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1

Streaming Transmission over Block Fading

Channels with Delay Constraint

Giuseppe Cocco†, Deniz G¨und¨uz∗and Christian Ibars†

†CTTC, Barcelona, Spain

∗Imperial College London, London, UK

giuseppe.cocco@cttc.es, d.gunduz@imperial.ac.uk, christian.ibars@cttc.es

Abstract—Streaming transmission over a block fading channel

is studied assuming that the transmitter receives a new message

at each channel block at a constant rate, which is ﬁxed by an

underlying application. A common deadline is assumed for all

the messages, at which point the receiver tries to decode as many

messages as possible. Various achievable schemes are proposed

and compared with an informed transmitter upper bound in

terms of average throughput. It is shown that the adaptive

joint encoding (aJE) scheme is asymptotically optimal; that is, it

achieves the ergodic capacity as the transmission deadline goes

to inﬁnity; and it closely follows the upper bound in the case of a

ﬁnite transmission deadline. On the other hand, in the presence

of multiple receivers with different signal-to-noise ratios (SNR),

memoryless transmission (MT), generalized time-sharing (gTS)

and superposition transmission (ST) schemes are shown to be

more robust than the joint encoding (JE) scheme as they have

gradual performance degradation with the decreasing SNR.

Index Terms—Block-fading; Delay-constrained transmission;

Multimedia streaming; Multiple access channel; Ergodic capac-

ity; Satellite broadcasting

I. INTRODUCTION

In a streaming transmitter data becomes available at the

transmitter over time rather than being available at the be-

ginning of transmission. Consider, for example, digital TV

satellite broadcasting. The satellite receives video packets

from a gateway on Earth at a ﬁxed data rate and has to

forward the received packets to the users within a certain

deadline. Hence, the transmission of the ﬁrst packet starts

before the following packets arrive at the transmitter. We

consider streaming transmission over a block fading channel

with channel state information (CSI) available only at the

receiver. This assumption results from practical constraints

when the receiver belongs to a large population of terminals,

or when the transmission delay is signiﬁcantly larger than

This is the pre-peer reviewed version of the following article: Giuseppe

Cocco, Deniz G¨und¨uz and Christian Ibars, “Streaming Transmission over

Block Fading Channels with Delay Constraint”, IEEE Transactions on

Wireless Communications, Vol. 12, Issue 9, September 2013. This work

was partially supported by the European Commission under project ICT-FP7-

258512 (EXALTED), by the Spanish Government under projects TEC2010-

17816 (JUNTOS) and TEC2010-21100 (SOFOCLES), and by the Govern-

ment of Catalonia under grant 2009-SGR-940. Giuseppe Cocco is partially

supported by the European Space Agency under the Networking/Partnering

Initiative. Deniz G¨und¨uz was supported by the European Commission’s Marie

Curie IRG Fellowship with reference number 256410 (COOPMEDIA) under

the Seventh Framework Programme.

the channel coherence time1[2]. The data that arrives at

the transmitter over a channel block can be modeled as an

independent message whose rate is ﬁxed by the quality of

the gateway-satellite link and the video encoding scheme

used for recording the event. We assume that the transmitter

cannot modify the contents of the packets to change the data

rate, as the satellite transmitter is oblivious to the underlying

video coding scheme adopted by the source, and considers

the accumulated data over each channel block as a single data

packet that can be either transmitted or dropped.

We further impose a delay constraint on the transmission

such that the receiver buffers the received messages for M

channel blocks before displaying the content, which is typical

of multimedia streaming applications (see Fig. I). As the

messages arrive at the transmitter gradually over Mchannel

blocks, the last message sees only a single channel realization,

while the ﬁrst message can be transmitted over the whole span

of Mchannel blocks. For a ﬁnite number Mof messages

and Mchannel blocks, it is not possible to average out the

effect of fading in the absence of CSI at the transmitter, and

there is always a non-zero outage probability [3]. Hence, the

performance measure we study is the average throughput, that

is, the average decoded data rate by the user.

Communication over fading channels has been extensively

studied [4]. The capacity of a fading channel depends on the

available information about the channel behavior [5]. When

both the transmitter and the receiver have CSI, the capacity

is achieved though waterﬁlling [6]. This is called the ergodic

capacity as the capacity is averaged over the fading distribu-

tion. In the case of a fast fading channel without CSI at the

transmitter ergodic capacity is achieved with constant power

transmission [4]. However, when there is a delay requirement

on the transmission as in our model, and the delay constraint is

short compared to the channel coherence time, we have a slow

fading channel. In a slow-fading channel, if only the receiver

can track the channel realization, outage becomes unavoidable

[4]. An alternative performance measure in this case is the ǫ-

1Transmission rate can be adjusted to the channel state through adaptive

coding and modulation (ACM) driven by a feedback channel. However,

in real-time broadcast systems with large delays and many receivers, such

as satellite systems, this is not practical. For instance, in real-time video

transmission the ACM bit-rate control-loop may drive the source bit-rate (e.g.,

variable bit rate video encoder), but this may lead to a large delay (hundreds

of milliseconds) in executing rate variation commands. In such cases the total

control loop delay is too large to allow real time compensation of fading [1,

Section 4.5.2.1].

2

W1W2W3WM

t

Deadline

0n2n(M−1)nMn

Ch. block 1 Ch. block 2 Ch. block M

Fig. 1. The transmitter receives message Wiof rate Rat the beginning of channel block i. All the Mmessages need to be transmitted to the receiver by

the end of channel block M.

outage capacity [7]. In general it is hard to characterize the

outage capacity exactly; hence, many works have focused on

either the high signal-to-noise ratio (SNR) [8] or low SNR [9]

asymptotic regimes. Another approach, which is also adopted

in this work, is to study the average transmission rate, i.e.,

average throughput, as in [10] and [11]. Outages may occur

even if the transmitter has access to CSI when the system is

required to sustain a constant transmission rate at all channel

states, called the delay-limited capacity [12], [13]. Due to the

constant rate of the arriving messages at all channel blocks,

our problem is similar to the delay-limited capacity concept.

However, here we neither assume CSI at the transmitter nor

require all arriving messages to be transmitted. Our work also

differs from the average rate optimization in [10] since the

transmitter in [10] can adapt the transmission rate based on the

channel characteristics and the delay constraint, whereas in our

model the message rate is ﬁxed by the underlying application.

The only degree-of-freedom the transmitter has in our setting

is the multiple channel blocks it can use for transmitting the

messages while being constrained by the causal arrival of the

messages and the total delay constraint of Mchannel blocks.

Streaming transmission has received signiﬁcant attention

recently especially with the increasing demand for multimedia

streaming applications [14]. Most of the work in this area

focus on practical code construction [15], [16], [17]. The

diversity-multiplexing tradeoff in a streaming transmission

system with a maximum delay constraint for each message

is studied in [18]. Unlike in [18], we assume that the whole

set of messages has a common deadline; hence, in our setting

the degree-of-freedom available to the ﬁrst message is higher

than the one available to the last.

