CINR gk,j (t) has a Log-normal distribution, it is obvious that both g k,j (t + 1) and g k,j (t) have Gaussian distribution. Moreover, sum of two Gaussian random processes is another Gaussian random process which concludes our proof. The signal-to-interference ratio (SINR) γk,j (t) is function of CINR gk,j (t), i.e., γk,j (t) = pk,j (t)gk,j (t) where pk,j (t) denotes the power. Therefore, Log-SINR denoted by γ k,j (t) = 10 log(γk,j ) can be obtained as follows γ k,j (t) = pk,j (t) + g k,j (t)

(6)

where pk,j (t) = 10 log(pk,j (t)). The linear dynamic of g k,j (t) and therefore of γ k,j (t) (i.e. with additive disturbance) will be used in the derivation of dynamics of power control and rate adaptation in the following subsection. B. Stochastic Linear Dynamic Model for Power Control and Rate Adaptation In this subsection, we develop a stochastic linear dynamic model for power control and rate adaptation. We introduce

∗ an auxiliary variable that we denote by γk,j (t). For a given ∗ source/video rate rk,j (t) (i.e., the arrival rate as defined in the ∗ previous section), γk,j (t) denotes the target SINR that should be achieved at the receiver so that the video bitstream, with rate ∗ rk,j (t), is successfully decoded (or decoded with a given/small target bit error rate)1 . In view of the objectives described in section II, our goal now becomes the development of a control strategy which allows the following: • the instantaneous actual SINR γk,j (t) tracks the target ∗ SINR γk,j (t) ∗ • the target SINR γk,j (t) is controlled by the streaming source such that a maximum video quality is provided fairly to all users ∗ • the target SINR γk,j (t) is feasible at any time instant t (feasible means that there exists a vector of powers that can be allocated to all users such that the SINR of each ∗ user approaches γk,j (t)). In order to achieve the above objectives, we start by ∗ studying the dynamics of γk,j (t) and γk,j (t). In the following, ∗ we show that the evolution of γk,j (t) and γk,j (t) can be captured by stochastic linear state space equations for a wide class of power control and rate adaptation strategies. However, the obtained state space equations are either unstable or do not achieve the above mentioned objectives. Therefore, in section IV, we will introduce a control strategy by using stochastic control techniques, in order to control the obtained state space equations and achieve the aforementioned objectives. 1) Stochastic Linear Dynamic Model for Power Control: We have the following proposition for the instantaneous power control:

Dynamic III.3. The fast power control can be written by the following linear stochastic dynamic equation: γ k,j (t + 1) = {1 − βk,j }γ k,j (t) + βk,j γ ∗k,j (t) + ng (t)

(7)

where βk,j is a given step size, ng (t) is a zero mean noise term, and the notation (.) stands for 10 log(.). In the following, we provide the interpretation for the use of the above mentioned dynamic equation. In order that the actual data rate approaches the video arrival rate, the users’ ∗ actual SINR γk,j (t) should approach the target SINR γk,j (t). This can be achieved by applying an appropriate power control strategy. The main idea (widely used in the literature) of power control in wireless communication consists simply in comparing the achieved SINR to a target value and adapting the transmit power accordingly. This principle is very simple which makes it suitable for practical implementation. In [42], the aforementioned power control is written as follows (we skip the derivation details here, and the readers can refer to [42] for more details on this algorithm): pk,j (t + 1) = pk,j (t) + βk,j {γ ∗k,j (t) − γ k,j (t)}

(8)

where βk,j is a given step size (between 0 and 1) that may vary from one node to another. Using Log-SINR expression 1 Any function which relates bit rate and SINR can be used to obtain ∗ (t), e.g., the widely used Shannon capacity formula i.e. r ∗ (t) = γk,j k,j ( ) 1 ∗ (t) . log2 1 + γk,j 2



(6), we have γ k,j (t + 1) = pk,j (t + 1) + g k,j (t + 1)

(9)

Now substituting the value of pk,j (t+1) from (8) and g k,j (t+ 1) from (5) into (9), we get γ k,j (t + 1) = {1 − βk,j }γ k,j (t) + βk,j γ ∗k,j (t) + ng (t) (10) It is well known that the power control (8) is unstable in stochastic environment. In addition, it does not meet the requirements of adaptive video rate services with maximum and fair video quality. Therefore, we will introduce a control strategy in the next section to meet these requirements. With the introduction of this control strategy, it is possible to drive the dynamics of a state space system to the required objectives. The distribution of ng (t) is estimated in the previous section. Although the ng (t) distribution is an approximation obtained for given powers of the interferes, using it in the control problem defined in the next section in the context of time varying transmit interferes’ powers will lead to performance improvement as one will see later in the paper. In the above power control algorithm, we have not introduced any notion of power constraint. A power constraint is important since the increase in the power level of a node may may severely affect other nodes by causing them harmful interference. To ensure that at any time instant, the nodes do not increase their power above a certain feasible level, we assume that each node should choose its power from a given feasible set of powers, pk,j (t) ≤ pmax

