Computing Quality of Experience of Video Streaming in Network with LongRangeDependent Traffic
Abstract
We take an analytical approach to study the Quality of user Experience (QoE) for video streaming applications. Our propose is to characterize buffer starvations for streaming video with LongRangeDependent (LRD) input traffic. Specifically we develop a new analytical framework to investigate Quality of user Experience (QoE) for streaming by considering a Markov Modulated Fluid Model (MMFM) that accurately approximates the Long Range Dependence (LRD) nature of network traffic. We drive the closeform expressions for calculating the distribution of starvation as well as startup delay using partial differential equations (PDEs) and solve them using the Laplace Transform. We illustrate the results with the cases of the twostate Markov Modulated Fluid Model that is commonly used in multimedia applications. We compare our analytical model with simulation results using ns3 under various operating parameters. We further adopt the model to analyze the effect of bitrate switching on the starvation probability and startup delay. Finally, we apply our analysis results to optimize the objective quality of experience (QoE) of media streaming realizing the tradeoff among different metrics incorporating user preferences on buffering ratio, startup delay and perceived quality.
I Introduction
It has been observed that people addicted to watching streaming videos constitute more than half of the Internet traffic. With the introduction of smartphones, mobile networks are witnessing an exponential traffic growth every year. This leads to scenario where Internet and wireless networks are pushed to operate close to their performance limits, dictated by current architectural considerations. Though much effort has been expended and in turn significant progress has been made in recent years to increase the capacity of mobile networks, there is little progress on dealing with the user satisfaction, which is strongly related to the Quality of Experience (QoE). This is the big challenge that the operators face today because they have to look at both the server side and the client side to make a link between the quality of service (QoS) of the network and the client satisfaction which depends on the QoE. Empirical studies in [7, 4, 11, 12] has identified critical metrics that affect the QoE through the user engagement:

Starvation probability. Denoting the probability that a streaming user sees frozen images.

Average bitrate. Denoting the mean video quality over the entire session.

Bitrate stability. Describing the jittering of video quality during the entire session.

