A Novel Adaptive Hybrid Error Correction Scheme for Wireless DVB Services

doi:10.4236/ijcns.2008.12023

I. J. Communications, Network and System Sciences, 2008, 2, 105-206

Published Online May 2008 in SciRes (http://www.SRPublishing.org/journal/ijcns/).

A Novel Adaptive Hybrid Error Correction Scheme for

Wireless DVB Services

Guoping TAN1, Thorsten HERFET2

1 Student Member, IEEE, 2 Senior Member, IEEE

FR 6.2 – Telecommunications Lab, Saarland University

Campus Building C6 3, Floor 10, 66123 Saarbrücken, Germany

E-mail: {tan, herfet}@nt.uni-saarland.de

Abstract

Real-time applications usually not only have a certain Packet Loss Ratio (PLR) requirement but also can

have strict delay constraints. In the past, we proposed a Hybrid Error Correction (HEC) scheme with Packet

Repetition (PR) technique for guaranteeing a certain PLR requirement under strict delay constraints.

Unfortunately, the HEC-PR scheme can only work efficiently in multicast scenarios with small group size

and small link PLR. Our further studies show that better performance can be obtained by combining the

HEC-PR scheme with other traditional HEC schemes such as Type I HARQ and Type II HARQ techniques.

Based on this idea, in this paper, a novel Adaptive HEC (AHEC) scheme combining the HEC-PR scheme

with Type I and Type II HARQ techniques is proposed to satisfy a certain PLR requirement for delay

bounded multicast services. Furthermore, the performance of the AHEC scheme is optimized by choosing

the scheme with the least needed redundancy information automatically among the three HEC schemes.

Finally, by applying the AHEC scheme in a typical wireless DVB scenario, we analyze the performances of

the AHEC scheme and compare it with the HEC-PR scheme and an Adaptive Forward Error Correction

(AFEC) scheme. The results show that the proposed AHEC scheme outperforms both the AFEC scheme and

the HEC-PR scheme.

Keywords: ARQ, Forward Error Correction (FEC), Hybrid ARQ (HARQ), Multicast, Wireless Networks

1. Introduction

With the rapid development of broadband wireless

networks, more and more attention has turned to

distributing real time multimedia services over wireless

networks. Many classes of mobile commerce

applications require or can benefit from real-time

multicast support in wireless networks: mobile auction

or reverse auction, mobile entertainment services and

interactive games, mobile distance educations etc. [1].

As a major example, most of our personal digital

assistants (PDAs) and laptops are factory-equipped with

a Wi-Fi interface. In recent years, more and more places

are covered by wireless LANs with the IEEE 802.11 [2]

family of protocols in hotspots like hotels, airports or

conference locations. This will allow travelers to use

their PDAs or laptops for watching television

broadcastings, enjoying games or participating in video

conferences etc. All these new real-time multicast

applications are very likely to appear soon with

upcoming WiMAX or DVB-H [3] enabled devices. In

the following, to show the packet loss issue in wireless

real-time multicast systems, we will take the practical

Digital Video Broadcasting (DVB) services over

wireless LANs as the example for illustration.

We know that the IEEE 802.11 has been expected to

be used for DVB services over home and nomadic

networks. Moreover, since IP multicast provides a

scalable and efficient means for distributing datagram to

a group of receivers [4], IP-based networks were

proposed for delivering DVB services [5]. The DVB

systems based on IP multicast typically employ an

application-level protocol to provide some information

about the set of receivers and reception quality statistics.

The Real-time Transport Protocol (RTP) [6] is usually

used for this purpose. RTP or MAC layer of the IEEE

802.11 does not, however, guarantee any Quality of

188 G.P. TAN ET AL.

Service (QoS) for real-time multicast applications,

although the amount of lost packets varies during the

day and depends on the multicast data rate [7]. Therefore,

it is essential to employ some error control techniques at

application layer to guarantee a certain Packet Loss

Ratio (PLR) requirement (e.g. 10-6, refer to [5]) needed

for DVB services. In this paper, we thus address the

packet loss issue in wireless real-time multicast systems.