, ∀j

(11)

where pmax is the maximum acceptable power a node can transmit and it is the amount of power for which the SINR level of that node reaches a given value γ max . Since the channels and interference gains are time varying, the corresponding value of maximum feasible power will also vary for each time slot and will be different for different nodes. Thus, in order to formulate this power constraint in our dynamic state space model, we introduce a new variable pfk,j (t) called the feasible power which denotes the maximum power a node j can transmit at time slot t. We model the feasible power variations according to the following equation; pfk,j (t + 1) = pfk,j (t) + ϵk,j {γ max − γ¯k,j (t)}

(12)

f where ϵk,j is a given step size. Let γk,j (t) denotes the value f of SINR level when pk,j (t) is transmitted, then by using (5) and (12), we obtain the following dynamic equation

Dynamic III.4. γ fk,j (t + 1) = γ fk,j (t) − ϵk,j γ¯k,j (t) + ϵk,j γ max + ng (t) (13) In order to guarantee that pk,j (t) ≤ pfk,j (t) at any time slot ∗ ∗ t, the arrival rate rk,j (t) should be adapted such that γk,j (t) ≤ f γk,j (t). The procedure of integrating the notion of feasible power into arrival rate adaptation is provided in the following subsection.

2) Stochastic Linear Dynamic Model for Rate Adaptation: ∗ As in video streaming, the video bitstream (i.e., rk,j (t)) for each node should be chosen by truncating/switching the bitstream according to its varying channel conditions and the available system resources. Due to the inter-dependent nature of the video frames in video transmission, the video quality can be better determined by the total resource allocation during a time window, which should be long enough for the user to determine the best bitstream for the video that could be supported by the physical layer. One way of dealing with this sort of problems is to consider a time average model for resource allocation such that the average actual/achieved data rate during the given time window approaches the average arrival rate. This time average model is suitable for video rate adaptation. However, we aim to develop a joint framework for instantaneous power control and video rate adaptation. Thus, we will propose a model that will enable us to adapt the arrival rate in an average sense while allowing the instantaneous power control. We will also integrate the satisfaction/fairness criterion among the nodes to our arrival rate adaptation model. To this end, we propose the following dynamic equation: Dynamic III.5. The video rate/arrival rate can be adapted using the following stochastic linear equation, { } γ ∗k,j (t + 1) = γ ∗k,j (t) + ξk,j (t) γ fk,j (t) − γ ∗k,j (t) + { T } ξk,j (t) fk,j − fk,j (t) γ ∗k,j (t) + δˆt nt (t)(14) with the step size ξk,j (t) defined as follows, { 1 if t = mW ξk,j (t) = 0 elsewhere

(15)

where W is the time window during which the video rate/arrival rate should not change; and δˆt , and nt (t) are small positive numbers. The second term on the R.H.S of the above equation ensures that the video arrival rate of the node should be in the feasible region such that the node’s transmitted power does not exceed the maximum value of its feasible power whereas the third term guarantees the fairness among nodes. If even both the feasible power constraint and the fairness criterion are satisfied, the arrival rate is still increased by δˆt nt (t) which has a very small value. This reflects the desire of each receiver node to have the best possible quality video, and will drive the the system to converge at high possible arrival rates. The term nt (t) can be assumed as a Gaussian-distributed variable of a positive mean and a very small variance so that its value is always positive. Furthermore, ξk,j (t) = 1 for t = mW where m is a positive integer, whereas ξk,j (t) = 0 for any other value of t. With the above choice of step size, for a positive integer m, the arrival rate will vary at t = mW in real sense whereas its variation will be negligibly small between t1 = mt, and t2 = mt+W − 1. This instantaneous arrival rate control approach ensures that the rate is adapted only after every W time slots (i.e., after a large enough time window) according to the achieved fairness and achieved data rate. This allows us to consider two different time scales for the updates of power control and video rate.



[ Thus even having a very bad instantaneous channel conditions, a node will be provided with enough resources to utilize its promised portion of the system capacity. On the other hand, a node with a very good instantaneous channel conditions will be forced to reduce its arrival rate in order not to deprive the nodes with bad channel conditions from the system resources. If the dynamic equations (10), (13), and (14) are directly used, the system will be completely unstable and the fairness and video quality requirements will not be met. Therefore, in the following section, we introduce a risk-sensitive control approach and develop a joint framework for dynamically adapting γ fk,j (t), and γ ∗k,j (t), and adjusting the power such that γ k,j (t) approaches to γ ∗k,j (t). IV. R ISK - SENSITIVE C ONTROL P ROBLEM AND ITS O PTIMAL S OLUTION In this section, we formulate the above design problem as a risk-sensitive control problem with exponential cost function. The objective is to formulate the problem as a control problem in standard form in order to use techniques from control theory and provide the optimal solution to it. A. State Space Equation The above problem is not a standard control problem. Thus, we transform the problem, and formulate it as a linear stochastic problem in standard form. To this end, we introduce a state vector defined as follows. zk,j (t) = [γ ∗k,j (t)