Startup delay. Denoting the waiting duration between the time that the user requests streaming service and the time that media player starts to play.
Paper [7] pointed out that the buffering ratio is most critical across genres. For example, 1% increase in buffering reduces 3 minutes for a 90minutes live video streaming. [16] showed that the total time spent rebuffering and the frequency of rebuffering events have substantial impact on QoE. Under this context, media servers and network operators face a crucial challenge on how to avoid the degradation of user perceived media quality based on these metrics. However, development of new models as function of these metrics can help operators and content publishers to better invest their network and server resources toward optimizing these metrics that really matter for QoE.
In this paper we focus on a setting in which a video is streamed over a wireless network which is subject to a lot of constraints like bandwidth limitation and rate fluctuations due to the frequent changes of channel states and mobility [5]. Indeed, time varying network capacity is especially relevant when considering wireless networks where such variations can be caused by fast fading and slow fading due to shadowing, dynamic interference, and changing loads. To addresse this issue, we focus on performance modelling and analysis of a streaming video with Long Range Dependence (LRD) traffic and variable service capacity. Due to the inherent difficulty and complexity of modelling fractallike LRD traffic, we assume that the arrival of packets at player buffer are characterized by a Markov Modulated Fluid Model, which accurately approximates the traffic exhibiting LRD behaviour and mimics the real behaviour of multimedia traffic with shortterm and longterm correlation [10]. In comparison to related works, our whole analysis is on transient regime. We construct sets of Partial Differential Equations (PDEs) to derive the starvation probability generating function using the external environment, which is described by the Continuous Time Markov Chain (CTMC). This approach predicts the starvation probability as function of the file size as well as the prefetching threshold. Moreover we provide relevant results to understand on how the starvation probabilities are impacted by the variation of traffic load and prefetching threshold. We do simulations to show the accuracy of our model using ns3. Achieving this goal, we are able to identify through our model the dependencies between quality metrics. For example, startup delay can reduce rebuffing ratio. Similarly bitrate rate switching can reduce buffering.
With the results developed in this work, we are able to answer the fundamental questions: How many frames should the media player prefetch to optimize the users’ quality of experience? From what file size the adaptive coding is relevant to avoid the starvation? How bitrate switching impacts the QoE metrics? Knowing these answers enables the user to maximize his QoE realising the tradeoffs among different metrics incorporating user preferences on rebuffering ratio, startup delay and quality [13, 14, 15]. We further introduce an optimization problem which takes these key factors in order to achieve the optimal tradeoff between them. We adopt a more flexible method by defining an objective of QoE by associated a weight for each metric based on user preferences [13].
Ii Related works
QoE analysis over wireless networks has been studied for many years. In [1], authors study the QoE in a shared
fastfading channel using an analytical framework based on Takacs Ballot theorem. They use a GI/D/1 queue to model the
system, so they assume that the arrival process is independent and identically distributed (i.i.d). In [9],
the analysis of buffer starvation using M/M/1 queue is performed. They use a recursive method to compare the results with the Ballot
theorem method even if the recursive method did not offer explicit results. They assume an i.i.d arrival process that is a
rough model of streaming services over the wireless networks. Since the performance measures depend on the autocorrelation
structure of the traffic, a consensus exists about the limitation of the Poisson process to model the traffic behaviour.
In [17], authors develop an analytical framework to investigate the impact of network dynamics on the user
perceived video quality, they model the playback buffer by a G/G/1 queue and use the diffusion approximation method to
compute the QoE. The QoE of streaming from the perspective of the network flow dynamics is studied in [5]. The throughput of a tagged user is governed by the number of the other users in the network. This study shows
that the network flow dynamics is the fundamental reason for playback starvation.
The rest of this paper is organized as follows: In section III, we describe the system model while section IV presents the analysis of the queuing system model. Section V describes the performance analysis of the quality of experience and section VI presents explicit results for two states MMFM. Section VII shows numerical results and section VIII concludes this paper.
Iii System Model Description
We consider a single user receiving a media file with finite size in streaming. Generally, media files are divided into blocks of frames. When a user makes a request the server segments this media into frames and transfers them to the user through the network (wired and wireless links). When frames traverse the internet, their arrivals are not deterministic due to the dynamics of the available bandwidth. One of the main characteristics of wireless traffic and Internet traffic in general is the rate fluctuation caused by fast fading and slow fading due to shadowing, dynamic interference, and changing load. Moreover, data packet arrivals in cellular networks are found to be correlated over both short and longtime scales. This is generally due to the arrival of packets bursts of comparable size, often leading to high instantaneous arrival rates. Hence, video flows through the Internet with fluctuating speed. In this paper, we assume that frames arrive to the playout buffer with a rate that can take values from finite set . The rate of arrival frames is governed by a ContinuousTime Markov Chain (CTMC) with infinitesimal generator .
where .
The maximum buffer size is assumed to be large enough so that the whole file can be stored. At the user side, incoming frames are stocked in a buffer and from there they are played with a rate (e.g., 25 frames per second (fps)) in the TV and moviemaking business. We quantify the user perceived media quality using two measures called startup delay and starvation. There is ongoing research on mapping these two measures on standard human evaluated QoE measures. As explained earlier, the media player wants to avoid the starvation by prefetching packets. However, this action might incur a long waiting time. In what follows, we reveal the relationship between the startup delay and the starvation behaviour, with the consideration of the file size.
We consider a fluid model that has been proven to be a powerful modeling paradigm in many applications and relevant to capture the key characteristics that determine the performance of networks. Let denote the effective input rate in state . Hence the matrix of the effective rates is , which is a diagonal matrix . We denote by the length of playout buffer of playback at time t. Let be the first time the buffer is empty before reaching the end of the file, i.e., and be the startup delay where is the prefetching threshold. In the next section we provide mathematical analysis to compute the distribution of the number of starvation and startup delay for a general bursty arrival process.
Iv Analysis of the Queuing System Model
Iva Laplace Transform of the Starvation Probability