In the present paper we extend our work in [19] by

presenting analytical results and introducing more effective

transmission schemes. We study joint encoding (JE), which

encodes all the available messages into a single codeword at

each channel block, and the more classical time-sharing (TS)

and superposition (ST) coding schemes. The main contribu-

tions of the present work can be summarized as follows:

1) We introduce a channel model for streaming transmission

over block fading channels with a common decoding

deadline to study real-time multimedia streaming in net-

works with large delays, such as digital satellite broad-

casting systems.

2) We introduce an informed transmitter upper bound on the

performance assuming the availability of perfect CSI at

the transmitter.

3) We ﬁnd a simpliﬁed expression for the average through-

put of the JE scheme, and use this expression to show

that, in the limit of inﬁnite channel blocks, the JE scheme

has a threshold behavior that depends on the average

channel quality.

4) We propose the adaptive JE (aJE) scheme, which drops

certain packets depending on the average channel SNR,

and show that it performs very close to the informed

transmitter upper bound for a ﬁnite number of messages,

and approaches the ergodic capacity as the number of

channel blocks goes to inﬁnity.

5) We propose a generalized time-sharing (gTS) scheme,

in which each message is transmitted over a window of

Wchannel blocks through time-sharing. We show that

optimizing the window size Wsigniﬁcantly improves the

average throughput in the high SNR regime.

6) We show that the gTS and the ST schemes provide

gradual performance improvement with increasing aver-

age SNR as opposed to the threshold behavior of the

JE scheme. Focusing on the gTS scheme with equal

time allocation and the ST scheme with equal power

allocation, we show, through numerical simulations, that

either scheme can outperform the other depending on the

average SNR. Both schemes can be further improved by

optimizing the time or power allocation, respectively.

7) We show that the aJE scheme is advantageous in a single

receiver system whereas the simple gTS and ST schemes

can be attractive when broadcasting to multiple users with

a wide range of SNR values, or in a point-to-point system

with inaccurate CSI.

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

describe the system model. In Section III we provide an upper

bound on the average throughput. In Section IV we describe

the proposed transmission schemes in detail. Section V is

devoted to the numerical results. Finally, Section VI contains

the conclusions.

II. SYSTEM MODEL

We consider streaming transmission 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

transmitter accumulates the data that arrives at a ﬁxed rate

during a channel block, and considers the accumulated data

as a single message to be transmitted during the following

channel blocks. We consider streaming of Mmessages over

Mchannel blocks, such that message Wtbecomes available

at the beginning of channel block t, for t= 1,...,M (see Fig.

I). Each message Wthas rate Rbits per channel use (bpcu),

i.e., Wtis chosen randomly with uniform distribution from

the set Wt={1,...,2nR}, where nis the number of channel

uses per channel block. Following a typical assumption in the

3

W1

W2

W3

WM

x[1]

x[2]

x[3]

x[M]

h[1]

h[2]

h[3]

h[M]

z[1]

z[2]

z[3]

z[M]

y[1]

y[2]

y[3]

y[M]

(ˆ

W1,ˆ

W2,..., ˆ

WM)

Fig. 2. Equivalent channel model for the sequential transmission of M

messages over Mchannel blocks to a single receiver.

literature (see, e.g., [10]), we assume that n, though still large

(as to give rise to the notion of reliable communication [20]),

is much shorter than the dynamics of the slow fading process.

The channel in block tis given by

y[t] = h[t]x[t] + z[t],(1)

where h[t]∈Cis the channel state, x[t]∈Cnis the channel

input, z[t]∈Cnis the i.i.d. unit-variance Gaussian noise, and

y[t]∈Cnis the channel output. The instantaneous channel

gains are known at the receiver, while the transmitter only

has knowledge of the statistics of the process 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], and E[x]is the mean value of x. As we

assume a unitary noise power, in the following we will use

interchangeably the quantities Pand SN R.

This point to point channel can be seen as an orthogonal

multiple access channel (MAC) with a special message hier-

archy [21], in which the encoder at each channel block acts

as a separate virtual transmitter (see Fig. 2). The receiver

tries to decode as many of the messages as possible, and

the performance measure is the average throughput, denoted

by Rand deﬁned as the normalized average number of

received messages multiplied by the transmission rate. We

denote the instantaneous channel capacity over channel block

tby Ct,log2(1 + φ[t]P), where φ[t]is a random variable

distributed according to a generic probability density function

(pdf) fΦ(φ). Note that Ctis also a random variable. We

deﬁne C,E[log2(1 + φP )], where the expectation is taken

over fΦ(φ).Cis the ergodic capacity when there is no delay

constraint on the transmission.

III. INFORMED TRANSMITTER UPPER BOUND

We ﬁrst provide an upper bound on the performance by as-

suming that the transmitter is informed about the exact channel

realizations at the beginning of the transmission. This allows

the transmitter to optimally allocate the resources among

messages so that the average throughput Ris maximized.

Assume that C1,...,CMare known by the transmitter and

the maximum number of messages that can be decoded is

m≤M. We can always have the ﬁrst mmessages to be the

successfully decoded ones by reordering. When the channel

state is known at the transmitter, the ﬁrst mmessages can be

decoded successfully if and only if [21]:

iR ≤Cm−i+1 +Cm−i+2 +···+CM,for i= 1,...,m.

We can equivalently write these conditions as

R≤min

i∈{1,...,m}

1

m−i+ 1

M

X

j=i

Cj

.(2)

Then, for each channel realization {h[1],...,h[M]}, the upper

bound on the average throughput is given by m∗

MR, where m∗

is the greatest mvalue that satisﬁes (2). This is an upper

bound for each speciﬁc channel realization. An upper bound

on Rcan be obtained by averaging this upper bound over the

distribution of channel realizations.

Another upper bound on Rcan be found from the ergodic

capacity assuming all messages are available at the encoder at

the beginning, and letting Mgo to inﬁnity. Finally, the bound

R≤Rfollows naturally from the data arrival rate. Thus, R

is bounded above by min R, C .

IV. TRANSMISSION SCHEMES

The most straightforward transmission scheme consists in

transmitting each message only within the channel block

following its arrival, and discard it. This is called memoryless

transmission (MT). Due to the i.i.d. nature of the channel,

successful decoding probability is constant over messages. De-

noting this probability by p,P r {Ct≥R}, the probability

that exactly mmessages are decoded is

η(m),M

mpm(1 −p)M−m.(3)

The average throughput of the MT scheme RMT is

R

MPM

m=1 mη(m) = Rp. The MT scheme treats all messages

equally. However, depending on the average channel condi-

tions, it might be more beneﬁcial to allocate more resources

to some of the messages in order to increase the average

throughput. In the following, we will consider three transmis-

sion schemes based on different types of resource allocation.

We will ﬁnd the average throughput for these schemes and

compare them with the upper bound introduced in Section III.