γ k,j (t)

γ fk,j (t)]T

(16)

Now, by combining (10), (13), and (14), we get the following state-space model, ˆ k,j (t)zk,j (t) + fk,j (t) + n ˆ k,j (t) zk,j (t + 1) = A where fk,j (t) [ δˆt nt (t) ng (t)

T

[0 0 ϵk,j γ max ] , ]T ng (t) , and =

ˆ k,j (t) n

ˆ k,j (t) = A { }  T 1 − ξk,j (t) + ξk,j (t) fk,j − fk,j (t) 0   βk,j 1 − βk,j 

(17) =

] [ ] ˆ k,j (t) ˆ k,j (t) u n , and nk,j (t) = . nk,j is a gaussian 0 0 disturbance. Note that, xk,j (t), uk,j (t) and nk,j (t) are 4 × 1 column vectors while, Ak,j (t) and Bk,j (t) are 4 × 4 matrices. B. Cost Function Formulation

We now define the following quadratic cost function for user k over a finite time horizon τ : τ }2 { }2 1 ∑{ ∗ Jk,j = γ k,j (t) − γ k,j (t) + u∗k,j (t) τ t=1 { }2 + upk,j (t) (20) In function, the first term i.e., { ∗ the above }cost 2 γ k,j (t) − γ k,j (t) is equivalent to the rate deviation, and its minimization is the main objective. The other two terms in the cost function represent the additional cost incurred due to control actions. We can write the cost function (20) in standard form as given by Jk,j =

τ } 1 ∑{ T xk,j (t)Qxk,j (t) + uTk,j (t)Ruk,j (t) τ t=1

(21)

where, R is a 4 × 4 identity matrix, and,   1 −1 0 0    −1 1 0 0    Q= 0 0 0   0  0 0 0 0 It is obvious that minimizing E(Jk,j ), (where E(.) is the expectation over time of Jk,j ) will minimize the average rate deviation (hence the average delays). This objective function therefore seems unsuitable for real time video transmissions with stringent delay constraints. We therefore construct the following exponential cost function, Jk,j = E {exp(Jk,j )}

(22)

 The main idea of introducing the exponential cost function is to amplify the impact of rate deviation so that the controller ξk,j (t)  should try to keep J k,j very small which will not only  0  minimize the rate deviation but will also minimize the jitter in 0 −ϵk,j 1 the video transmission by smoothly changing the video quality. In fact, the exponential cost function can be replaced by a ˆ k,j (t) = Furthermore, we introduce a control vector u p more general cost function by introducing a parameter called ∗ T [uk,j (t) uk,j (t) 0] into (17) in order to drive γ k,j (t) “risk-sensitive” parameter [43], [44]. By varying the risk∗ towards γ k,j (t) and is given as follows, sensitive parameter the cost function can be changed. A large ˆ k,j (t)zk,j (t) + fk,j (t) + Bˆ ˆ uk,j (t) + n ˆ k,j (t) (18)value of this parameter can even make the cost function infinite zk,j (t + 1) = A irrespective of the control strategies. In context to our problem, ˆ is a 3 × 3 identity matrix and n ˆ k,j is a gaussian where B this parameter can be regarded as a design parameter which disturbance. The above state space model can be written in can be varied according to a desired criterion and which in the following standard form: turn can give less or more weight to the rate deviation term xk,j (t + 1) = Ak,j (t)xk,j (t) + Buk,j (t) + nk,j (t) (19) in the cost function. The control problem with exponential [ ] cost function and the risk-sensitive parameter is called a “riskzk,j (t) where xk,j (t) = , Ak,j (t) = sensitive control problem”. We therefore re-formulate our 1 problem as a risk-sensitive control problem where the value [ ] [ ] ˆ k,j (t) fk,j (t) ˆ 0 function is defined as, A B { } , B = , uk,j (t) = 0 1 0 0 (23) Vk,j = E eµJk,j 0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


where µ > 0 is the risk-sensitive parameter. By applying the logarithmic transformation we have, Wk,j =

inf

{uk,j (0)...,uk,j (T )}

1 log Vk,j µ

(24)

Now, our goal is to find the sequence of control variables {uk,j (0)..., uk,j (τ )} that minimizes the cost function (24) subject to the linear stochastic state equation (19). Intuitive View of the Risk-Sensitive Criterion: In order to have an intuitive view } of the risk-sensitive criterion, let, { Rµ := µ1 log E eµJk,j . We look at the Taylor’s expansion of Rµ for small value of µ which is given by µ Rµ = E{Jk,j } + var(Jk,j ) + o(µ2 ) (25) 2 This means that the risk-sensitive criterion considers the expectation as well as the variance. In other words, by minimizing the cost function Wk,j in (24), we can simultaneously minimize the average value of Jk,j (i.e. the average rate deviation) and the variance of the rate deviation. Besides, one can notice the following, • If µ −→ 0, our problem becomes risk-neutral. In other words, the minimization of Wk,j is equivalent to the minimization of the average delay. • If µ > 0, our problem is a risk-averse, i.e. depending on the value of µ, we can increase or decrease the sensitivity of the system to the delay. Notice that the cost Wk,j is not linear in µ.