We compute the Laplace transform of the probability of starvation given the ContinuousTime Markov Chain . We define to be the probability of starvation in state before time , given the initial state and the initial queue length .
(1) 
for and . It is clear that the CTMC cannot be in a state at time if . Hence
Let be the steady state probability vector of the CTMC where is the probability to be in the state at the stationary regime. The expected input and output rates are and respectively. The buffer is stable if . Conditioning on the first transition from the state at time we have
(2) 
Taking the limit and after some algebraic simplification we obtain the following partial differential equation
(3) 
with the initial conditions
where .
The Laplace Stieljes Transform (LST) of is
for and . Taking the LST of Equation (3) and using the fact that for all , we find
(4) 
For a fixed value of , we take
as a solution to Equation (4). Substituting in (4) we get
where the scalar and the vector are to be determined. The theorem from [6] gives
(5) 
where the coefficients are obtained by solving
are the roots with negative real parts of and are the corresponding eigenvectors satisfying the equation
(6) 
IvB Laplace Transform of the Startup delay
We consider the previous system during the prefetching process and we denote by the length playout buffer of playback at time . Let
be the first time that the length playout buffer reaches .
is the time that the system will take to accumulate content in the buffer. This distribution is difficult to solve directly, so we resort to the following duality problem:
Duality problem: What is the starvation probability by time if the queue is depleted with rate and the duration of prefetching contents is ?
This duality problem allows us to compute the prefetching delay as a probability of starvation. We define to be the probability of starvation before time at the state , conditioning on the initial state and the initial prefetching content , i.e., the startup threshold.
(7) 
for and .
Conditioning on the first transition from the state at time ,
(8) 
Taking the limit and after some algebraic simplification we obtain the following partial differential equation
(9) 
with the same initial conditions as in section IVA,
where and .
(10) 
where the coefficients are obtained by solving
and are the roots of and are the corresponding eigenvectors.
In what follows, we compute the probability that the prefetching ends at a given state [3]. For this purpose, we define
(11) 
to be the probability that the prefetching ends at state given the initial state i and the initial queue length where is the prefetching threshold. In the time interval , conditioning on the first transition from the state at time , we have
(12) 
After some algebraic simplification and letting yields the differential equation
(13) 
with the boundary condition
Let
Eq. (13) becomes: . is given by
(14) 
Using Eq.(14) and the initial conditions, we get
(15) 
where , is the diagonal matrix containing all the eigenvalues of and is an invertible matrix.
IvC The Probability of Starvation and the Startup Delay
In the previous sections we derived explicit expressions for the LaplaceStieltjes Transform of the probability of starvation and the startup delay. In this section, we present theoretical models to find the corresponding probability of starvation and startup delay. The Laplace Stieljes Transform of is
where is the probability density function of .
Lemma 1 (Bromwich inversion integral).
Given the Laplace transform , the function value can be recovered from the contour integral
(16) 
where is a real number to the right of all singularities of , , and the contour integral yields the value for .
It is shown in [8] that for real value functions, has the following form
(17) 
According to the Bromwich inversion integral, can be calculated from the transform by performing a numerical integration (quadrature). We use a specific algorithm based on the Bromwich inversion integral. It is based on a variant of the Fourierseries method  the trapezoidal rule  which proves to be remarkably effective. If we use a step size , then the trapezoidal rules gives
(18) 
where since is real. Replacing by which is the Laplace transform of , we get the probability of starvation before time
(19) 
The infinite series in (19) can simply be calculated by simple truncating because it converges, but more efficient algorithm can be obtained by applying a summation acceleration method. An acceleration technique that has proven to be effective in our context is Euler summation, after transforming the infinite sum into a nearly alternating series in which successive summands alternate in sign. We convert (19) into a nearly alternating series by letting and
where
Let be the approximation with the infinite series truncated to terms, i.e.,
where t is suppressed in the notation and . We apply Euler summation to terms after an initial , so that the Euler sum approximation is
(20) 
Euler summation can be very simply described as the weighted average of the last partial sums by a binomial probability with parameter and . Hence, (20) is the binomial average of the terms . The implementation of the algorithm takes into account the values of . As in [8], we use where . After simplification and letting be 1, we get the Euler approximation of the inverse which seems to be a good approximation
(21) 
Eq. (21) looks complicated, but it consists of only additions, that is a low computation level. To have the cdf , we just replace by . The same formula holds for the startup delay distribution in replacing by .
V Performances Analysis of the Quality of Experience
In this section we compute the QoE metrics based on the analysis derived in the previous section.
Va The Probability of Starvation
We consider a single user receiving a media file with size . The necessary time to play the whole video if there is no starvation is . Hence, using the first passage time distribution , the probability of starvation happened in state before reaching the end of file given the initial state , is given by
(22) 
The starvation of probability before time gives an idea of the severity of the starvation during the video session. Let be the mean continuous playback time if the initial state is , the prefetching threshold is and the starvation happens in state . is an important measure for the severity of starvations. A small means that the starvation events happen frequently. We find by taking derivatives of in .
When the user starts the video session, the initial state is unknown to the system. The video starts playing when the prefetching process is finished. Conditioning on the distribution of the entry states , the distribution of the states that the playback process begins (or prefetching process ends) is computed by . Recalling that is the probability that the prefetching phase ends at state knowing that the video session starts at state . Then the starvation probability with the prefetching threshold is obtained by
(23) 
where H is a column vector, and . is called the overall starvation probability. The probability of no starvation is .
VB The distribution of the Startup delay
The startup delay is proportional to the startup threshold. But in the QoE literature, it is more practical to consider the delay rather than the threshold because the delay has a direct impact on the streaming user behaviour. Using the results of the sections IVC and IVB, we derive the cumulative distribution function of the startup delay
(24) 
where is the startup threshold, is the file size and are the Euler Summation Algorithm parameters.
VC The generating function of the starvation events
When a starvation event happens, the media player pauses until contents are rebuffered. We are interested in the probability distribution of the starvations, given the file size . We define a path as a complete sequence of frames arrivals and departures. We illustrate a typical path with starvations in Fig. 1. The path can be decomposed into three types of mutually exclusive events as follows:

