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1

Throughput and Delay Analysis in Video

Streaming over Block-Fading Channels

G. Cocco‡, D. G¨

und¨

uz∗and C. Ibars†

‡German Aerospace Center (DLR), Weßling, Germany

∗Imperial College, London, United Kingdom

†Intel Corporation, Santa Clara, CA, USA

giuseppe.cocco@dlr.de, d.gunduz@imperial.ac.uk, christian.ibars.casas@intel.com

Abstract

We study video streaming over a slow fading wireless channel. In a streaming application video

packets are required to be decoded and displayed in the order they are transmitted as the transmission

goes on. This results in per-packet delay constraints, and the resulting channel can be modeled as a

physically degraded fading broadcast channel with as many virtual users as the number of packets. In

this paper we study two important quality of user experience (QoE) metrics, namely throughput and

inter-decoding delay. We introduce several transmission schemes, and compare their throughput and

maximum inter-decoding delay performances. We also introduce a genie-aided scheme, which provides

theoretical bounds on the achievable performance. We observe that adapting the transmission rate at

the packet level, i.e., periodically dropping a subset of the packets, leads to a good tradeoff between

the throughput and the maximum inter-decoding delay. We also show that an approach based on initial

buffering leads to an asymptotically vanishing packet loss rate at the expense of a relatively large initial

delay. For this scheme we derive a condition on the buffering time that leads to throughput maximization.

I. INTRODUCTION

Video trafﬁc constitutes a large portion of today’s Internet data ﬂow, and it is foreseen to

exceed 70% of the total IP trafﬁc within the next ﬁve years [1]. A signiﬁcant portion of the

video trafﬁc is generated by streaming applications, such as YouTube and Netﬂix. This, together

with the increasing utilization of mobile terminals for streaming high-deﬁnition video content,

poses growing challenges to mobile network operators in terms of bandwidth availability and

quality of user experience (QoE).

DRAFT

2

Mobile wireless channels are often modelled with block fading, where the channel gain stays

constant during the channel coherence time, and changes independently across channel blocks

according to a certain probability distribution [2]. From the extensive literature on fading channels

(see, e.g., [3]-[9]), it emerges that a pivotal role for reliable communications is played by the

delay constraint, which is a critical design parameter in streaming applications.

In [10] and [11] the broadcast strategy proposed in [12] is used to improve the end-to-end

quality in multimedia transmission. However, the broadcast strategy requires encoding bits into

multiple superposed messages of increasing rates, and this level of ﬁne adaptation is not possible

in practical multimedia communication systems, in which the encoding rate is ﬁxed by a higher

layer application1[13]. Moreover, practical network architectures are strictly layered, and the

channel encoder is typically oblivious to the video coding scheme used by the application layer;

and therefore, rate adaptation is usually not possible at the code level. Video packets received

by the channel encoder are already video-encoded at a ﬁxed rate, which cannot be changed. On

the other hand, the channel encoder can choose to drop some of the video packets, and achieve

rate adaptation at the packet level at the expense of inter-decoding delay at the receiver.

In the Moving Picture Experts Group (MPEG) standard, the video encoder output units are

called group of pictures (GOP). Each GOP consists of an I- frame and a number of P- and B-

frames [14]. A GOP can be decoded and displayed independently of the previous and following

GOPs. We assume that a whole GOP (or an integer number of GOPs) forms one video packet,

and the coding rate is normalized such that the display time of a GOP (or an integer number of

GOPs) is equal to the channel coherence time2.

We consider streaming over a Gaussian block fading channel, in which the transmitter has no

channel state information (CSIT), which is the case for networks with large round trip delay (like

satellite networks), or wireless broadcast networks with a large number of users3. Due to the

lack of CSIT, the transmitter uses a ﬁxed transmission rate. In order to minimize the probability

of packet loss over the channel, the transmission rate is kept at the minimum value that allows

1Some streaming protocols, such as HTTP Live Streaming, allow rate adaption among only a limited number of available

rates.

2With this we implicitly assume a slow varying channel, for example, a mobile terminal moving at pedestrian speed.

3In the downlink channel with many receiving terminals, acquisition of CSIT is not viable, since this requires the transmission

of an extensive amount of information which may result in the feedback implosion problem [15].

DRAFT

3

no freezing in the display process at the receiver provided no packet is lost. This implies that

the transmission time of a packet is equal to its display time (assuming that the time needed to

process the packet at the receiver is negligible), which is assumed to be constant for all packets.

In the streaming scenario, this imposes a different decoding deadline for each video packet, i.e.,

the ﬁrst packet needs to be received by the end of the ﬁrst channel block, the second packet by

the end of the second block, and so on. Modeling the decoder at each channel block as a distinct

virtual receiver, this channel can be seen as a physically degraded fading broadcast channel with

as many virtual users as the number of channel blocks.

The loss of a data packet implies the loss of the corresponding GOP; and hence, an interruption

in the playback of the video at the end user, which lasts until the next packet is received. In [16]

the quality degradation due to GOP losses as perceived by the end user has been assessed by

streaming pre-recorded videos while introducing video segment losses in a controlled fashion.

The results illustrate that users are more tolerant to long freezes with respect to choppy playback,

that is, few long freezing events are on average preferred to many short freezing events. However,

this is no longer true if the transmission is for a live event, such as a sport event or news video.

In this case, the loss of a large chunk of video content, which may lead to loss of important

information, is much worse than choppy playback quality. In this paper we target the latter kind

of video content, and consider the interdecoding delay as a performance measure.

The effect of GOP loss in video streaming has been studied in [17], [18] and [19]. In the

video streaming literature, the problem is usually tackled at the network level, focusing on the

effect of packet loss rate, delay and jitter [20]. However, these parameters are usually assumed

to be given as ﬁxed values to the system designer, or studied from a networking perspective,

where packet losses are mainly due to buffer overﬂow, while jitter is due to the congestion level

of the network, link failures and dynamic routing. The problem of radio resource allocation in

wireless multimedia transmission over frequency selective channels is studied in [21] and [22].