˜ µk,j (t) can be according to the risk sensitive control theory, P computed recursively using the following equations (it can be noticed that these equations depend on matrices Ak,j and B that are known), ˜ µk,j (t + 1) P

=

Pµk,j (t + 1) + [ ]−1 1 µ µ Pk,j (t + 1) I − Pk,j (t + 1) Pµk,j (t + 1) µ (27)

Pµk,j (t) is a 4 × 4 matrix obtained by the following backward update equation, µ

˜ k,j (t + 1)Ak,j (t + 1) − Pµk,j (t) = Q + Ak,j (t + 1)T P µ ˜ k,j (t + 1)BMBT P ˜ µk,j (t + 1)Ak,j (t + 1); Ak,j (t + 1)T P ˜ µk,j (τ ) = 0 P (28) [ ] −1 ˜ µk,j (t + 1)B where M = R + BT P . Notice that in the case ˜ µk,j (t) = P ˜ 0k,j (t) of risk-neutral problem, i.e. when µ −→ 0, P is given by the following update equation (i.e. no need for equation (27)), ˜ 0k,j (t + 1)Ak,j (t + 1) − ˜ 0k,j (t) = Q + Ak,j (t + 1)T P P 0

0

˜ k,j (t + 1)BN BT P ˜ k,j (t + 1)Ak,j (t + 1); Ak,j (t + 1)T P

˜ 0k,j (τ ) = 0 P (29) ] [ −1 ˜ 0k,j (t + 1)B . Based on the above where N = R + BT P optimal control and update equations, we develop Algorithm 1. C. Complete Problem Representation This is a joint power control and rate adaptation algorithm that Based on the derivation/development in subsections IV.A is implemented by each transmitter-receiver pair k, j in the netand IV.B (19)-(24), the joint rate and power control problem work. In this algorithm, in every fairness window p comprising is formulated as a risk sensitive control problem in stan- of τ = N time slots, equations (27-29) can be implemented dard form. The objective is to find the sequence of control offline since the matrices involved in the computation are {uk,j (0)..., uk,j (τ )} that minimizes the cost function (24) sub- either identity matrices or they comprise of constant entries ject to the linear stochastic state equation (19). Mathematically, which are known. Therefore, inµ steps 2-5, each transmitter ˜ k,j (τ ) = 0, and solves the k in the network starts from P the problem can be represented as follows. backward equations (27-29) offline to determine the values of { ˜ µk,j are ˜ µk,j (i), ∀i = 1, . . . , N time slots. Once the matrices P P P := min Jk,j = E {exp(Jk,j )} computed, the online part of the algorithm is performed to {uk,j (0)...,uk,j (τ )} } determine the power and rate allocations described as follows. | xk,j (t + 1) = Ak,j (t)xk,j (t) + Buk,j (t) + nk,j (t) At the end of each time slot t, each transmitter computes the optimal control vector in step 8 of this algorithm. Note that the current value of state vector xk,j (t) is required for this computation. Steps 9 and 10 in of the algorithm update the D. Solution of the Problem power and target SINR. These updates come from the fact that Since our developed framework (19)-(24) has a standard introducing control uk,j (t) in the state equation (19) is nothing form, we can find the optimal sequence of control by using but the introduction of upk,j (t) and u∗k,j (t) respectively into (8) the risk sensitive control theory [45]. The optimal controller and (14). The obtained power and target SINR equations (30) is given as, and (31) equipped with a control strategy allows the dynamics uk,j (t) = of these stochastic equations to meet the streaming and power control requirements. In steps 11 and 12 we update the values ]−1 [ ˜ µk,j (t + 1)Ak,j (t)xk,j (t) (26) of γ k,j (t + 1) and γ fk,j (t + 1). Step 13 updates the state vector ˜ µk,j (t + 1)B BT P − R + BT P xk,j (t + 1) in the next time slot. Video arrival rate is then ˜ µk,j (t) is a 4 × 4 matrix. It can be seen that the adapted in step 14 of this algorithm. It can be noticed that in where P implementation of the above optimal control is simple and can step 7 of this algorithm, we are updating the value of ξk,j (t) ˜ µk,j (t) is known. Fortunately, using (15) which ensures that the rate adaptation occurs every be done online provided that P 0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


Algorithm 1 Distributed Joint Power Control and Rate Adaptation Algorithm Initialize: µ, p = 1, fk,j (p), τ = N , xk,j (t) = [γ ∗k,j (t) γ k,j (t) γ fk,j (t) 1]T , pk,j (t), ϵk,j , γ max , δˆt , nt (t), ξk,j , βk,j . 1: for p = 1, 2, . . . . . . do 2: for i = τ, . . . . . . , 1 do 3: Determine the value of ξk,j (i) using (15). 4: Use the backward update equations (27) and (28) to ˜ µk,j (i). compute the value of P 5: end for 6: for t = 1, . . . . . . , τ do 7: Determine the value of ξk,j (t) using (15). 8: Use (26) to determine the optimum control vector uk,j (t) = [u∗k,j (t) upk,j (t) 0 0]T . 9: Determine the transmit power using,