Event : the buffer becoming empty for the first time in the entire path.

Event : the empty buffer after the instant given that the previous empty buffer happens at .

Event : the last empty buffer observed after the instant .
Obviously, a path with starvations is composed of a succession of events
We let , and be the probabilities of events , and respectively. The probability distribution of event is expressed as
(25) 
where V and h are x and x matrices respectively. The first starvation cannot happen at the departure of first contents because of the prefetching of contents. It cannot happen after all contents have been served because this empty buffer is not a starvation. For the starvation happens at time conditioned on the states that the playback process begins. The probability distribution of event is given by
(26) 
where H is a column vector. is the time of the th starvation. The extreme case is that these starvations take place consecutively.Then should be greater than . Otherwise there cannot have starvations. If is no less than , the media player resumes until all the remaining content is stored in the buffer. Then, starvation will not appear afterwards. In the remaining case, it is the probability of having no starvation after time . We denote by the probability of having starvations. The case with one starvation is given by
(27) 
To compute the probability of having more than one starvation, we need to find the probability of event . should not be less than in order to have starvations. Given that the buffer is empty just after time , the starvation cannot happen at because of the prefetching process. Since there are starvations in total, the starvation must satisfy . We next compute the remaining case that the and the starvations happen at time and respectively. We compute this probability using the first passage time density when the starvation happens at time and the initial time was with a prefetching process. is expressed as
(28) 
We use in this method a trick that concerns the time scale. Every time the player resumes for the prefetching process we resume also the time scale, that means if the starvation happens at time , the player will start playing at the same time with initial contents in the buffer. The probability of having starvations is given by
(29) 
In the next section, we provide explicit expressions of QoE metrics where CTMC has two states.
Vi The 2state MMFM Source
In this section, we consider a special case in which the CTMC has two states : (see Fig. 2) with infinitesimal generator and rate matrix
Our objective is to understand the interaction between the parameters of arrival process and the probability of starvation. Using the results of section IVA irrespective of the condition of stability of the queue , we get:
(30) 
It is a polynomial of degree 2 in where the two zeros are given by:
(31) 
(32) 
where . Equation (5) contains terms with only . So we have to determine the signs of and . The next propositions give the placement of these two zeros in the complex plane.
Proposition 1.
Let ,

, so and then no starvation.

and , so and then and .

and , so and then and .

then no starvation.

, and .
Proposition 2.
Let ,

, so and then no starvation.

and , so and then and .

and , so and then and .

then no starvation.

, and .
Proposition 3.
Let ,

, no starvation.

, and .

, no starvation because of the prefetching.
The LST of the distribution is given in the next theorem.
Theorem 1.