A. Joint Encoding (JE) Transmission

In the joint encoding (JE) scheme we generate a single

multiple-index codebook for each channel block. JE is also

studied in [18] in terms of the diversity-multiplexing tradeoff

(DMT) with a per-message delay constraint rather than the

common deadline constraint we consider. Moreover, unlike in

[18], here we study the JE scheme in terms of its average

throughput for a ﬁxed message rate R.

For channel block t, we generate a tdimensional codebook

of size s1×···×st,si= 2nR,∀i∈ {1,...,t}, with Gaussian

distribution, and index the codewords as xt(W1,...,Wt),

where Wi∈ W ={1,...,2nR}for i= 1,...,t. The receiver

4

R

R

R

R

R

R

R

2R

2R

2R

2R

2R

2RC1

C1

C2

C2

0

0

Fig. 3. Total decoded rate regions illustrated on the (C1, C2)plane with

M= 2 messages for MT (on the left) and JE (on the right) schemes.

uses joint typicality decoder and tries to estimate as many

messages as possible at the end of block M. The decoder will

be able to decode the ﬁrst mmessages correctly if [21]:

(m−j+ 1)R≤

m

X

t=j

Ct,∀j= 1,2,...,m. (4)

As a comparison, we illustrate the achievable rate regions

for MT and JE schemes for M= 2 in Fig. 3. In the case of

MT, a total rate of 2Rcan be decoded successfully if both

C1and C2are above R. We achieve a total rate of Rif only

C1or C2is above R. On the other hand, in the case of joint

encoding, we tradeoff a part of the region of rate Rfor rate

2R.

Using the conditions in Eqn. (4) we deﬁne functions gm(R),

for m= 0,1,...,M, as follows:

gm(R) = (1,if (m−j+ 1)R≤Pm

t=jCt, j = 1,...,m,

0,otherwise.

Then the probability of decoding exactly mmessages can be

written as,

η(m) = P r gm(R) = 1 and gm+1(R) = 0.(5)

After some manipulation, it is possible to prove that exactly

mmessages, m= 0,1,...,M, can be decoded if:

Cm−i+1 +···+Cm≥iR, i = 1,...,m, (6)

Cm+1 +···+Cm+i< iR, i = 1,...,M −m. (7)

Then η(m)can be calculated as in Eqn. (8) at the bottom of

the next page, where we have deﬁned x+= max{0, x}, and

fC1···Cm(c1,...,cm)as the joint pdf of C1,...,Cm, which is

equal to the product of the marginal pdf’s due to independence.

The probability in Eqn. (8) cannot be easily evaluated for

a generic M. However, we provide a much simpler way to

calculate the average throughput RJ E . This simpliﬁcation is

valid not only for i.i.d. but also for conditionally i.i.d. channels.

Random variables {C1,···, CM}are said to be conditionally

i.i.d. given a random variable Uif the joint distribution is of

the form

fC1,···,CM,U (c1,...,cM, u)

=fC1|U(c1|u)× · ·· × fCM|U(cM|u)fU(u),

(9)

where

fCi|U(ci|u) = fCj|U(cj|u),∀i, j ∈ {1,...M}.(10)

Note that i.i.d. channels is a particular case of conditionally

i.i.d. channels where Uis a constant.

Theorem 1: The average throughput for the JE scheme in

the case of conditionally i.i.d. channel capacities is given by:

RJE =R

M

M

X

m=1

P r{C1+···+Cm≥mR}.(11)

Proof: See Appendix A.

In general it is still difﬁcult to ﬁnd an exact expression for

RJE , but Theorem 1simpliﬁes the numerical analysis signif-

icantly. Moreover, it is possible to show that RJ E approaches

Rfor large Mif C > R. To prove this, we rewrite Eqn. (11)

as:

RJE =R−R

M

M

X

m=1

am,(12)

where we have deﬁned

am,P r C1+···+Cm

m< R.(13)

It is sufﬁcient to prove that, if C > R, then

limM→∞ PM

m=1 am=c, for some 0< c < ∞. We start

by noting that limm→+∞am= 0, since, by the law of large

numbers, C1+···+Cm

mconverges to Cin probability as mgoes

to inﬁnity. To prove the convergence of the series sum we

show that

lim

m→+∞

am+1

am

=λ, (14)

with 0< λ < 1.

We deﬁne:

lm,√mC−C1+···+Cm

m, m = 1,2,...,M, (15)

where each lmis a random variable with zero mean and

variance σ2

c, which corresponds to the variance of the channel.

From the central limit theorem we can write:

lim

m→+∞

am+1

am

= lim

m→+∞

P r nlm+1 >C−r

1/√m+1 o

P r nlm>C−r

1/√mo(16)

= lim

m→+∞

QC−R

σc/√m+1

QC−R

σc/√m(17)

≤lim

m→+∞

σc/√m+1

(C−R)√2πe−1

2C−R

σc/√m+1 2

C−R

σc/√m

1+C−R

σc/√m21

√2πe−1

2C−R

σc/√m2(18)

= lim

m→+∞

σ2

c+m(C−R)2

pm(m+ 1)(C−R)2e−(C−R)2

2m+1

σ2

c−m

σ2

c(19)

=e−(C−R)2

2σ2

c<1,(20)

5

where equality (17) follows from the fact that lmconverges in

distribution to a Gaussian random variable with zero mean and

variance σ2

c, while inequality (18) follows from the bounds on

the Q-function:

x

(1 + x2)√2πe−x2

2< Q(x)<1

x√2πe−x2

2,(21)

for x > 0. Similarly, we prove that if C < R, the average

rate tends to zero asymptotically with M. To see this, we

consider the series in Eqn. (11) deﬁning bm=P r{C1+···+

Cm≥mR}. We want to prove that RJ E =M−1PM

m=1 bm

converges to zero. It is sufﬁcient to prove that PM

m=1 bm

converges to a constant. We ﬁrst notice that limm→+∞bm= 0

by the law of large numbers, as M−1PM

m=1 Cmconverges in

probability to C, which is lower than Rby hypothesis. One

can also show that limm→+∞bm+1

bm= 0; and hence, RJ E goes

to zero with M. Overall we see that the average rate of the

JE scheme shows a threshold behavior, i.e., we have:

lim

M→∞ RJE =(R, if R < C

0,if R > C. (22)

1) Adaptive Joint Encoding (aJE) Transmission: Eqn. (22)

indicates a phase transition behavior such that RJ E is zero

even for large Mif R > C. While the transmission rate R

cannot be modiﬁed by the transmitter, it may choose to trans-

mit only a fraction α=M′

M<1of the messages, allocating the

extra M−M′channel blocks to the M′messages, effectively

controlling the transmission rate. In other words, the M′

messages are encoded and transmitted in M′channel blocks

as described above, while each of the remaining M−M′

blocks is divided into M′equal parts, and the encoding process

used for the ﬁrst M′blocks is repeated, using independent

codewords, across the M′parts of each block. For instance,

if M= 3 and M′= 2,x1(W1)and x2(W1, W2)are

transmitted in the ﬁrst and second channel blocks, respectively.