We study the interaction between the physical layer and the display process of the received

video data. In particular, we study different communication strategies, each of which adopts

a different policy to select the subset of messages to be transmitted, as well as the amount of

resources (in terms of transmission time) dedicated to each message, which has an impact on the

successful decoding probability. The performance of these strategies is evaluated based on two

ﬁgures of merit: average throughput and maximum inter-decoding delay [23]. The interaction

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4

between the display process and the lower layers is of fundamental importance for streaming

services such as Dynamic Adaptive Streaming over HTTP (DASH), that need an estimation of the

link quality in order to provide an adequate QoE to the end users. In its current implementation

DASH uses the information about the link status at each user in order to optimize the QoE that can

be provided with the available resources [24]. However, DASH systems require a feedback link

that instructs the transmitter on the highgest bit-rate that can be received in the current channel

condition, whereas we assume no information on the current channel state at the transmitter, and

thus the optimisation of the transmission strategy at the transmitter has to be done independently

of the current channel condition.

While there is an extensive literature on the higher layer analysis of video streaming applica-

tions [25], research on the physical layer aspects of streaming focus mostly on code construction

[26], [27], [28]. The diversity-multiplexing trade-off for a streaming system is studied in [29].

The channel model we study here is the dual of the streaming transmitter model studied in [30],

[31], where the data packets, rather than being available at the transmitter in advance and having

a per-packet delay constraint, arrive gradually over time, and have a global delay constraint.

We propose four different transmission schemes based on time-sharing4. More elaborate

transmission techniques have been previously studied in literature such as in [10]. In [33]

the problem of still images transmitting over slow fading channel using a FEC-based multiple

description encoder over an OFDM modulation was studied. Unlike in such previous works, we

exclusively focus on time-sharing transmission because of its applicability in practical systems,

as it leads to lower complexity decoding schemes with respect to, for example, successive

interference cancellation, which is required in the case of superposition transmission. Moreover,

the throughput and delay analysis is not completely understood even for this relatively simpler

transmission scheme. In particular, we consider memoryless transmission (MT),equal time-

sharing (eTS),pre-buffering (PB) and windowed time-sharing (wTS) schemes. We also consider

an informed transmitter (IT) bound on the achievable throughput and delay performances, as-

suming perfect CSIT. We compare these achievable schemes and the informed transmitter bound

in terms of both throughput and maximum inter-decoding delay. Our results provide fundamental

performance bounds as well as an insight for the design of practical video streaming systems

4Part of the present work has been presented in [32].

DRAFT

5

over wireless fading channels.

The rest of the paper is organized as follows. In Section II we present the system model. In

Section III we derive informed transmitter bounds on throughput and average maximum delay.

In Section IV we presents four different transmission schemes and, for each of them, we analyze

throughput and delay. Section V contains the numerical results, while the conclusions are drawn

in Section VI.

II. SYSTEM MODEL

We consider a video streaming system over a block fading channel. The channel is constant

for a block of nchannel uses and changes in an independent and identically distributed (i.i.d.)

manner from one block to the next. We assume that the ﬁle to be streamed to the receiver consists

of Mindependent packets denoted by W1,...,WM, all available at the transmitter at the very

beginning. The receiver wants to decode these packets gradually as the transmitter continues

its transmission. We assume that the packet Wtneeds to be decoded by the end of channel

block t,t= 1,...,M, otherwise it becomes useless. The data packets all have the same size;

and it is assumed that each packet is generated at rate Rbits per channel use (bpcu), which is

ﬁxed by the application layer, i.e., Wtis chosen randomly with uniform distribution from the

set Wt={1,...,2nR}[34]. The channel in block tis given by

y[t] = h[t]x[t] + z[t],

where h[t]is the channel state, x[t]is the length-nchannel input vector, z[t]is a vector of i.i.d.

zero mean unit-variance Gaussian noise, and y[t]is the length-nchannel output vector at the

receiver. Instantaneous channel states are known only at the receiver, while the transmitter has

only statistical channel knowledge, i.e., it knows the probability density function (pdf) of h(t).

We have a short-term average power constraint of P, i.e., E[x[t]x[t]†]≤nP for t= 1,...,M,

where x[t]†represents the Hermitian transpose of x[t].

The channel from the source to the receiver can be seen as a physically degraded broadcast

channel, such that the decoder at each channel block acts as a virtual receiver trying to decode

the packet corresponding to that channel block. See Fig. 1 for an illustration of this channel

model. We denote the instantaneous channel capacity over channel block tby Ct:

Ct,log2(1 + φ[t]P),(1)

DRAFT

6

Receiver

channel

block M

Receiver

channel

block 2

Receiver

channel

block 1.

Transmitter

Fig. 1. Equivalent channel model for streaming a video ﬁle composed of Mpackets over Mblocks of the fading channel to

a single receiver with a per packet delay constraint.

where φ[t] = |h[t]|2is a random variable distributed according to a zero-mean pdf fΦ(φ). We

deﬁne C,E{Ct},E{x}being the mean value of x.

We deﬁne the average throughput, T, as the average decoded rate at the end of Mchannel

blocks:

T,R

M

M

X

m=1

m·η(m),(2)

where η(m)is the probability of decoding exactly mmessages out of M.

In addition to the average throughput, we also study the frame delay, which is deﬁned as

the maximum number of consecutive channel blocks in which the corresponding message is not

decoded, denoted by Dmax. When a video packet over a channel block is not decoded at the

receiver, video playback at the receiver’s device stalls, and the user continues to see the same

video frame until a new GOP is successfully received. Since Dmax is also a random variable

whose realization depends on the channel, we consider the average maximum delay Dmax as our

performance measure. We have:

Dmax ,

M

X

d=1

d·P r{Dmax =d}=

M

X

d=1

P r{Dmax ≥d}.(3)

In the next section, we ﬁrst study an informed transmitter bound on the system performance,

assuming perfect CSIT about all the future channel realizations.