When , , and

When , , and

When , ,
where
The proof of this theorem can be find in the appendix of [18]. Taking gives the first passage time distribution for the ONOFF source.
Vii Numerical Analysis
Viia Simulation
We use ns3 simulator in order to compare the dynamics of the process with our model. The simulation topology consists on a server and a client in order to simulate the queue model. The server sends the traffic to the client following the Continuous Time Markov Chain. The client holds a buffer where the traffic is stored. The parameters of the traffic depend on the CTMC parameters. Then we analyze the behavior of the client buffer content which simulates the player. We run 10 simulations and compute the 95% confidence interval on all observed metrics, but it is not shown on all the figures for improving readability because it is very narrow. We first show the accuracy of the method that we use to invert the Laplace Transform. In Fig. 3, we plot the known inverse Laplace Transform of the function that is and the inverse using formula (21) of section IVC. Fig. 4 shows the starvation probability for a two states MMFM source for , and , that means the buffer size increases on state and decreases on state . This is done for states transitions and .
ViiB Performances Evaluation
Fig. 8 gives the CDF of the startup delay for different values of startup threshold . We can see that the startup delay increases with . On the other hand figure 6 illustrates the impact of the startup threshold on the probability of no starvation. When is large enough (near 300 pkts in the figure) no starvation will happen until the end of the video. Since, the curve grows sharply, it is clear that a slight increase in can greatly improve the starvation probability. In figure 9, we plot the probability of having no more than two starvations with , , irrespective to . When is large enough, no starvation will happen until the end of the video session. On the other side, figure 10 shows that the starvation happens for sure when the file size approaches infinity. The curves of the probability of having one starvation or two starvations increase first, and then decrease to zero. This means that the starvation can be avoided when is large enough. The two curves have a maximum value at a given startup threshold or a given file size. So one can choose the threshold to have exactly one or two starvations. This is an important measure because it allows one to achieve the buffer requirements in setting up the desired values. Indeed, a very small threshold do not help to reduce the starvation probability and very large thresholds do not further reduce the starvation probability. Hence the analytical model aims to predict the player buffer behavior in video streaming sessions. The network parameters and the video size are the framework inputs that can be used to improve the QoE related to user preferences.
ViiC Optimization of the QoE
In this section, we introduce an optimization problem of the QoE by including different metrics and incorporating user preferences by associated a weight to each metric. We denote by the cost of a user watching the media stream,
where is the number of starvation, is the startup delay, is the lost on video quality and is the fraction of the total session
time spent in low bitrate. , and are depending on the user preferences of the three metrics (starvation, startup delay and video quality). Based on the user preferences, we compare the cost of QoE for two scenarios. In scenario 1, the adaptive bitrate streaming is not used and for the second scenario, the adaptive bitrate streaming is used in order to adjust the quality of a video stream according the available bandwidth.
We consider a network that throughput varies between 200Kbps and 400Kbps. For the adaptive bitrate streaming, we have two coding rates depending on the throughput. This leads to two different frame sizes (10kbits, 20kbits). Then we compute the cost for and .
In fig. 11 and 12, we compare the cost for the two scenarios. For short video duration, the adaptive bitrate streaming is not benefit because there is a less number of starvation and the quality of the video is degraded. But, for the long video duration, the adaptive bitrate streaming becomes interesting because the low coding rate decreases the number of the starvation. In fig. 11 and 12, we can see that the value of the parameter changes the preference of the user for the quality of the video. For , , we use the adaptive bitrate streaming when the size of the file is more than 400, 600 frames respectively. Otherwise, the adaptive bitrate streaming is not necessary.
Viii Conclusion
In this paper, we have proposed a new analytical framework to compute the QoE of video streaming in the network modeled by the Markov Modulated Fluid Model. We found the probability of starvation and the startup delay in solving Partial Differential Equations through the Laplace Transform method. This allowed us to compute the number of starvation during the video session that is an important metric of the quality of experience of the user. In addition, we have presented simulation results using ns3 to show the correctness of our model. We have proposed a method to optimize the quality of experience given a tradeoff between the player starvation and the quality of the video. These results show that the adaptive bitrate streaming could impact negatively on the quality of the short video duration.
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