TABLE II S IMULATION PARAMETERS Parameter ϵk,j βk,j δˆ ϵ nt T fk,j wk,j µ

Value 0.01 0.01 0.01 0.0001 0.01 equal for all users 1 0.05

desired value of the video rate (video quality), and the transmit power needed for video transmission at this rate. The video transmitter then transmits video to the corresponding receiver with the newly obtained transmit power and video rate.

pk,j (t + 1) = pk,j (t) + βk,j {γ ∗k,j (t) − γ k,j (t)} +upk,j (t)

(30)

V. S IMULATION R ESULTS

We consider a network with 14 transmitter-receiver pairs uniformly distributed in a region of 1km radius. The trans{ } γ ∗k,j (t + 1) = γ ∗k,j (t) + ξk,j (t) γ fk,j (t) − γ ∗k,j (t) mitters are assumed to use the same bandwidth of 1MHz where a transmitter’s multimedia data is destined to only { T } +ξk,j (t) fk,j − fk,j (t) γ ∗k,j (t) one receiver. Each node has a peak power constraint pmax ∗ +δˆt nt (t) + uk,j (t) (31) which corresponds to the amount of power for which its SINR level reaches to 20 dBm. We consider a frequency selective 11: Update γ k,j (t + 1) = pk,j (t + 1) + g k,j (t) for next Rayleigh fading wireless channel with an exponential delay time slot. profile. The CINR of each node is not assumed to follow a 12: Update γ fk,j (t + 1) using (13) for next time slot. given distribution but each node compute its SINR at each 13: State vector for next time slot is then: time based on the received interference from other nodes. The power spectral density of noise is -124 dBm/Hz. Path xk,j (t+1) = [γ ∗k,j (t+1) γ k,j (t+1) γ fk,j (t+1) 1]T losses are calculated according to Cost-Hata Model [46]. The ∗ 14: Determine the equivalent arrival rate rk,j (t+1) using time space is divided into slots where the duration of each Transmission Time Interval (TTI) is 1ms. The length of the γ ∗k,j (t + 1). rate update window is assumed to be 10 time slots while the 15: end for length of fairness window is assumed to be 200 time slots. 16: Update the fairness fk,j (p) using (3). Simulation is run for 10,000 time slots (which constitutes 50 17: end for fairness windows). Values of other parameters are given in the Table II. We consider 8 test videos sequences: News, Hall, Silent, m time slots, i.e., arrival rate adaptation occurs after every arrival rate window. Finally, in step 16, fairness is updated and City, Foreman, Crew, Harbour and Mobile, which are YUV sequences commonly used in the literature. These video sethen the algorithm is repeated for the next fairness window. quences can be downloaded from [47]. In our simulations, we use PSNR as a measure of video quality as done in E. Implementation reference [12]. In reference [12], these test video sequences The proposed controller for the joint power and rate al- are encoded using scalable video coding (SVC), which is a location is implemented at the video transmitter nodes. It is special extension of H.264/MPEG4-AVC [5], [48] and then a therefore distributed where each node performs locally its relationship between data rate and PSNR is derived. All the power and video rate control based on its estimated SINR. video sequences in [12] are encoded at 30Hz with GOP size At each fairness window, which is equal to multiple video of 16 pictures and an intra period of 64 frames. In our paper, frames or few hundred of ms, the nodes exchange their average we use the same encoding scheme and the same relationship received rates over this fairness window Tw (only the average between the data rates and the PSNR. Using our algorithm, we rate) so that each node can compute its fairness fk,j (see obtain the data rates achieved at the physical layer, which is definition II.2). Depending upon the target and instantaneous then converted into the resulting PSNR indicating the quality fairness of the corresponding receiver, a given target bit error of the encoded video sequences. In Fig. 2, we assume that all the users in the network the rate criterion, and the maximum acceptable/feasible transmit power level, each video transmitter node solves the above demand same video sequence and their fairness requirements control problem.The video transmitter node, thus, gets the are also the same. We vary the number of users in the 10:

Update target SINR γ ∗k,j using,



system and compare the performance of our algorithm with the dynamic scheduling algorithm presented in [12]. In this algorithm, the weighted sum of the video quality of all the users is maximized by determining a dynamic scheduling vector in every time slot that is also able to achieve the target data rate which is related to the video quality. In this figure, for different number of users in the system, we repeat the simulations for each video sequence and then plot the average PSNR over all the eight video sequences. As the number of users increases, the average PSNR of both the algorithms decrease due to the increased inter-user interference. However, it is obvious that the average PSNR achieved by our proposed algorithm is always greater than the algorithm presented in [12]. Since all the transmitters are using the same frequency band, it indicates that our proposed algorithm where we perform a joint power and rate adaptation can better manage the interference, thereby achieving a higher average PSNR and hence better video quality.