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7

III. INFORMED TRANSMITTER BOUND

An upper bound on the achievable average throughput and a lower bound on the average

maximum inter-decoding delay can be obtained by assuming that the transmitter is informed

about the exact channel realization over all the Mchannel blocks non-causally. This allows

the transmitter to optimally allocate the available resources among the messages. In particular,

knowing the channels a priori the transmitter can choose the optimal subset Sopt of messages to

be transmitted that maximizes Tand minimizes Dmax. Note that power allocation across channel

blocks is not possible due to short-term power constraint. In order to ﬁnd the set of messages

Sopt that minimizes the average maximum delay, we ﬁrst ﬁnd the maximum number of decodable

messages for the given channel realizations. It follows from the physically degraded broadcast

channel model depicted in Fig. 1 that the total number of messages that can be decoded up to

channel block t, denoted by Ψ(t),t= 1,...,M, is bounded as:

Ψ(t)≤min t, Itot(t)

R,(4)

where Itot(t),Pt

i=1 Ci,is the total mutual information (MI) accumulated up to and including

channel block t, while ⌊x⌋is the largest integer smaller than or equal to x. At each channel

= 1 1 0 0 1

Fig. 2. Itot(t)plotted against t, and the corresponding vector Vin case of throughput-optimal transmission. The light blue bars

represent the amount of MI accumulated in each of the 5channel blocks considered, while the dark blue rectangles indicate a

decoding event and represent the amount of MI that is used to decode a message.

block t, we check whether we can decode packet Wtin addition to the packets that have already

been decoded. Note that there is no gain in decoding a packet prior to its decoding deadline. Let

DRAFT

8

v(t)∈ {0,1}denote the decoding event for Wt, i.e., v(t) = 1, if Wtis decoded, and v(t) = 0

if not. We have Ψ(t) = v(1) + ···+v(t), and

v(t+ 1) =

1if Itot(t+ 1) ≥(Ψ(t) + 1) R,

0otherwise.

(5)

This recursion returns the M-length binary vector V= [v(1) ···v(M)], which corresponds to a

transmission scheme that maximizes the throughput. Although Vrepresents an optimal solution

in terms of T, it may be suboptimal in terms of D.max From the maximum delay perspective it

may be a better choice not to transmit some of the packets even if enough mutual information

could be accumulated by their deadlines, and instead to transmit packets that are further in the

sequence. This is equivalent to shifting rightwards some of the 1’s in Vso that the number

of consecutive 0’s in the vector is minimized. Note that this process leaves the throughput

unchanged.

Let us consider the example shown in Fig. 2, where the mutual information accumulated by

the receiver at the end of channel block t,Itot(t)is plotted against the channel block number.

The lines Itot(t) = jR,j= 1,...,4, indicate the threshold values of Itot(t)after which a new

message can be decoded. The vector Vhas entries equal to 1in correspondence to decoding

events (shadowed areas) and zero in correspondence to channel blocks in which the receiver

does not decode the corresponding message.

Ԣ= 1 0 1 0 1

Fig. 3. Itot(t)plotted against t, and the corresponding vector Vin case of throughput- and delay-optimal transmission. The light

blue bars represent the amount of MI accumulated in each of the 5channel blocks considered, while the dark blue rectangles

indicate a decoding event and represent the amount of MI that is used to decode a message.

DRAFT

9

With reference to Fig. 2, the iterative process described by Eqn. (5) returns the sequence

V= [11001]. This allocation achieves a throughput of 3/5and a maximum delay of 2. However,

a better choice for the transmitter is to transmit message W3instead of W2, as shown in Fig.

3. This gives the new allocation V′= [10101], which has the same throughput as Vbut a

maximum delay of Dmax = 1 instead of 2.

In order to minimize the maximum delay, the transmitter can choose to drop a message even

if it could be decoded with high probability. In other words, the resources are allocated to a

message with a higher index, which, if decoded, would lead to a lower maximum delay. Note that

the maximum delay is optimized without decreasing the average throughput. Next we provide

the necessary deﬁnitions and results to introduce the algorithm Min_Del_Max_Rate, which

optimizes both Tand Dmax.

Deﬁnition 3.1: Let Vlb,D denote the binary string of length Mwith maximum number of

consecutive zeros equal to D, which has the smallest number of 1’s and the smallest decimal

representation.

If M > D,Vlb,D can be constructed by taking a sequence of Mzeros and starting from the

(D+ 1)-th most signiﬁcant bit (i.e., the leftmost one), substituting a 0with a 1, every Dbits.

If M=D,Vlb,D is the all-zero string of length M.

Let us clarify the deﬁnition considering an example with M= 5. To each value of Din

the set {0,1,2,3,4,5}corresponds a different vector Vlb,D:Vlb,0= [11111] ,Vlb,1= [01010],

Vlb,2= [00100],Vlb,3= [00010],Vlb,4= [00001] and Vlb,5= [00000].

Deﬁnition 3.2: We deﬁne Ψ(t) = Pt

n=1 v(n)and Ψlb,D(t) = Pt

n=1 vlb,D(n), where v(n)and

vlb,D(n)are the n-th bits, starting from the most signiﬁcant ones, of V(tentative allocation

vector returned by recursion (5)) and Vlb,D (see Deﬁnition 1), respectively. In other words, Ψ(t)

and Ψlb,D(t)are the cumulative sum, from left, of the vectors Vand Vlb,D, respectively, up to

the t-th coordinate.