The third channel block is divided into M′= 2 equal parts

and the independent codewords x31 (W1)and x32(W1, W2)

are transmitted in the ﬁrst and in the second half of the block,

respectively. We call this variant of the JE scheme adaptive

JE (aJE) scheme. The conditions for decoding exactly m

messages, m= 0,1,...,M′, in aJE can be obtained from

those given in Eqn. (6) and Eqn. (7) by replacing Ciwith

C∗

i=Ci+1

M′PM

j=M′+1 Cj,i∈ {1, . . . , M′}. Note that the

random variables C∗

i,i∈ {1,...,M′}, are conditionally i.i.d.,

i.e., they are i.i.d. once the variable U=1

M′PM

j=M′+1 Cjis

ﬁxed. This implies that Theorem 1 holds.

In Appendix B we prove that, by choosing αsuch that αR ≤

C, we can have:

lim

M→∞ RaJE = min{R, C }.(23)

Eqn. (23) suggests that the average transmission rate can be

adapted at the message level while keeping a ﬁxed rate at the

physical layer. Comparing the bound min R, Cand Eqn.

(23) we see that the aJE scheme achieves the optimal average

throughput in the limit of inﬁnite M; hence, as the number

of messages and the channel blocks go to inﬁnity, the aJE

scheme achieves the optimal performance. We will show in

Section V through numerical analysis that near optimality of

the aJE scheme is valid even for ﬁnite M. However, when

there are multiple users or inaccuracy in the channel statistics

information at the transmitter, aJE performs very poorly for

users whose average received SNR is below the target value.

In the following we propose alternative transmission schemes

providing gradual performance degradation with decreasing

SNR.

B. Time-Sharing (TS) Transmission

One of the resources that the encoder can allocate among

different messages is the total number of channel uses within

each channel block. While the whole ﬁrst channel block has

to be dedicated to message W1(the only available message),

the second channel block can be divided among the messages

W1and W2, and so on so forth. Assume that the encoder

divides the channel block tinto tportions α1t,...,αtt such

that αit ≥0and Pt

i=1 αit = 1. In channel block t,αitn

channel uses are allocated to message Wi. A constant power

Pis used throughout the block. Then the total amount of

received mutual information (MI) relative to message Wiis

Itot

i,PM

t=iαitCt. Letting αit = 1 if t=iand αit = 0

otherwise, we obtain the MT scheme.

For simplicity, in the time-sharing (TS) scheme we assume

equal time allocation among all the available messages; that

is, for i= 1,...,M, we have αit =1

tfor t=i, i + 1,...,M,

and αit = 0 for t= 1,...,i. The messages that arrive earlier

are allocated more resources; and hence, are more likely to be

decoded. We have Itot

i> Itot

jfor 1≤i < j ≤M. Hence,

RT S =R

M

M

X

m=1

P r Cm

m+Cm+1

m+ 1 +···+CM

M≥R.

(24)

1) Generalized Time-Sharing (gTS) Transmission: Note

that, in TS transmission, message Wiis transmitted over

M−i+ 1 channel blocks, which allocates signiﬁcantly more

resources to the earlier messages. To balance the resource

allocation between the messages, we consider transmitting

each message over a limited window of channel blocks.

In generalized time-sharing transmission each message is

encoded with equal time allocation over Wconsecutive blocks

as long as the total deadline of Mchannel blocks is not met.

η(m) = Z∞

RZ∞

(2R−xm)+

· · · Z∞

(mR−xm−···−x2)+

fC1···Cm(x1,...,xm)dx1···dxm

×ZR

0Z2R−xm+1

0

···Z(M−m)R−xm+1−···−xM−1

0

fCm+1···CM(xm+1 ,...,xM)dxm+1 ···dxM(8)

6

20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

1

W

Average throughput

P= 2 dB (C > R)

P= 0 dB (C < R)

Fig. 4. Average throughput for the gTS scheme plotted against the window

size Wfor M= 104messages and R= 1 bpcu for two different average

SNR values.

Messages from W1to WM−W+1 are encoded over a window

of Wblocks, while messages Wi, for i∈ {M−W+ 2, M −

W+3,...,M}are encoded over M−i+1 blocks. In particular

we focus on the effect of variable Won the average throughput

RgT S . In case W≪M, most of the messages are transmitted

over Wslots together with W−1other messages. In this case

the MI accumulated for a generic message Wiis:

Itot

i=1

W

i+W−1

X

t=i

Ct.(25)

By the law of large numbers, Eqn. (25) converges in proba-

bility to the average channel capacity Cas W→ ∞. Thus,

we expect that, when the transmission rate Ris above C, the

gTS scheme shows poor performance for large W(and hence,

large M), while almost all messages are received successfully

if R < C. We conﬁrm this by analyzing the effect of Won

RgT S numerically in Fig. 4 for M= 104and R= 1 bpcu. For

P= 0 dB we have C < R, which leads to a decreasing RgT S

with increasing window size W. On the other hand, for P= 2

dB, we have C > R, and accordingly Rg T S approaches 1as

Wincreases.

The same reasoning cannot be applied if the window size

is on the order of the number of messages, as the number of

initial messages which share the channel with less than W−1

other messages and the number of ﬁnal messages which share

the channel with more than W−1messages are no longer

negligible with respect to M. In Fig. 5(a), we plot RgT S as

a function of Wfor relatively small number of messages and

C≥R. As seen in the ﬁgure, for a given Man optimal value

of Wcan be chosen to maximize RgT S . Optimal Wincreases

with Mwhen R < ¯

C. We plot RgT S for C < R in Fig. 5(b).

From the ﬁgure we see that RgT S decreases monotonically

with Wup to a minimum, after which it increases almost

linearly. The initial decrease in the average throughput is due

to the averaging effect described above, which is relevant for

W≪M. The following increase in RgT S is because the

messages which are transmitted earlier (i.e., Wiwith i≪W)

get an increasing amount of resources as Wincreases, and so

the probability to be decoded increases. As a matter of fact,

for each ﬁnite i, the average MI accumulated for message Wi

grows indeﬁnitely with W, i.e.:

lim

W→∞ E(i+W−1

X

t=i

Ct

min{t, W })

=Clim

W→∞

i+W−1

X

t=i

1

min{t, W }= +∞.(26)

Thus, for a ﬁxed i, letting Wgo to inﬁnity leads to an inﬁnite

average MI, which translates into a higher RgT S . Note that

this is valid only for relatively small iand large W, i.e., only

messages transmitted earlier beneﬁt from increasing W, while

the rest of the messages are penalized. If Wis small compared

to M, as in the plot of Fig. 4for P= 0 dB, the fraction of

messages which beneﬁt from the increasing W(i.e., messages

W1,...,WLwith L≪W) remains small compared to M.

For this reason RgT S does not increase with Wfor the range

of Wconsidered in Fig. 4, while it does for the same range

in Fig. 5(b).

Although the idea of encoding a message over a fraction

of the available consecutive slots (e.g., W < M for message

W1in gTS) can be applied to all the schemes considered in

this paper, the analysis becomes quite cumbersome. Hence,

we restrict our analysis to the TS scheme as explained above.