38

Average PSNR (dBs)

36

34

32

30

28 Proposed Joint Power Control and Rate Adaptation Algorithm Dynamic Scheduling Algorithm [12]

26 0

2

4

6 8 User Index

10

12

14

Fig. 3. Fairness Comparison: Average PSNR vs User index (14 users in the system and averaged over all the eight video sequences 38 40 39

38

36 37

Average PSNR (dBs)

Average PSNR (dBs)

37

35

34

33

35

34

33

Proposed Joint Power Control and Rate Adaptation Algorithm Dynamic Scheduling Algorithm [12]

32 4

36

5

6

7

8 9 10 Number of Users

11

32

12

13

14

Fig. 2. Average PSNR vs Number of Users in the network: Averaged over all the eight video sequences

Proposed Algorithm: NEWS Video Sequence Proposed Algorithm: HALL Video Sequence Proposed Algorithm: SILENT Video Sequence Proposed Algorithm: CITY Video Sequence Proposed Algorithm: FOREMAN Video Sequence Proposed Algorithm: CREW Video Sequence Proposed Algorithm: HARBOUR Video Sequence Proposed Algorithm: MOBILE Video Sequence

31

30 −20

−15

−10

−5 0 Maximum SINR (dBs)

5

10

Fig. 4. Average PSNR vs γ max

In Fig. 3, we compare the fairness among users. We consider 14 users and again the results are averaged over all the eight video sequences. The simulation is run over 1000 fairness windows. We can observe that the average PSNR achieved by our algorithm is almost same for all the users in the network. In fact, the difference between maximum average PSNR and minimum average PSNR is only (36.1170 − 35.0050) = 1.1120 dBs. On the other hand, the average PSNR achieved by the algorithm presented in [12] for different users have very large variations and the difference between maximum average PSNR and minimum average PSNR is (37.1011 − 27.0050) = 10.0961 dBs. In this case, the video quality of user 7 which has the worst average PSNR is very poor and might be unacceptable. Our algorithm does not suffer from these issues due to the fact that we have explicitly incorporated the fairness constraint in our equations. In Fig. 4, we plot the average PSNR achieved by our algo-

rithm for individual video sequences vs γ max (γ max is defined as the SINR level that is achieved when a node transmits at maximum acceptable power pmax ). In these simulations, we consider 14 users in the network. The difference in the achieved PSNR of individual video sequences is due to their different video rate requirements as explained in [12]. We can see that for some video sequences e.g. Mobile and Harbor, the achieved PSNR is poor when the allowable transmit power is small and it increases when we increase the value of γ max . For Mobile and Harbor video sequences, when achieved rate at physical layer is low, the average PSNR is also low and the quality of video only improves at sufficiently higher values of achieved data rate which happens when the value of γ max is increased. On the contrary, for some video sequences, e.g., News, Hall, etc., the achieved PSNR is almost constant and does not change much when the value of γ max is increased.



In fact, for such video sequences, at low value of pmax , one can provide sufficient video quality (the relationship between rate and average PSNR can be seen in Figure 1 in [12]). Our algorithm is able to maintain sufficient video quality even when the allowable power is low.

1 LQG R.S with µ = 0.15 R.S with µ = 0.31

0.9 0.8

1 LQG RS with µ = 0.35 RS with µ = 0.31 RS with µ = 0.21

0.9

Normalized Cost

0.8

Pr(Delay > DTh)

0.7 0.6 0.5 0.4

0.7

0.3

0.6

0.2

0.5

0.1

0.4

0

5 D

Th

0.3

10 15 = Target Delay (TTI)

20

Fig. 6. Probability that the experienced delay is greater than a given target delay

0.2 0.1 0 0

100

200

300

400 t

500

600

700

800

Fig. 5. Cost Comparison: Proposed Risk-Sensitive (RS) scheme with different values of risk-sensitive parameter µ

In Fig. 5, the normalized cost versus time for several values of risk-sensitive parameter µ is plotted which illustrates the performance of the proposed risk-sensitive (RS) approach in terms of the cost incurred that is a function of the difference/deviation between the target and the actual SINR levels. Again the results are averaged over all the video sequences. Recall that when µ −→ 0 the controller becomes a Linear Quadratic Gaussian (LQG) controller. We therefore denote the case when µ −→ 0 by LQG. The figure highlights the performance improvement of the RS approach by varying the value of the risk-sensitive parameter. In Fig. 6, we plot the probability that the experienced delay is greater then a given target delay. In video streaming the stringent delay constraint should be satisfied and the packets that arrive at the queue of the node will be dropped if they are not transmitted in the given target delay, DTh . The target delay is given in terms of Transmit Time Intervals (TTIs) where each TTI is equal to 1ms. The results in the figure provides guidelines for choosing the appropriate value of the risk-sensitive parameter for achieving a given target delay constraint. VI. C ONCLUSION In this paper, we studied the challenging problem of power allocation and video bitstream adaptation for video streaming in multi-node wireless networks with time-varying interference. We developed a cross-layer optimization framework that performs instantaneous power control at the PHY/MAC layer joint with video rate adaptation (in an average manner) at the video coding layer (upper layer). The proposed joint power control and rate adaption framework exploits the time diversity