With reference to the example in Fig. 2, we have Ψ(1),...,Ψ(5) = 1,2,2,2,3. For D= 2,

we have Vlb,2= [00100], and Ψlb,2(1),...,Ψlb,2(5) = 0,0,1,1,1.

DRAFT

10

Theorem 1 Given the allocation vector Vreturned by recursion (5), a maximum delay less

than or equal to D∗is achievable if the following holds: Ψ(t)≥Ψlb,D ∗(t),∀t∈ {1,...,M}.

Proof We recall that Ψlb,D (t)is the total number of 1’s among the leftmost tbits of the

sequence Vlb,D (see Deﬁnition 1), while Ψ(t)is the total number of 1’s among the leftmost t

bits of the sequence V.Ψ(t)≥Ψlb,D (t),∀t∈ {1,...,M}, implies that Vhas at least as many

1’s as Vlb,D among the leftmost tpositions, ∀t∈ {1,...,M}, which, in turn, implies that V

achieves a maximum delay that is no greater than D∗, which concludes the proof.

In order to ﬁnd the minimum possible maximum delay starting from a given sequence V, one

can start with a delay D∗= 0 and check if the condition of Theorem 1is satisﬁed. If not, the

maximum delay is increased by 1, and so on.

Using Theorem 1, the Min_Del_Max_Rate algorithm (Algorithm 1) has been obtained.

The algorithm takes as input the vector V, which is obtained using the recursion in Eqn. (5).

First the algorithm calculates the minimum achievable maximum delay Dmax

IT (see Theorem 1

and the following note) and derives the vector Vlb,Dmax

IT . Then it calculates the difference in the

number of ones between Vand Vlb,Dmax

IT (excess_0 in the algorithm). By deﬁnition of Dmax

IT ,

excess_0 is greater than or equal to zero. Using Vlb,Dmax

IT as an initialization allocation vector,

the vector Sopt is then constructed by simply substituting the rightmost excess_0 zeros with

ones. The output of the algorithm is the set of messages Sopt (containing a 1or a 0in position

tif message Wtis to be transmitted, or not) that constitutes the optimal transmission choice in

terms of both throughput and maximum delay. It can be easily shown that Algorithm 1 has a

complexity which is quadratic in M.

In order to clarify the procedure just described, let us consider again the example in Fig. 2.

The recursion in Eqn. (5) returns the vector V= [11001], which corresponds to Ψ = [12223].

The algorithm starts with a tentative delay Dmax

IT = 0, and generates the corresponding sequence

Vlb,0= [11111], with Ψlb,0= [12345]. Since the condition of Theorem 1is not satisﬁed (Ψ(3) <

Ψlb,0(3)), a minimum maximum delay Dmax

IT = 0 cannot be achieved, and the tentative delay is

increased by 1, i.e., Dmax

IT = 1. The corresponding sequences Vlb,1= [01010] and Ψlb,1= [01122]

are then calculated. The cumulative function Ψlb,1satisﬁes the condition of Theorem 1, which

implies that the minimum achievable maximum delay is Dmax

IT = 1. At this point the algorithm

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11

Algorithm 1 Min_Del_Max_Rate(V)

M= length(V)

if V== [0,...,0] then // if no packet can be decoded return the all zero sequence

Sopt = [0,...,0]

return Sopt

end if

D,k= 0

while found == 0 do

found = 1

Vlb,D = [0,...,0] // vector of Mzeros

for i= 1 to jM

D+1 kdo

Vlb,D[i(D+ 1)] = 1 // assign 1 to the i(D+ 1)-th component

end for

cumsum d = 0

cumsum lb = 0

for j= 1 to Mdo

cumsum d = cumsum d+V[j]// calculate Ψ(j)

cumsum lb = cumsum lb+Vlb[j]// calculate Ψlb,D (j)

if cumsum d <cumsum lb then // if cumulative sum is lower, start again increasing delay

found = 0

exit for

end if

end for

if found == 1 then

Dmax

IT =D

exit while

end if

D=D+ 1

end while

Sopt =Vlb,Dmax

IT

excess 0 = sum(V)−sum(Vlb,D )

while k < excess 0do // assign 1 to the rightmost excess 0zeros of Vlb,Dmax

IT

if Sopt[M−k] == 0 then

Sopt[M−k] = 1

k=k+ 1

end if

end while

return Sopt

DRAFT

12

calculates the optimal allocation vector. First, the difference in the number of ones between vector

Vlb,1and vector V(excess_0) is computed, which in the example is equal to excess_0=1.

Finally, the rightmost excess_0 zeros in Vlb,1are set to 1, which leads to the allocation

sequence Sopt = [01011].

IV. TRANSMISSION SCHEMES

In this section we introduce four different transmission schemes based on time-sharing. Each

channel block is divided among the messages for which the deadline has not yet expired. Thus,

while the ﬁrst channel block is divided among all the messages W1,...,WM, the second channel

block is divided among messages W2,...,WM, as the deadline of message W1expires at the

end of the ﬁrst block. In general the encoder divides channel block tinto M−t+ 1 portions

αtt,...,αM t, such that αmt ≥0and PM

m=tαmt = 1. In channel block t,αmtnchannel uses are

allocated for the transmission of message Wm. We assume that Gaussian codebooks are used in

each portion for each message, and the corresponding codelengths are sufﬁcient to achieve the

instantaneous capacity. Then the total amount of received mutual information relative to message

Wmis:

Itot

m,

m

X

t=1

αmtCt.(6)

The proposed schemes differ in the way the channel uses are allocated among the messages

for which the deadline has not yet expired. Different time allocations lead to different average

throughput and average maximum delay performances.