C. Superposition Transmission (ST)

In superposition transmission (ST) the superposition of

tcodewords, chosen from tindependent Gaussian code-

books of size 2nR , corresponding to the available mes-

sages {W1,...,Wt}is transmitted in channel block t,t∈

{1,...,M}. The codewords are scaled such that the average

total transmit power in each block is P. In the ﬁrst block, only

information about message W1is transmitted with average

power P11 =P; in the second block we divide the total power

Pamong the two messages, allocating P12 and P22 for W1and

W2, respectively. In general, over channel block twe allocate

an average power Pit for Wi, while Pt

i=1 Pit =P.

Let Sbe any subset of the set of messages M=

{1,...,M}. We deﬁne C(S)as follows:

C(S),

M

X

t=1

log2 1 + φ[t]Ps∈S Pst

1 + φ[t]Ps∈M\S Pst !.(27)

This provides an upper bound on the total rate of messages

in set Sthat can be decoded jointly at the user considering

the codewords corresponding to the remaining messages as

noise. The receiver ﬁrst checks if any of the messages can

be decoded alone by considering the other transmissions as

noise. If a message can be decoded, the corresponding signal

is subtracted and the process is repeated over the remaining

signal. If no message can be decoded alone, then the receiver

considers joint decoding of message pairs, followed by triplets,

and so on so forth. This algorithm gives us the maximum

throughput. However, it is challenging in general to ﬁnd

a closed form expression for the average throughput, and

optimize the power allocation. Hence, we focus here on the

special case of equal power allocation, where we divide the

total average power Pamong all the available messages at

each channel block. The performance of the ST scheme will

7

0 50 100 150 200 250 300

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

W

Average throughput

M= 500

M= 200

M= 100

M= 50

M= 20

(a) R= 1 bpcu, P= 5 dB (C > R).

1 50 100 150 200 250 300 350 400 450 500

0

0.05

0.1

0.15

W

Average throughput

M=500

M=200

M=100

M=50

M=20

(b) R= 1 bpcu, P=−3 dB (C < R).

Fig. 5. Average throughput for the gTS scheme plotted against the window size Wfor different values of M.

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Number of decoded messages

CMF

Joint encoding (JE)

Time−sharing (TS)

Generalized TS (gTS)

Memoryless (MT)

Superposition (ST)

Upper bound

(a) P= 1.44 dB (C > R).

0 10 20 30 40 50

0

0.2

0.4

0.6

0.8

1

Number of decoded messages

CMF

Joint encoding (JE)

Time−sharing (TS)

Generalized TS (gTS)

Memoryless (MT)

Superposition (ST)

Upper bound

(b) P= 0 dB (C < R).

Fig. 6. The cumulative mass function (cmf) of the number of decoded messages for R= 1 bpcu and M= 50.

be studied in Section V numerically and compared with the

other transmission schemes and the upper bound.

V. NUMERICAL RESULTS

In this section we provide numerical results comparing

the proposed transmission schemes. For the simulations we

assume that the channel is Rayleigh fading, i.e., the channel

gain φ(t)is exponentially distributed with parameter 1, i.e.,

fΦ(φ) = e−φfor φ > 0, and zero otherwise.

In Fig. 6(a) the cumulative mass function (cmf) of the

number of decoded messages is shown for the proposed

transmission techniques for R= 1,M= 50 and P= 1.44

dB, which correspond to an average channel capacity C≃

1.07 > R. We see that MT outperforms ST and TS schemes,

as its cmf lays below the other two. The gTS scheme improves

signiﬁcantly compared to the ordinary TS scheme. On the

other hand, the comparison with the JE scheme depends on the

performance metric we choose. For instance, JE has the lowest

probability to decode more than mmessages, for m≤15,

while it has the highest probability for m≥22.

In Fig. 6(b) the cmf’s for P= 0 dB are shown. In this

case the average capacity is C≃0.86. Comparing Fig. 6(b)

and Fig. 6(a), we see how the cmf of the JE scheme has

different behaviors depending on whether Cis above or below

R. We see from Fig. 6(b) that for the JE scheme there is a

probability of about 0.3not to decode any message, while in

all the other schemes such probability is zero. However, the JE

scheme also has the highest probability to decode more than

23 messages. Furthermore, we note that the cmf of the gTS

scheme converges to the cmf of the TS scheme at low SNR.

This is because, as shown in Section IV-B1, when C < R, the

optimal window size Wis equal to M, which is nothing but

the TS scheme.

In Fig. 7(a) and Fig. 7(b) the average throughput is plotted

against the delay constraint Mfor SNR values of −3 dB and

2 dB, respectively, and a message rate of R= 1 bpcu. While

JE outperforms the other schemes at SNR = 2 dB, it has

the poorest performance at SNR =−3 dB. This behavior is

expected based on the threshold behavior of the JE scheme

that we have outlined in Section IV-A. Note that the average

capacity corresponding to SN R =−3dB and 2dB are C=

0.522 < R and C= 1.158 > R, respectively.

When C > R the average throughput of the JE scheme gets

close to the upper bound as the delay constraint Mincreases.

It can be seen from Fig. 7(b) that, with a relatively short delay

constraint of M= 100, the JE scheme achieves 90% of the

average throughput of the upper bound. Also the gTS scheme

improves its performance as the delay constraint gets larger,

8

10 20 30 40 50 60 70 80 90 100

0

0.1

0.2

0.3

0.4

0.5

0.6

Total number of messages (M)

Average throughput

Joint encoding (JE)

Time−sharing (TS)

Generalized TS (gTS)

Memoryless (MT)

Superposition (ST)

Upper bound

(a) P=−3 dB (C < R).

10 20 30 40 50 60 70 80 90 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Total number of messages (M)

Average throughput

Joint encoding (JE)

Time−sharing (TS)

Generalized TS (gTS)

Memoryless (MT)

Superposition (ST)

Upper bound

(b) P= 2 dB (C > R).

Fig. 7. Average throughput vs delay constraint (total number of messages) Mfor R= 1 bpcu.

while the other schemes achieve an average throughput that is

almost constant with M.

We observe that, in the low SNR regime, i.e., when C < R,

all the proposed schemes other than JE perform very close to

each other, and signiﬁcantly below the upper bound. The large

gap to the informed transmitter upper bound is mostly due to

the looseness of this bound in the low SNR regime.

Although none of the schemes dominates the others at all

SNR values, it is interesting to note how, for a given SNR,

the best scheme does not depend on the delay constraint. This

may be useful in case of a practical implementation of the

proposed schemes for the case of a single receiver or multiple

receivers with similar average SNRs, as indicates that the SNR,

rather than the delay, is the main design parameter. Further in

this section we consider the case in which the receivers have

different average SNR values.

In Fig. 8 the average throughput Ris plotted against the

transmission rate Rfor the case of M= 100 and P= 20 dB.