Traditionally, the packet loss issue can be treated as

erasure errors. As it is well known, there are mainly two

categories of erasure error control techniques: Automatic

Repeat Request (ARQ) that retransmits the lost packets

and packet-level Forward Error Correction (FEC) that

transmits redundant packets. Recently, many researchers

have been studying how to efficiently improve the

reliability of real-time multimedia multicast over

wireless networks by using these two techniques. Some

approaches focus on only one of the two schemes, ARQ

alone [8] or FEC alone [7]. Many other approaches

consider the combination of both to improve the

performance (see, e.g., [9–13]). The integrated

FEC/ARQ schemes are referred to as Hybrid Error

Correction (HEC) schemes in this paper. The studies

indicate that HEC schemes are much more efficient for

recovering data packets than the schemes with either

FEC or ARQ alone. In HEC schemes, many authors

employ powerful FEC erasure coding techniques (e.g.

Graph Codes [9] or Reed-Solomon (RS) codes

[10,11,12]). In addition, different retransmission-based

schemes for error control in multicast protocols geared

toward real-time multimedia applications are analyzed in

[14]. It is found that retransmission schemes are

appropriate for such applications, and actually can be

quite effective. In fact, the studies have shown that the

retransmission based error control schemes for point-to-

point communication or single receiver in multicast

scenario can outperform all the existing point-to-point

schemes [15]. Therefore, using retransmission based

error control mechanism with a Packet Repetition (PR)

technique; we developed an HEC-PR scheme for

satisfying the target PLR requirement under strict delay

constraints and optimized its performance in [13] for

DVB distribution in home networks. Even though the

scheme works perfectly in the in-home scenario and

additionally has the merit of being backward compatible

(so that conventional receivers with input buffers can

benefit from this scheme without modifications), it’s not

fully scalable for applications with larger group sizes. To

overcome its shortage, we proposed a Type I HARQ

scheme in [12] for those multicast scenarios with large

number of receivers.

However, the previous works mentioned above only

show that the good performance can be obtained by

combining the HEC-PR scheme and the Type I HARQ

scheme, while the question on how to combine them has

been left unanswered. In this paper, we thus try to

answer this question by proposing an adaptive HEC

(AHEC) scheme combining the HEC-PR scheme with

traditional Type I and Type II HARQ scheme. Following

the idea, we focus on developing one framework for the

AHEC scheme and then optimizing its performance. In

this paper, our main contributions include: (i) A novel

Adaptive HEC scheme combing the HEC-PR scheme

with other two traditionally HARQ schemes is proposed.

This scheme is suitable for any delay bounded multicast

application. (ii) By building a general mathematical

framework for the AHEC scheme under strict delay

constraints, we optimize its performance by minimizing

the total needed Redundancy Information (RI). To the

best of our knowledge, no general frameworks

combining those HEC schemes have been proposed

before for optimizing the performance of those schemes

under strict delay constraints.

The rest of the paper is organized as follows. In

Section 2, the performance of the packet level FEC

scheme is introduced. In Section 3, we present a general

mathematical framework on our novel AHEC scheme

and introduce a method to optimize its performances.

Applying the AHEC scheme in a typical DVB scenario

over wireless LANs, we analyze its performances and

compare it with the HEC-PR scheme and an Adaptive

FEC scheme in Section 4. Finally, conclusions are given

in Section 5.

Notation: Throughout of the paper, E(X) denotes the

expected value of a random variable X; and we keep in

mind that ⎟

⎟

⎠

⎞

⎜

⎝

⎛

h

mis the number of ways h objects can be

chosen from among m objects without repetition.