A. Memoryless Transmission (MT)

In memoryless transmission (MT) each message is transmitted only within the channel block

just before its expiration, that is, message Wtis transmitted over channel block t. Equivalently

we have αmt = 1, if t=m, and αmt = 0, otherwise. In MT message Wtcan be decoded if

and only if Ct≥R. Due to the i.i.d. nature of the channel state over blocks, the successful

decoding probability p,P r{Ct≥R}is constant over messages. The probability that exactly

mmessages are decoded is given by:

η(m),M

mpm(1 −p)M−m.(7)

The average number of decoded messages for the MT scheme is TMT =M p.

DRAFT

13

Next we derive the exact expression for the average maximum delay for MT, denoted by

Dmax

MT . The term P r{Dmax ≥d}in the summation in Eqn. (3) is the probability that a sequence

of MBernoulli random variables with parameter pcontains at least dconsecutive zeros. This

probability can be evaluated by modeling the number of consecutive zeros as a Markov chain,

and ﬁnding the probability of reaching the ﬁnal absorbing state of dconsecutive zeros. This

probability is given in the following theorem:

Theorem 2: Let x1,··· , xMbe a sequence of i.i.d. Bernoulli random variables with parameter

p= E[xi]. The probability of having at least dconsecutive zeros in the sequence is given by:

P r{Dmax ≥d}=

k

X

i=0

si

X

ri=1

ad,riM+ri−1

ri−1 1

ϕdi M

,(8)

where k∈ {0,...,M},k≤d+ 1 is the number of distinct zeros of the polynomial (1 −z)qd(z)

where:

qd(z) = 1 −p

d

X

j=1

zj(1 −p)j−1,(9)

ϕdi,i∈ {0,...,k}, are the zeros of (1 −z)qd(z)with multiplicity si,ad,ri,ri∈ {1,...,si}, are

constants derived from the partial fraction expansion of

(zp)d

(1 −z)qd(z).(10)

Proof: See Appendix.

Finally, by plugging (8) into (3) we ﬁnd:

Dmax

MT =

M

X

d=1 "k

X

i=0

si

X

ri=1

ad,riM+ri−1

ri−1 1

ϕdi M#.(11)

B. Equal Time-Sharing (eTS) Transmission

In the equal time-sharing (eTS) transmission scheme each channel block is equally divided

among all the messages whose deadline has not expired yet, that is, for m= 1,...,M, we have

αmt =1

M−t+1 for t= 1, . . . , m, and αmt = 0, for t=m+ 1,...,M.

DRAFT

14

In eTS, messages whose deadlines are later in time are allocated more resources; and hence,

are more likely to be decoded. We have Itot

i< Itot

jfor 1≤i < j ≤M. Hence, the probability

of decoding exactly mmessages is:

η(m),P r{Itot

m≥R≥Itot

m−1},(12)

for m= 0,1,...,M, where we deﬁne Itot

0= 0 and Itot

M+1 =∞. Since the decoded messages in

eTS are always the last ones, we can express the average maximum delay of eTS, Dmax

eTS, as a

function of its average throughput TeTS as follows:

Dmax

eTS ,

M

X

m=0

(M−m)·η(m)

=

M

X

m=0

M·η(m)−

M

X

m=0

m·η(m)

=M1−TeTS

R.(13)

The numerical analysis of eTS, together with other schemes is presented in Section V.

C. Pre-Buffering (PB) Transmission

In most practical streaming systems the receiver ﬁrst accumulates GOPs in the playout buffer

and then starts displaying them at a constant frame rate once a sufﬁcient portion of the video has

been received, in order to compensate for the delay jitter of arriving packets [35]. We consider

a slightly different version of this type of streaming transmission in which only the last B

messages are transmitted while the ﬁrst packets are not transmitted at all. The ﬁrst M−B+ 1

channel blocks are used to convey information relative to the last Bpackets as explained in the

following. We call this method pre-buffering (PB) transmission.

The initial buffering phase introduces a start-up delay of M−Bchannel blocks. On the

other hand, if a sufﬁciently large buffering period is chosen, all the transmitted messages can be

received correctly, achieving an average throughput of RB

M. Transmitted messages are encoded

with equal time allocation over the ﬁrst M−B+ 1 blocks. Due to the delay constraint, message

WM−B+1 is transmitted up to channel block M−B+ 1. Hence, in block M−B+ 2 the

last B−1messages are transmitted with equal time allocation. The process continues up until

channel block M, in which only message WMis transmitted. Next we indicate with TPB(B)

and Dmax

PB (B)the average throughput and the average maximum delay achieved by the scheme

DRAFT

15

using a buffering period of Bchannel blocks, respectively. The number Bopt of messages to be

transmitted is chosen so that

Bopt = arg min

B∈{1,...,M}nDmax(B)o.(14)

Next we show that the Bopt, as deﬁned in Eqn. (14), also maximizes the average throughput.

The average throughput when transmitting only the last Bmessages is given by:

TPB(B) = R

M

B

X

m=1

P r {decode at least mmessages}

=R

M

B

X

m=1

P r Itot

M−m+1 ≥R,(15)

where the mutual information accumulated by the receiver for message Wm, for m=M−B+ 1,

M−B+ 2,...,M, is given by:

Itot

m=1

B

M−B+1

X

t=1

Ct+

m

X

t=M−B+2

Ct

M−t+ 1.(16)

From Eqn. (15) we have:

TPB(B) = R

M"B−

B

X

m=1

P r Itot

M−m+1 < R#

=R

M"B−

B

X

m=1

P r {Dmax ≥M−m+ 1}#.(17)

The average maximum delay when only the last Bmessages are transmitted is:

Dmax

PB (B) = M−B+PB

d=1 P r {Dmax ≥M−B+d}.(18)

From (17) and (18) we ﬁnd

TPB(B) = R 1−Dmax(B)

M!,

and ﬁnally

arg min

B∈{1,··· ,M}nDmax

PB (B)o= arg max

B∈{1,··· ,M}TPB(B).(19)

This proves that the average throughput and the maximum delay can be optimized simultaneously.