of the nodes’ channels, takes into account the stringent delay constraints of the video services, and guarantees fair sharing of the limited available resources among the users. In view of the time varying channels and interferences, stringent delay constraints, and a certain fairness/satisfaction criterion, we modeled our problem as a stochastic control problem. We then developed a risk-sensitive control approach for this problem by introducing a non-linear cost function called risk-sensitive cost function. We then provided the optimal solution to the risksensitive problem and provided simulation results to assess the performance of our proposed framework. Simulation results show that a given target delay can be achieved by choosing an appropriate value for the risk-sensitive parameter. Simulation results also show that our proposed framework overwhelmingly outperforms the existing frameworks. A PPENDIX A. Proof of Proposition III.1 We start our analysis by writing the expression of the CINR at the tth time slot between nodes k and j (we consider that receiver k is connected to transmitter j), gk,j (t) =

Gk,j (t) J ∑ η+ pl (t)Gk,l (t)

(32)

l=1,l̸=j µ sk,j /10 In general, Gk,j (t) is proportional to d−˜ |hk,j (t)|2 k,j 10 where dk,j is the distance between receiver k and transmitter j, µ ˜ is the path loss slope (˜ µ = 3, 4 in macro cell and µ ˜ = 2 in micro cell) and 10sk,j /10 corresponds to Log-normal shadowing. Owing to the fact that 10sk,j /10 is Log-normally distributed, the variable sk,j has a Gaussian distribution with zero mean and standard deviation σs (σs2 ranges from 8 to 12 dB) [49]. The coefficient hk,j (t) denotes the fast fading at the tth time slot for the channel between receiver k and transmitter j. Since we consider the instantaneous CINR, the time index



t will be ignored in this analysis for simplicity. The CINR can then be written as 1 gk,j = (33) J ∑ dk,l −˜µ sk,l /10 2 pl ( ) 10 |hk,l | η dk,j −µ ˜ d l=1,l̸=j k,j + sk,j /10 sk,j /10 2 2 10

|hk,j |

10

|hk,j |

There are different approaches in the literature to approximate the distribution of SIR (e.g. [36]–[38], and the references therein). In this paper, we use the following approach. We approximate the product of lognormal and Chi-Square variables by a lognormal variable and then approximate the sum of lognormal variables by another lognormal variable using Fenton-Wilkinson approximation [39] [37] [40] [41]. The proof therefore consists in several steps. Step 1 - Let us first consider the variable yk,l = 10sk,l /10 |hk,l |2 . which is the product of two variables: a lognormal and a chi-square variable (|hk,j |2 has a central ChiSquare distribution with two degrees of freedom since |hk,j | has a Rayleigh distribution). Notice that 10sk,l /10 = eθsk,l where θ = log(10)/10. X = 10sk,l /10 has lognormal dis−

(log(x)−θµs )2

2 1 2θ 2 σs tribution given by √2πθσ . In order to find e sx the probability distribution function for yk,l , one can proceed by considering the cumulative distribution function and then differentiate to find the pdf. However, in our case, we simply use the definition of the pdf of the product of two independent random variables. In fact, if X and Y are two independent positive random variables with pdf fX (x) and ∫ ∞ fY (y) respectively, 1 then the pdf of Z = XY is fZ (z) = 0 fX (x)fY ( xz ) |x| dx. From this result, the probability distribution function of yk,l can be written as ∫ (log(t)−θµs )2 1 1 ∞ − yk,l 1 − 2 2θ 2 σs e 2t √ dt (34) pdf (yk,l ) = e 2 0 t 2πθσs t

Now, the objective is to provide an approximation of the closed form expression of the above integral. For that, we use the following approximation (from [39]) ∫ ∞ (log(x)−µx )2 1 1 − 2 2σx dx ≃ e−z/x √ e x 2πσ x 0 x ( ) 1 √ e 2πσz z

−

(log(z)−µz )2 2 2σz

(35)

where µz = −C + µx and σz2 = ζ(2) + σx2 . C is the Euler Constant (C=0.5772) and ζ(2) = π 2 /6 is the Riemann-Zeta function. Using the above approximation and after making the variable change, 2t = t′ in (34), the pdf of yk,l is obtained as given by − 1 e pdf (yk,l ) = √ 2πσk,l yk,l