2. Performance of Packet Level FEC

In this paper, it is assumed that a perfect forward erasure

error correcting code (e.g. RS code) is used for the

AFEC scheme and the AHEC scheme. More recent

codes like LDPC [16] or Fountain / Raptor-codes [17]

do scale well for long blocks and offer advantages

concerning computational efficiency; in this paper,

however, a perfect erasure code is taken as the “upper

anchor”, and the block sizes for the used application

scenario (DVB over WLAN) can well be solved by RS

codes with acceptable computational complexity. For the

convenience of description, the perfect FEC code is

denoted by (n, k) code here, where k denotes the number

of data symbols per codeword and n denotes the code

word length. Figure 1 shows the structure of the coding

block transmitted within packets protected by the ideal

(n, k) code.

As shown in Figure1, the source data packet stream is

divided into blocks each consisting of k consecutive data

packets with a length of l bytes. The (n, k) code is

applied to each row containing k data packets in order to

produce a group of (n-k) parity packets. Without loss of

generality, it is assumed in this paper that the symbol

A NOVEL ADAPTIVE HYBRID ERROR CORRECTION SCHEME FOR 189

WIRELESS DVB SERVICES

size is always one byte for the FEC code. The coding

block is transmitted by packets in the form of columns.

Assuming exactly one packet per column, the receiver

only needs to correctly receive any k of these n columns

to be able to recover all the k data packets. Therefore,

the PLR performance of this scheme is exactly the same

as the performance of the ideal (n, k) code.

Figure 1. Applying ideal (n, k) code at the packet level,

which forms an FEC coding block in n packets

To simplify the analysis, in this paper, the packet loss

channel in wireless networks is modeled as the erasure

error channel with independently and identically

distributed (i.i.d.) losses with uniform distribution.

Firstly, we define the probability of b packets lost in a

sequence of n packets in the erasure channel with link

PLR of Pe as P(b,n,Pe). Since all of the n packets have

the same loss probability of Pe, the probability P(b,n,Pe)

is given by:

b

e

bn

ee PP

b

n

PnbP )()1(),,( −

−

⎟

⎠

⎞

⎜

⎝

⎛

= (1)

In the following, let the random variable Ik represent

the number of data packets lost in a group of k data

packets after decoding using the (n,k) code. Upon the

definition of Ik, the PLR performance of the (n,k) code

actually can be computed by E(Ik)/k. That means we only

need to calculate the expected value of I

k. To obtain

E(Ik), we firstly have to find out the Probability

Distribution Function (PDF) of Ik. For the convenience

of description, here we assume that the value of Ik is i

and there are b packets lost in a group of n packets. If b

is not more than n-k, the number of packets received in a

group of n packets will be at least k so that all of the k

data packets can be recovered. Obviously, the value of Ik

is zero in this case. On the other hand, if the value of Ik

is more than zero, there are exactly i (where 1≤i≤k) data

packets lost in the group of n packets after decoding

with the (n,k) code. It indicates that at least max (n-k+1,i)

and at most (n-k+i) packets are lost in this group. That is,

the value of b is in the range of [max(n-k+1,i), n-k+i].

Let Pd(i,b) denote the probability of i data packets lost

under the condition of b packets lost in a group of n

packets. In other words: Among all of the b packets lost

in the group, there are i data packets lost among all of

the k data packets and b-i parity packets lost among all

of the n-k parity packets. Let Pd(i,b) denote the

probability of i data packets lost among all of those b

packets lost. Note that all packets in a group of n packets

have the same loss probability in the i.i.d erasure error

channel, we thus have:

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛

=

b

n

ib

kn

i

k

biPd),( (2)

Based on the analysis above, using (1) and (2), we

then obtain the PDF of Ik as follows:

.,...,2,1),,(),,()I(Pr

),1max(

kibiPPnbPi de

ikn

iknb

k=== ∑

+−

+−=

(3)

Following (3), the expected value of Ik thus can be

calculated by:

∑∑

=

+−

+−=

×=

k

i

ikn

iknb

dek biPPnbPiE

1),1max(

),(),,()I( (4)

Finally, when the ideal (n,k) code is applied in the

erasure error channel with link PLR of Pe, the PLR

achieved will be:

∑∑

=

+−

+−=

×=

=

k

i

ikn

iknb

de

k

kn

biPPnbPi

k

E

PLR

1),1max(

),(

),(),,(

1

)I(

(5)

From the analysis above it follows that (5) is also the

PLR performance of the packet level FEC scheme with

an ideal (n,k) code over the erasure error channel with

link PLR of Pe.