The aJE scheme outperforms all the other schemes, performing

very close to the upper bound, illustrating its rate adaptation

capability. The number M′of messages transmitted in the

aJE scheme is chosen so that M′

M= 0.95 C

R. In the ﬁgure we

also show the upper bound obtained from the ergodic capacity

min(R, C). It can be seen how it closely approximates the

informed transmitter upper bound for R < 6. The JE scheme

performs better than the others up to a certain transmission

rate, beyond which rapidly becomes the worst one. This is

due to the phase transition behavior exposed in Section IV-A

in the case of asymptotically large delay, and observed here

even for a relatively small M. Among the other schemes, MT

achieves the highest average throughput in the region R < 6.8,

while TS has the worst performance. The opposite is true in

the region R > 6.8, where the curve of ST scheme is upper

and lower bounded by the curves of the MT and TS schemes.

We have repeated the simulations with different parameters

(i.e., changing Pand M) with similar results, that is, MT, TS,

and ST schemes meet approximately at the same point, below

which MT has the best performance of the three while above

the intersection TS has the best performance. At the moment

we have no analytical explanation for this observation, which

would mean that there is always a scheme outperforming ST.

We next study the performance of the considered schemes as

a function of the distance from the transmitter.

We scale the average received power at the receiver with

d−αpl , where dis the distance from the transmitter to the

receiver and αpl is the path loss exponent. The results are

shown in Fig. 9 for P= 20 dB, M= 100,R= 1 bpcu and

a path loss exponent αpl = 3. The dependence of Ron the

distance is important, for instance, in the context of broadcast

transmission in cellular networks, in which case the receiving

terminals may have different distances from the transmitter.

In such a scenario the range of the average channel SNR

values at the receivers becomes important, and the transmitter

should use a transmission scheme that performs well over this

range. For instance, in a system in which all users have the

same average SNR, such as a narrow-beam satellite system

where the SNR within the beam footprint has variations of

at most a few dB’s on average [22], the transmission scheme

should perform well around the average SNR of the beam. A

similar situation may occur in a microcell, where the relatively

small radius of the cell implies a limited variation in the

average SNR. Instead, in the case of a macrocell, in which

the received SNR may vary signiﬁcantly from the proximity

of the transmitter to the edge of the cell, the transmitter should

adopt a scheme which performs well over a larger range

of SNR values. In the range up to d= 4 the JE scheme

achieves the highest average throughput while for d≥6the

TS scheme outperforms the others. The drop in the average

throughput in the JE scheme when passing from d= 4 to

d= 5 is similar to what we observe in Fig. 8 when the rate

increases beyond R= 6 bpcu. In both cases the transition

takes place as the transmission rate surpasses the average

channel capacity. The aJE scheme, which selects the fraction

of messages to transmit based on C, outperforms all other

schemes and gets relatively close to the informed transmitter

upper bound and the ergodic capacity. It is interesting to

observe that the behavior of the JE and the aJE schemes in

case of ﬁnite delay constraint (M= 100) closely follows the

results shown for the asymptotic case in Section IV-A. The

aJE scheme adapts the average transmission rate at message

9

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

R (bpcu)

Average throughput

Joint encoding (JE)

Adaptive JE (aJE)

Time-sharing (TS)

Memoryless (MT)

Superpositio n (ST)

Upper bound

min(R, C)

Fig. 8. Average throughput vs Rfor P= 20 dB and M= 100 messages.

The upper bound min(R, C)is also shown.

level to the average channel capacity. We recall that, in the

aJE scheme, the transmitter only has a statistical knowledge

of the channel, and yet gets pretty close to the performance of

a genie-aided transmitter even for a reasonably low number of

channel blocks. We further notice how the adaptive JE scheme

closely approaches the ergodic capacity, even though data

arrives gradually at the transmitter during the transmission,

instead of being available at the beginning, which is generally

assumed for the achievability of the ergodic capacity [6].

We should note that in Fig. 9 the average transmission rate

is optimized for each given distance for the aJE scheme,

while such optimization is not done for the other schemes.

Thus, when two (or more) terminals have different distances

from the transmitter, the performance for the receivers might

not be optimized simultaneously by the same scheme, and a

tradeoff between the average throughputs of the two nodes is

required. The performance can be improved by considering a

combination of the aJE scheme with the TS or ST schemes.

The plots in Fig. 9 show how TS, MT and ST schemes are

more robust compared to the JE scheme, as their average

throughput decreases smoothly with the distance, unlike the

JE scheme, which has a sudden drop. This provides robustness

in the case of multiple receivers with different average SNRs

or when the channel statistics information at the transmitter is

not accurate.

VI. CONCLUSIONS

We have considered a transmitter streaming data to a

receiver over a block fading channel, such that the transmitter

is provided with an independent message at a ﬁxed rate at the

beginning of each channel block. We have used the average

throughput as our performance metric. We have proposed

several new transmission schemes based on joint encoding,

time-division and superposition encoding. A general upper

bound on the average throughput has also been introduced

assuming the availability of CSI at the transmitter.

We have shown analytically that the joint encoding (JE)

scheme has a threshold behavior. It performs well when the

target rate is below the average channel capacity C, while

its performance drops sharply when the target rate surpasses

C. To adapt to an average channel capacity that is below

the ﬁxed message rate R, the adaptive joint encoding (aJE)

scheme transmits only some of the messages. We have proved

analytically that the aJE scheme is asymptotically optimal as

the number of channel blocks goes to inﬁnity, even though data

arrives gradually over time at a ﬁxed rate, rather than being

available initially. We have also shown numerically that, even

for a ﬁnite number of messages, the aJE scheme outperforms

other schemes in all the considered settings and performs close

to the upper bound.

The JE and the aJE schemes create an M-block long

concatenated code, and imposes a certain structure on the way

messages can be decoded. For example, message Wkcan be

decoded only if message Wk−1can also be decoded. This is

useful when the underlying application has a minimum rate re-

quirement that needs to be satisﬁed over Mchannel blocks, or

when the average SNRs of the users vary over a limited range

of SNR values. Independent encoding used in time-sharing

based schemes (TS, gTS, MT), instead, makes messages less

dependent on the decoding of the other messages. The ST

scheme, based on message superposition, collocates itself

between JE- and TS-based schemes, as messages are encoded

independently, but the probability of correctly decoding each

one is affected by decoding of others.

We conclude that the aJE scheme is advantageous in systems

with a single receiver or with multiple receivers having similar

average SNR values, as the performance of the user with

the highest average SNR is limited by the user with the

lowest average SNR. On contrast, the gTS and ST schemes

can be attractive when broadcasting to multiple users with a

wide range of SNR values, or in a point-to-point system with

inaccurate CSI, as the average throughput decreases gradually

with decreasing SNR in these transmission schemes.

APPENDIX A

Proof of Theorem 1

Let Bkdenote the event “the ﬁrst kmessages can be

decoded at the end of channel block k”, while Bc

kdenotes

the complementary event. The event Bkholds if and only if

Ck−i+1 +Ck−i+2 +···+Ck≥iR (28)

is satisﬁed for all i= 1,...,k. Let Ek,j denote the event

“the j-th inequality needed to decode the ﬁrst kmessages in

kchannel blocks is satisﬁed”, that is:

Ek,j ,{Ck−j+1 +···+Ck≥jR},(29)

for j= 1,...,k, while Ec

k,j denotes the complementary event.