It is not straightforward to come up with an analytical expression for the optimal value of B

in the PB scheme for the general case. In the following theorem we derive the optimal fraction

DRAFT

16

of messages αopt =Bopt

M, such that almost all of the transmitted messages can be decoded with

probability that approaches 1asymptotically as Mgoes to inﬁnity, if a fraction α′< αopt of the

messages is transmitted, while a fraction smaller than αopt of the messages can be decoded if

α′> αopt.

Theorem 3 Average throughput of αR can be achieved in the limit of inﬁnite Mby transmitting

αM +o(M)messages as long as

α < αopt ,1

R

C+ 1.

If α > αopt, the achieved average throughput is smaller than αoptR.

Proof Assume that the last Bmessages, i.e., WM−B+1 ,...,WM, are transmitted, with B=

Mα +o(M),α≤1. Message WM−B+1, for which the deadline expires ﬁrst, is the one that

accumulates the least amount of mutual information, that is:

IM−B+1 =1

B

M−B+1

X

t=1

Ct.(20)

The probability of decoding all the transmitted messages is then:

P r {IM−B+1 ≥R}=P r n1

BPM−B+1

t=1 Ct≥Ro

=P r nPM−B+1

t=1

Ct

M−B+1 −C≥B

M−B+1 R−Co

=P r SM−B+1 −C≥B

M−B+1 R−C,(21)

where SM−B+1 ,PM−B+1

t=1

Ct

M−B+1 , is the sample mean of the instantaneous channel capacity

over the ﬁrst M−B+ 1 channel blocks. From the law of large numbers it follows that:

lim

M→∞ P r nSM(1−α−o(M)

M)−C> δo= 0,∀δ > 0.(22)

Using equations (21) and (22) we ﬁnd:

lim

M→∞ P r {IM−B+1 ≥R}=

1,if limM→∞ B

M−B+1 R < C

0,if limM→∞ B

M−B+1 R > C.

(23)

We can write:

lim

M→∞

B

M−B+ 1R= lim

M→∞

Mα +o(M)

M−Mα +o(M)R

=α

1−αR. (24)

DRAFT

17

Finally, using Eqn. (24) in Eqn. (23) we ﬁnd:

lim

M→∞ P r {IM−B+1 ≥R}=

1,if α < αopt

0,if α > αopt.

(25)

Eqn. (25) implies that if a fraction of messages α′larger than αopt is transmitted, then the average

throughput is less than αoptR, which concludes the proof.

In Section V, we provide a numerical optimization of the PB scheme, and compare it with the

other proposed transmission strategies and the upper bound. As we will see from the numerical

results, this buffering approach can improve the average throughput signiﬁcantly as it provides

rate adaptation at the packet level by eliminating some of the packets, thus increasing the correct

decoding probability of the remaining packets.

D. Windowed Time Sharing (wTS)

We have seen in the PB scheme that transmitting only a subset of the messages can improve

the system throughput by allowing rate adaptation at the packet level. However, in the PB scheme

only the last Bpackets are transmitted leading to a minimum delay of M−Bchannel blocks.

In the next scheme, called the windowed time-sharing (wTS) scheme, ⌈M/B⌉messages are

transmitted, where ⌈x⌉is the smallest integer greater than or equal to x; however, unlike in

PB, the transmitted messages are distributed among the whole set of available messages, that is,

only one from Bconsecutive packets is transmitted over Bconsecutive channel blocks. So, for

instance, if B= 3, the ﬁrst message to be transmitted is W3, which is repeated over channel

blocks 1,2and 3, followed by message W6, which is transmitted in the next three channel

blocks, and so on.

The parameter Bcan be optimized according to two different criteria, namely to maximize

the average throughput or to minimize the delay, which leads to the two variants of the wTS

scheme, which we call throughput-wTS (T-wTS) and delay-wTS (D-wTS), respectively. In wTS

a message is decoded with probability pBgiven below:

pB=P r {IkB ≥R}=P r

min{kB,M }

X

t=kB−W+1

Ck≥R

,(26)

for k∈ {1,...,M

B}. A lower bound on Dmax

wT S can be found by substituting M

Bfor Min

Eqn. (11), pBfor pin equations (9) and (10) and multiplying Eqn. (11) with B. An upper bound

DRAFT

18

can be found in a similar way by using M

Binstead of M

B. Similarly, an upper and a lower

bound on TwT S are given by M

B·pBand M

B·pB, respectively. Analytical optimization of

parameter Bin both the T-wTS and D-wTS schemes is elusive and we resort to the numerical

analysis presented in the next section.

V. NUMERICAL RESULTS

In this section we compare the average throughput and the average maximum delay of the

proposed schemes numerically. The channel model used in the simulations is a Rayleigh block

fading channel, in which the channel gain φ[t]in block number t,t= 1,...,M (see Eqn. 1) is a

unit-mean exponential random variable that changes in an i.i.d. fashion at the beginning of each

channel block and stays constant until the beginning of the next one. Fig. 4 and Fig. 5 show

the average throughput and the average maximum delay for the proposed schemes, respectively,

for R= 1 and SNR = −5 dB. Both variants of the wTS scheme perform close to the informed

transmitter lower bound in terms of the maximum delay, while the PB scheme is the one with

the highest average throughput, followed by T-wTS and D-wTS. The eTS scheme shows quite

poor performance in terms of both the delay and the throughput. From the plots it emerges that

wTS in its two variants T-wTS and D-wTS, can help to reduce the inter-decoding delay while

achieving a relatively good average throughput in the low SNR regime. The transmitter can

choose between the two schemes based on its preference between higher throughput and lower

inter-decoding delay. While PB provides the highest throughput among the proposed schemes,

its inter-decoding delay is signiﬁcantly high, due to the initial buffering time. PB might be a

particularly attractive choice for video streams of long duration, for which the users would be

willing to have a larger startup delay to enjoy a higher throughput for the rest of the video.