3. Proposed AHEC Scheme

In this section, first, we introduce the system model of

the proposed Adaptive Hybrid Error Correction (AHEC)

scheme combing the HEC-PR scheme and the traditional

Type I and Type II HARQ schemes. Then, we present a

mathematical framework for the AHEC scheme. Based

on the mathematical framework, we finally present how

to design the optimum parameters for the AHEC scheme

guaranteeing a certain PLR requirement under strict

delay constraints.

At the beginning, for the AHEC scheme using

retransmission technique, rather than focus on a

particular transport protocol, we shall consider a generic

retransmission based scheme with the following features:

y A selective repetition, NACK-only retransmission

scheme is used;

y The transmitter multicasts the required packets

immediately to all receivers upon getting a NACK.

In addition, to simplify the analysis we make the

190 G.P. TAN ET AL.

following assumptions for the retransmission based

schemes:

y The feedback channel for NACKs is assumed to be

error-free. Since NACKs are control messages and

real systems usually provide mechanisms to

guarantee the reliable transmission of control signals,

this assumption is realistic in many cases. Or

alternately, the effect of NACKs loss can be

overcome by setting a margin for the PLR

performance of the AHEC scheme.

y All of the receivers experience erasure error channel

with i.i.d of uniform distribution. This means we do

not consider the effect of temporal correlation of the

channel and spatial correlation among different

receivers in this paper; this, however, is actually

ongoing work, which will come soon in [18].

Now, the essential symbols are defined and summed

up in Table 1.

Table 1. Symbols definitions

Symbol Definition

PLRtarget target PLR requirement

Dtarget target Delay requirement

Pe (j) the PLR for the j-th receiver

RTT(j) average round trip time for the j-th receiver,

one way delay is RTT(j) /2

Nrecv number of receivers in the multicast scenario1

ts the average interval between two continuous

original data packets at the transmitter2

trw the average waiting time at each receiver,

which is the time between the detection of a

packet loss and the time when the

corresponding NACK is sent3

tsw the average waiting time at the transmitter,

which is the time between receiving a NACK

message and the time when the corresponding

packets required by the NACK message are

retransmitted

tlp(j) the duration from the time the j-th receiver

detected packets lost to the time it possibly

receives the required packets, which is

RTT(j)+ tsw+ trw

3.1. System Model

From [13] we know that the HEC-PR scheme has a

major drawback: the total needed RI raises with the

increase of the group size linearly in a multicast scenario,

because the receivers can not share common

retransmission packets for repairing different missing

data packets. For overcoming this shortage, we proposed

a Type I HARQ scheme in [12] for those multicast

1The parameter Nrecv is also viewed as the group size in a multicast

scenario in this paper.

2 In this paper, it is assumed that the interval is same to the

retransmission interval for different copies of retransmission packets.

3 The average waiting time at each receiver is identical due to the same

process for all of the receivers.

scenarios with large number of receivers. However, the

works mentioned above only show that good

performance can be obtained by combining the HEC-PR

scheme and the Type I HARQ scheme, while the

question on how to combine them has been left

unanswered. In this paper, we thus propose an adaptive

HEC (AHEC) scheme combining the HEC-PR scheme

with traditional Type I and Type II HARQ scheme. The

system model of the AHEC scheme is shown in Figure 2.