Note that in the JE scheme if mmessages are decoded these

are the ﬁrst mmessages. Let nddenote the number of decoded

messages at the end of channel block M. Then the average

throughput is

RJE =R[P r {nd≥1}+P r{nd≥2}+

···+ Pr{nd≥M−1}+ Pr{nd≥M}].(30)

The k-th term in the sum of Eqn. (30) is the probability of

decoding at least k(i.e. kor more) messages. Each term in

Eqn. (30) can be expressed as the sum of two terms as:

P r{nd≥k}=P r{Bk, nd≥k}+P r{Bc

k, nd≥k}(31)

10

3 4 5 6 7 8 9 10

10−4

10−3

10−2

10−1

100

d

Average throughput

Joint encoding (JE)

Adaptive JE (aJE)

Time -sharing (TS)

Memoryless (MT)

Superposition (ST)

Upper bound

Ergodic capacity (C)

Fig. 9. Average throughput Rvs distance from the transmitter for R= 1

bpcu, M= 100,P= 20 dB and αpl = 3.

The ﬁrst term in Eqn. (31) is the probability of “decoding k

messages at the end of channel block kand decoding at least

kmessages at the end of Mchannel blocks”. If Bkholds, the

event “decode at least kmessages at the end of channel block

M” is satisﬁed; hence, we have:

P r{Bk, nd≥k}=P r{Bk}=P r{Ek,1,...,Ek,k }.(32)

As for the second term in Eqn. (31), it is the probability of

decoding at least kmessages but not kat the end of channel

block k. It can be further decomposed into the sum of two

terms:

P r{Bc

k, nd≥k}=P r{Bc

k, Bk+1, nd≥k}

+P r{Bc

k, Bc

k+1, nd≥k}.(33)

The event nd≥kholds if the condition Bk+1 is satisﬁed (i.e.,

if k+ 1 messages are decoded at the end of block k+ 1, then

more than kmessages are decoded at the end of channel block

M); hence, we have:

P r{Bc

k, Bk+1, nd≥k}=P r {Bc

k, Bk+1}.

Plugging these into Eqn. (31), we obtain

P r{nd≥k}=P r{Bk}+P r{Bc

k, Bk+1}

+P r{Bc

k, Bc

k+1, nd≥k}.(34)

We can continue in a similar fashion, so that, the event “at

least kmessages are decoded” can be written as the union of

the disjoint events (“kmessages are decoded in kslots”) S

(“kmessages are not decoded in kslots but k+ 1 messages

are decoded in k+ 1 slots”) S··· S(“no message can be

decoded before slot Mbut Mmessages are decoded in slot

M”). Hence, by the law of total probability, we have:

P r{nd≥k}=

M

X

j=k

P r{Bc

k, Bc

k+1, . . . , B c

j−1, Bj}.(35)

Note that each term of the sum in Eqn. (35) says nothing about

what happens to messages beyond the j-th, which can either

be decoded or not. Plugging Eqn. (35) in Eqn. (30) we ﬁnd:

E[m] =

M

X

k=1

M

X

j=k

P r{Bc

k, Bc

k+1,...,Bc

j−1, Bj}

=

M

X

j=1

j

X

k=1

P r{Bc

k, Bc

k+1,...,Bc

j−1, Bj}.(36)

We can rewrite each of these events as the intersection of

events of the kind Ek,i and Ec

k,i. Each term of the sum in

Eqn. (36) can be written as:

P r{Bc

k, Bc

k+1,...,Bc

j−1, Bj}

=P r{Ek,1, Bc

k, Bk+1,...,Bc

j−1, Bj}

+P r{Ec

k,1, Bc

k, Bc

k+1,...,Bc

j−1, Bj}.(37)

As the event Ec

k,1implies the event Bc

k, this can be removed

from the second term on the right hand side of Eqn. (37).

Note that, in general, the event Ec

k,i,i∈ {1,··· , k }implies

the event Bc

k. In order to remove the event Bc

kfrom the ﬁrst

term as well, we write it as the sum of probabilities of two

disjoint events: one intersecting with Ek,2and the other with

Ec

k,2. Then we get:

P r{Bc

k,Bc

k+1,...,Bc

j−1, Bj}

=P r{Ek,1, Ek,2, Bc

k,...,Bc

j−1, Bj}

+P r{Ek,1, Ec

k,2, Bc

k,...,Bc

j−1, Bj}

+P r{Ec

k,1, Bc

k+1,...,Bc

j−1, Bj}.(38)

Now Bc

kcan be removed from the second term of the sum

thanks to the presence of Ec

k,2. Each of the terms in the right

hand side of Eqn. (38) can be further written as the sum of the

probabilities of two disjoint events, and so on so forth. The

process is iterated until all the Bc

d,d < j events are eliminated

and we are left with the intersections of events only of the type

Ep,q and Ec

p,q, for some p, q ∈ {k, k +1,...,M}and Bj. The

iteration is done as follows:

For each term of the summation, we take the Bc

levent

with the lowest index. If any Ec

l,j event is present, then Bc

l

can be eliminated. If not, we write the term as the sum of

the two probabilities corresponding to the events which are

the intersections of the Bc

levent with El,d+1 and Ec

l,d+1,

respectively, where dis the highest index jamong the events

in which El,j is already present. The iteration process stops

when l=j.

At the end of the process all the probabilities involving

events Bc

k,...,Bc

j−1will be removed and replaced by se-

quences of the kind:

{Ek,1, Ek,2,...,Ec

k,ik, Ek+1,ik+1,...,

Ec

k+1,ik+1 ,...,Ej−1,ij−2+1, Ec

j−1,ij−1, Bj},

(39)

where ij−1∈ {j−1−k,...,j−1}is the index corresponding

to the last inequality required to decode j−1messages, which

is not satisﬁed. Note that exactly one Ec

l,r event for each Bc

l

is present after the iteration.

For Bjto hold, all the events Ej,1,...,Ej,j must hold. It is

easy to show that, after the iterative process used to remove the

11

Bc

l’s, the event Ej,ij−1+1 ensures that all the events required

for Bjwith indices lower than or equal to ij−1automatically

hold. Thus, we can add the events {Ej,ij−1+1,...,Ej,j }to

guarantee that Bjholds, and remove it from the list. It is

important to notice that the term Ej,j is always present. At

this point we are left with the sum of probabilities of events,

which we call E-events, each of which is the intersection of

events of the form Ei,j and Ec

i,j . Thus, an E-event Sj

khas the

following form:

Sj

k,{Ek,1, Ek,2,...,Ec

k,ik, Ek+1,ik+1,...,Ec

k+1,ik+1 ,

...,Ej−1,ij−2+1, Ec

j−1,ij−1, Ej,ij−1+1,...,Ej,j }.(40)

By construction, the number of E-events for the generic term

jof the sum in Eqn. (36) is equal to the number of possible

dispositions of j−k Ec’s over j−1positions. As the number

of events of type Ecis different for the E-events of different

terms in Eqn. (36), the E-events relative to two different terms

of Eqn. (36) are different. We deﬁne Sjas the set of all

E-events which contain the event Ej,j . The elements of Sj

correspond to all the possible ways in which jmessages can

be decoded at the end of block number j. The cardinality of

Sjis equal to:

|Sj|=

j

X

k=1

(j−1)!