Fig. 6 and Fig. 7 show the average throughput and the average maximum delay, respectively,

for the proposed schemes for R= 1 and SNR = 5 dB. Also for this SNR level the two

variants of the wTS scheme perform close to the informed transmitter lower bound in terms

of maximum delay. The highest average throughput is achieved by the T-wTS scheme together

with the MT scheme, followed by the PB, D-wTS and eTS schemes. From Fig. 6 and Fig. 7 we

see that, when the SNR is high, the MT scheme, together with the T-wTS scheme, achieves the

best performances in terms of both delay and average throughput. This suggests that a simple

memoryless approach is sufﬁcient when the channel SNR is sufﬁciently high, while at low SNR

DRAFT

19

10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

T

M

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 4. Average throughput Tplotted against the number of messages transmitted for SN R =−5 dB and R= 1 bpcu.

0 10 20 30 40 50 60

0

10

20

30

40

50

60

M

Dmax

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 5. Average maximum delay Dmax plotted against the number of transmitted messages for S N R =−5 dB and R= 1

bpcu.

more complex encoding techniques can help to signiﬁcantly improve the performance. The D-

wTS scheme shows a sudden decrease in the average throughput, which, with reference to Fig.

6, also corresponds to a decrease in the slope of the curve at points corresponding to M= 7

and M= 48. This is due to the optimization of the window size B. We recall that in D-wTS the

window size represents the number of channel blocks dedicated to a message, and is optimized

so as to achieve the minimum average maximum delay. While a large Bleads to a high decoding

probability, it implies a small number of transmitted messages, which bounds from below the

minimum delay by B. As a matter of fact, only ⌈M

B⌉messages are transmitted in the wTS

scheme, which implies that the maximum delay, in a given realization, is a multiple of B. If, for

DRAFT

20

10 20 30 40 50 60

0

0.2

0.4

0.6

0.8

1

1.2

T

M

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 6. Average throughput Tplotted against the number of messages transmitted for SN R = 5 dB and R= 1 bpcu.

5 10 15 20 25 30 35 40 45 50 55 60

0

5

10

15

20

25

M

Dmax

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 7. Average maximum delay Dmax plotted against the number of transmitted messages for SN R = 5 dB and R= 1 bpcu.

instance, B= 2 and m= 3 consecutive messages are lost, the corresponding delay is m·B= 6.

Formally, given a window size B∗there is a certain probability pl

B∗of not decoding a message.

For any ﬁxed m∈ {0, . . . , M}, using Eqn. (8) it can be easily shown that the probability of

losing at least mconsecutive messages increases with M. Thus a value B∗which is optimal

for a certain M, may not be the optimal for a larger number of messages, as the probability

that more than one consecutive messages get lost increases with M. The optimal choice may

be to increase B, so that the probability of losing consecutive messages is decreased. This is

conﬁrmed by Fig. 8, where the optimal window size, obtained numerically, is plotted against

the total number of messages. An increase in Bimplies a decrease in the slope of the average

number of decoded messages, since a smaller fraction of messages is transmitted, as shown in

DRAFT

21

0 10 20 30 40 50 60 70 80

0

1

2

3

M

Optimal window size for wTS

Fig. 8. Optimal window size (B) for the T-wTS scheme plotted versus the total number of messages (M) for SN R = 5 dB .

−20 −15 −10 −5 0 5 10 15 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

SNR [dB]

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 9. Average throughput Tplotted against the SN R for M= 40 packets and R= 1 bpcu.

the plots. The T-wTS scheme, in which Bis optimized so as to achieve the maximum average

throughput, shows a good tradeoff between the average throughput, which, unlike D-wTS, is

almost independent of the number of messages, and the average maximum delay, performing

close to the D-wTS scheme.

In Figures 9 and 10 the average throughput and the average maximum delay, respectively, are

plotted against average SNR. The plots were obtained for M= 40 packets and R= 1 bpcu. As

observed in Figures 4 and 6, for M= 40, the PB scheme outperforms all other schemes in terms

of throughput at low SNR (lower than 2 dB), while T-wTS and MT achieve almost the same

performance, and outperform the PB scheme at higher SN Rs. From the ﬁgures we observe that

the PB scheme is the most robust one against packet losses at low SNR, while at higher SNR

DRAFT

22

−20 −15 −10 −5 0 5 10 15 20

0

5

10

15

20

25

30

35

40

Dmax

SNR [dB]

MT

eTS

PB

T−wTS

D−wTS

IT

Fig. 10. Average maximum delay Dmax plotted against the SN R for M= 40 packets and R= 1 bpcu.

it is outperformed by all the schemes but the trivial MT. In terms of maximum delay, PB shows

relatively poor performance for most of the considered SN R range, which is due to the initial

buffering phase. Note that, if, unlike assumed in this paper, the loss of large consecutive chunks

of the content were not an issue, and choppy playback were to be avoided, the PB scheme would

be the best among the considered schemes since it guarantees that, once the buffering phase is

ﬁnished, no additional packet is lost, as proven in Theorem 3 for the asymptotic case.

VI. CONCLUSIONS

We have studied the streaming of stored video content over slow fading channels with per-

packet delay constraints. In addition to the classical throughput metric, we have also considered

the inter-decoding delay, i.e., the number of consecutive video GOPs that cannot be decoded

successfully, as a performance measure. We have proposed four different transmission schemes

based on time-sharing. We have carried out theoretical as well as numerical analysis for the

average throughput and maximum delay performances. We have also derived bounds on both the

average throughput and maximum inter-decoding delay by introducing an informed transmitter

bound, in which the transmitter is assumed to know the channel states in advance. We have

seen that the wTS scheme can provide a good trade-off between the average throughput and

the maximum inter-decoding delay by deciding on the proportion of transmitted video packets.