(log(yk,l )−µk,l )2 2σ 2 k,l

(36)

where θ = log(10)/10, µk,l = −C + log(2) + θµs = 2 = ζ(2) + θ2 σs2 . This approximation −C + log(2) and σk,l results from the fact that the product or the sum of a log-normal variable with other variables of sharper frequency distributions (e.g., exponential, Chi-square,

etc.) is dominated at the higher order moments by the log-normal distribution with largest logarithm variance [50]. Similarly, yk,j = 10sk,j /10 |hk,j |2 can be approximated by a log-normal variable with parameters µk,j = −C + log(2) + θµs = −C + log(2) and 2 σk,j = ζ(2) + θ2 σs2 . Now, yk,j can be approximated vk,j as e where vk,j is a Gaussian variable with mean 2 µvk,j = µk,j and variance σv2k,j = σk,j . Step 2 We now consider Xk = ∑J dk,l −˜ µ 10sk,l /10 |hk,l |2 . l=1,l̸=j pl ( dk,j ) k,l −˜ The variable (pl ( dk,j ) µ 10sk,l /10 |hk,l |2 ) has a log-normal

d

2

k,l −˜ distribution with mean value Ek,l = pl ( dk,j ) µ eµk,l +σk,l /2

d

2

2

k,l −2µ and variance Vk,l = p2l ( dk,j ) (eσk,l − 1)e2µk,l +σk,l . Therefore, Xk is the sum of independent log-normal variables. According to [40], using the approximation of Fenton-Wilkinson, the sum of a finite number of independent log-normal variables can be approximated by a log-normal variable. The first two (1) (2) order moments (respectively mk and mk ) of the equivalent log-normal variable can be computed, using the FentonWilkinson (FW) approximation, as follows [40] [41]: ∑J (1) mk = l=1,l̸=j Ek,l (37) ∑J ∑J−1 ∑J (2) 2 mk = l=1,l̸=j (Ek,l +Vk,l )+2 l=1,l̸=j l′ =l+1,l′ ̸=j Ek,l Ek,l′ (38) ∑J dk,l −µ ) 10sk,l /10 can be modTherefore, Xk = l=1,l̸=j pl ( dk,j ′ eled/represented by evk where vk′ has a Gaussian distribution with mean µvk′ and variance σv2′ [41] given by k [ ] [ ] 1 (2) (1) (39) µvk′ = 2Log mk − Log mk 2 [ ] [ ] (2) (1) σv2′ = Log mk − 2Log mk (40)

d

k

Step

3

- It is straightforward from above that

µ ˜ d− k,j sk,j /10 |hk,j |2 η 10

can be approximated by a lognormal ˜ d− µ

2

µk,j +σk,j /2 ′ variable with mean Ek,j = k,j and variance η e ( −µ˜ )2 2 2 d k,j ′ Vk,j = (eσk,l − 1)e2µk,l +σk,l . It can then be written η ′′

′′ as evk,j where vk,j is a gaussian variable with mean µv′′ and k,j σv2′′ given by k,j

′ Vk,j 1 ′ µv′′ k ,j = Log(Ek,j ) − Log(1 + ′ )2 ) 2 (Ek,j

σv2′′ = Log(1 + k,j

′ Vk,j ′ )2 ) (Ek,j

(41) (42)

Step 4 - The CINR expression can now be approximated as follows: 1 gk,j = −v′′ (43) ′ k,j e + e(vk −vk,j ) ′

′′′

where e(vk −vk,j ) = evk,j has also a log-normal distribution 2 ′′′ and σ ′′′ given by with parameters µvk,j v k,j

µv′′′ k ,j = µvk′ − µvk,j

(44)

σv2′′′ = σv2′ + σv2k,j

(45)

k,j

k



One can see clearly that v ′′ k , j and v ′′′ k , j are correlated. The sum of two correlated lognormal variables can also be approximated by a lognormal distribution using the Fenton-Wilkinson approximation [40] as follows. Let ρk,j = ′′ ′′′ E[(vk,j −µv′′ ,j )(vk,j −µv′′′ ,j )] k k σv′′ σv′′′ k,j

k,j

be the correlation coefficient be′′

tween v k , j and v k , j. Recall that the lognormal e−vk,j has parameters −µv′′ k ,j and σv2′′ and the correlation coefficient k,j between −v ′′ k , j and v ′′′ k , j is −ρk,j . The first two moments of the equivalent lognormal are, respectively: ′′

(1) m′ k

′′′

=e

m′ k = e (2)

+2e

k,j

σ 2 ′′ v k,j 2

+e

−2µv′′ +2σv2′′

−µv′′ +µv′′′ k,j

−µv′′ +

k,j

k,j

e

k,j

µv′′′ +

+e

k,j

σ 2 ′′′ v k,j 2

(46)

2µv′′′ +2σv2′′′ k,j

k,j

2 2 1 ′′ σv ′′′ ) 2 (σv ′′ +σv ′′′ −2ρk,j σvk,j k,j k,j k,j

(47)

The CINR can then be approximated as follows 1

gk,j = e

f vk,j

(48)

f where vk,j is a Gaussian variable with mean and variance given, respectively by [40] [41] [ ] 1 [ ] (1) (2) µvf = 2Log m′ k − Log m′ k (49) k,j 2 [ ] [ ] (2) (1) (50) σv2f = Log m′ k − 2Log m′ k k,j

The probability distribution function of CINR can therefore be approximated by a log-normal distribution with parameters −µvf and σv2f . This completes the proof. k,j

k,j

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