(k−1)!(j−k)! = 2j−1.(41)

Now we want to prove that

X

Sj

k∈Sj

P r{Sj

k}=P r{Ej,j }.(42)

Note that each Ek,l corresponds to a different event if the

index kis different, even for the same index l; thus, the law

of total probability can not be directly applied to prove Eqn.

(42). However, we will prove in the following preposition that

P r{Ek1,l}=P r {Ek2,l},∀k1, k2.

Proposition 1: Let us consider a set of random variables

C1,···, Cjthat are conditionally i.i.d. given U. Given any

two ordering vectors i=i1, i2,···, ijand l=l1, l2,···, lj,

we have

P r{Ci1≷R,...,Ci1+···+Cij≷jR}

=P r{Cl1≷R,...,Cl1+···+Clj≷jR}.(43)

Proof: The left hand side of Eqn. (43) can be rewritten as:

P r{Ci1≷R,...,Ci1+···+Cij≷jR}

=Z+∞

−∞

du Zθup

1

θlow

1

dci1...Zθup

j

θlow

j

dcijfCi|U(ci|u)fU(u),

(44)

where Ci=Ci1,...,Cijand ci=ci1,...,cij, while θlow

h

and θup

hare the lower and upper extremes of the integration

interval. θlow

his either equal to −∞ or to hR−ci1−· ··−cih−1,

∀h∈ {1,...,j}, depending on whether there is a <or a ≥in

the h-th inequality within brackets in Eqn. (44), respectively,

while θup

his either equal to hR −ci1−· ·· −cih−1or to +∞

depending on whether there is a <or a ≥in the h-th inequality

of Eqn. (44), respectively. By using Eqn. (9) and Eqn. (10) we

can write:

P r{Ci1≷R,...,Ci1+···+Cij≷jR}

=Z+∞

−∞

dufU(u)Zθup

1

θlow

1

dci1...Zθup

j

θlow

j

dcijfCi|U

=Z+∞

−∞

dufU(u)Zθup

1

θlow

1

dcl1...Zθup

j

θlow

j

dcljfCl|U

=P r{Cl1≷R,...,Cl1+···+Clj≷jR},(45)

where we deﬁned fCi|U,fCi1|U(ci1|u)···fCij|U(cij|u)and

fCl|U,fCl1|U(cl1|u)× ·· · × fClj|U(clj|u)The proposition

above guarantees that, although these events do not partition

the whole probability space of Ej,j, their probabilities add up

to that of Ej,j , i.e.:

2j−1

X

k=1

P r{Sj

k}=P r{Ej,j }

=P r{C1+···+Cj≥jR}.(46)

Finally, plugging Eqn. (46) into Eqn. (36) we can write:

E[m] =

M

X

j=1

j

X

k=1

P r{Bc

k, Bc

k+1,...,Bc

j−1, Bj}(47)

=

M

X

j=1 X

Sj

k∈Sj

P r{Sj

k}

=

M

X

j=1

P r{C1+···+Cj≥jR}.(48)

APPENDIX B

In the following we prove that the average throughput of the

aJE scheme RaJE approaches αR for large Mif C > αR.

Similarly to the JE scheme, it is sufﬁcient to prove that, if

C > αR,

lim

M→∞

Mα

X

m=1

a∗

m=c, (49)

for some 0< c < ∞, where a∗

m,P r nC∗

1+···+C∗

m

m< Ro.

We can rewrite a∗

mas:

a∗

m=P r

lm>C/α −R

σcq1

m+1−α

Mα2

,(50)

where

lm,C/α −C1+···+Cm

m−(1−α)

α

1

M(1−α)PM

j=Mα+1 Cj

σcq1

m+1−α

Mα2

(51)

is a random variable with zero mean and unit variance. From

the law of large numbers applied to Eqn. (50), we have

limm→+∞a∗

m= 0. First we show that

lim

m→+∞a∗

m

dm=c′,(52)

12

for some 0< c′<+∞where we have deﬁned:

dm,P r

l′

m>C/α −R

σcq1

m+1−α

mα2

,(53)

and

l′

m,C/α −C1+···+Cm

m−(1−α)

α

1

m(1−α)PMα+m

j=Mα+1 Cj

σcq1

m+1−α

mα2

(54)

such that l′

mis a random variable with zero mean and unit

variance. We have

lim

m→+∞a∗

m

dm= lim

m→+∞

P r (lm>C /α−R

σcq(1

m+1−α

Mα2))

P r (l′

m>C/α−R

σcq(1

m+1−α

mα2))

= lim

m→+∞

Q C/α−R

σcq(1

m+1−α

Mα2)!

Q C/α−R

σcq(1

m+1−α

mα2)!(55)

≤1,(56)

where inequality (56) follows from the fact that m < M and

Q(x)is monotonically decreasing in x. Then we show that

lim

M→∞

Mα

X

m=1

dm=c′′,(57)

for some 0< c′′ <+∞. To prove the convergence of the

series sum we show that limm→+∞dm+1

dm=λ′, for some 0<

λ′<1. From the central limit theorem we can write:

lim

m→+∞

dm+1

dm

= lim

m→+∞

P r

lm+1 >C/α−R

σcr1

m+1 +1−α

(m+1)α2

P r (lm>C /α−R

σcq(1

m+1−α

mα2))

= lim

m→+∞

Q

C/α−R

σcr1

m+1 +1−α

(m+1)α2

Q C/α−R

σcq(1

m+1−α

mα2)!

≤lim

m→+∞

σcr1

m+1 +1−α

(m+1)α2

(C/α−R)√2πe

−1

2

C/α−R

σcs1

m+1 +1−α

(m+1)α2

2

C/α−R

σcr(1

m+1−α

mα2)

1+

C/α−R

σcr(1

m+1−α

mα2)

21

√2πe−1

2

C/α−R

σcr(1

m+1−α

mα2)

2

(58)

= lim

m→+∞r1

m+1 +1−α

(m+1)α2

(C/α −R)2

·σ2

c1

m+1−α

mα2+ (C/α −R)2

q1

m+1−α

mα2

e−(C/α−R)2

2σ2

cα2

α2−α+1

(59)

=e−(C/α−R)2

2σ2

cα2

α2−α+1 <1,(60)

where inequality (58) follows from Eqn. (21).

From Eqn. (60) it follows that limM→∞ RaJ E =Rif αR <

C. Similarly, it can be easily shown that limM→∞ RaJE = 0

if αR > C.

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