In practice this corresponds to reducing the coding rate of the video at the packet level. We

have also proved that in the PB scheme almost all transmitted messages can be decoded with a

probability that goes to 1as Mgoes to inﬁnity if only a fraction of the messages smaller than

DRAFT

23

a threshold value, which depends on the transmission rate and the average channel capacity, are

transmitted.

APPENDIX

Proof of Theorem 1

The probability of having a run of at least d,d∈ {0,...,M}, consecutive zeros in the

sequence is equivalent to ﬁnding the probability of state dafter Msteps in the Markov chain

depicted in Fig. 11. The state dis an absorbing state, i.e., once the process reaches that state, it

TXGGXGWGXTGXTGXTGGGGGXG

Fig. 11. Markov chain for the calculation of the average maximum delay in memoryless transmission.

remains there with probability 1. Let ptbe a d-length probability mass function, where pt(i),

i= 0,...,d, denotes the probability of being in state iat step t. The vector ptof state occupancy

at step tfor the Markov chain in Fig. 11 can be obtained as:

pt=pt−1H=p0Ht,(27)

where p0= [1 0 ··· 0] and His the (d+ 1) ×(d+ 1) transition matrix of the chain which has

the following structure:

H=

1−p p 0 0 ··· 0 0

1−p0p0··· 0 0

.

.

..

.

..

.

..

.

.

1−p000··· 0p

0 0 0 0 ··· 0 1

.(28)

The probability of being in state dafter Msteps, pM(d), can be found from Eqn. (27). Since

p0= [1 0 ··· 0] we have:

pM(d) = HM(1, d + 1).(29)

DRAFT

24

In order to evaluate HM(1, d + 1), we apply the Z-transform to Eqn. (27), taking into account

that the recursive formula is deﬁned only for t≥1. The Z-transform P(z)of a discrete vector

function ptis deﬁned as [36]:

Pz,Z(pt) =

+∞

X

t=0

ptzt.(30)

To account for the fact that t≥1in Eqn. (27) we can write:

+∞

X

t=1

ptzt=

+∞

X

t=0

ptzt−p0=Pz−p0,(31)

and

+∞

X

t=1

pt−1Hzt=z

+∞

X

t=1

pt−1Hzt−1

=z

+∞

X

t=0

ptHzt

=zPzH.(32)

Plugging Eqn. (31) and Eqn. (32) into Eqn. (27) we ﬁnd:

Pz=p0(I−zH)−1,(33)

where Iis the (d+ 1) ×(d+ 1) identity matrix.

The Z-transform Czof a matrix Ctof functions in the discrete variable tis deﬁned as:

Cz,Z(Ct) =

+∞

X

t=0

Ctzt.(34)

Note that in Eqn. (34) the term ztis a scalar function of zand twhich is multiplied to each

of the elements of matrix Ct. By comparing Eqn. (33) with Eqn. (27), we see that (I−zH)−1

is the Z-transform of the matrix Hthaving functions in the discrete variable tas elements. We

have:

I−zH=

1−z(1 −p)−zp 0 0 ··· 0 0

−z(1 −p) 1 −zp 0··· 0 0

.

.

..

.

..

.

..

.

.

−z(1 −p) 0 0 0 ··· 1−zp

0 0 0 0 ··· 0 1 −z

.(35)

DRAFT

25

(I−zH)−1

[1,:] =1

(1 −z)qd(z)h(1 −z) (1 −z)(zp) (1 −z)(zp)2··· (1 −z)(zp)d−1(zp)di.(36)

Once (I−zH)−1is known, it is sufﬁcient to inversely transform it and get Ht. We ﬁnd the

inverse of matrix (35) for a generic dusing Gauss-Jordan elimination. As we only need the

element HM(1, d + 1), we only report the ﬁrst row of (I−zH)−1in Eqn. (36) at the top of the

next page, where

qd(z),1−p

d

X

j=1

zj(1 −p)j−1.(37)

The probability of being in state dat step Mis the inverse Z-transform of element (1, d + 1) of

matrix (I−zH)−1, i.e.:

pM(d+ 1) = Z−1(zp)d

(1 −z)qd(z)t=M

,(38)

where Z−1{Pz}is the inverse Z-transform of Pzdeﬁned as [36]:

Z−1{Pz}=−1

2πj Iγ

Pzz−t−1dz =pt,(39)

γbeing a counterclockwise-oriented circle around the origin of the complex plane. An easier

way to solve Eqn. (38) is to decompose the Z-transform using partial fraction decomposition,

i.e., rewriting Pzas:

Pz=(zp)d

(1 −z)qd(z)=

k

X

i=0

si

X

ri=1

ad,ri 1

1−z

ϕd,i !ri

,(40)

where ϕd,i,i∈ {0,...,k}, are the k≤d+ 1 distinct zeros with degree d+ 1 and multiplicity

siof the polynomial (1 −z)qd(z), while ad,ri,ri∈ {1,...,si}, are constants deriving from the

partial fraction expansion of Pz. Once in the form of Eqn. (40), Pzcan be inversely transformed

using the linearity of the inverse Z-transform and the fact that:

Z−1( 1

1−z

ϕd,i !ri)=t+ri−1

ri−1 1

ϕd,i t

.(41)

DRAFT

26

Eqn. (41) follows from the fact that:

Z(1

ϕt),

∞

X

t=0 1

ϕt

zt

=

∞

X

t=0 z

ϕt

=1

1−z/ϕ ,(42)

for |z|< ϕ, and from the fact that the Z-transform of the convolution of sequences is the product

of the Z-transform of the individual sequences (see [36, Appendix 1] for further details). Finally,

using Eqn. (42) and Eqn. (40) and putting t=M, we ﬁnd:

P r{Dmax ≥d}=pM(d+ 1)

=

k

X

i=0

si

X

ri=1

ad,riM+ri−1

ri−1 1

ϕdi M

.(43)

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