As shown in this figure, the transmitter firstly

transmits encoding blocks to all receivers using the

packet level FEC code. Here it is assumed that perfect

forward erasure error correction code (e.g. Reed-

Solomon code.) is used and the number of source data

packets is k in one encoding block. That is, upon

received any k packets of one encoding block, the

receiver can recover all the data packets. Otherwise, the

receiver will send Negative-Acknowledgments (NACKs)

to the transmitter for repairing the missing data packets.

Now we explain the AHEC scheme in more detail:

1) First, the sender sends a certain amount of redundant

packets or only k data packets to all of the receivers

immediately with the first transmission. Especially,

when k is set to one then the redundant packets

during all of the retransmissions are always multiple

copies of source data packets; this scheme acts

actually as the HEC-PR scheme proposed in [13].

2) If any k packets of one encoding block are received,

the receiver then can recover all of the k data packets

and forward them to the application immediately.

Otherwise, the receiver will transmit a NACK

message to the sender to require essential redundant

packets for recovering all of the missing data packets.

3) Upon getting the first NACK message for one

encoding block during each retransmission round, the

sender will multicast a certain number of redundant

packets to all of receivers immediately with one copy

(or multiple copies) of these retransmission packets;

afterwards, if other NACKs for the same encoding

block are received, the sender will decide if multicast

more redundant packets to all of the receivers

according to the requirements of NACKs. That is, if

those later NACKs require more redundant packets

than the fist NACK message, the sender will

multicast further redundant packets to all of receivers

immediately; otherwise, the sender will neglect those

NACKs. Similarly, all of receivers can do

suppression of NACKs by this rule if those NACKs

are transmitted by multicasting mode.

From above introduction, we know that the

performance of the AHEC scheme mainly depends on

three parameters: the number of retransmission rounds;

the number of redundant packets with the first

transmission and retransmissions and the number of

copies of redundant packets with retransmissions. Note

that if the redundant packets are parity packets, they

A NOVEL ADAPTIVE HYBRID ERROR CORRECTION SCHEME FOR 191

WIRELESS DVB SERVICES

should not be repeated, but possibly more new parity

packets should be retransmitted. The remaining task is to

find out those suitable parameters of the AHEC scheme

for satisfying the strict QoS requirements of real-time

services. In the following section, we will present a

mathematical framework for analyzing the performances

of the AHEC scheme.

3.2. Performance Analysis

Theoretically, we should also design parameters for the

AHEC scheme with each receiver separately as for

HEC-PR scheme as in [13]. However, it is very hard to

implement for practical systems if different FEC codes

used for different receiver with the first transmission. To

simplify the implementation, therefore, the AHEC

scheme will adopt the same parameters for every

receiver. Since the assumed erasure code is perfect, a

suitable choice of the code rate can guarantee that all

receivers with their different channel conditions can be

served, so this simplification doesn’t negatively

influence the overhead.

Figure 2. System model of the AHEC scheme

The parameters for the AHEC scheme with

retransmissions are defined as follows:

y k: the number of source data packets in one encoding

block;

y Np: the number of redundant packets in one encoding

block with the first transmission;

y Ncc: a constant coefficient, which is the number of

additional new redundant packets for one encoding

block with different retransmission rounds;

y Nblk: the number of packets in one block with the first

transmission, which is k+Np;

y Nrr,max: the maximum possible number of

retransmission rounds;

y q

rt

N: the number of copies for each retransmission

packet at the sender during the q-th retransmission

round, where max,

1rr

Nq ≤≤ ;

y Nrt,max: the maximum possible number of copies for

each retransmission packet at the sender, which is:

∑

=

max,

1

max,

rr

N

q

rtrt NN .

For the convenience of description, two additional

random variables are defined as follows:

y Ik(j,w): a random variable representing the number of

missing data packets for the j-th receiver in one

encoding block of k source data packets after

experiencing w retransmission rounds, where 1≤w≤

Nrr,max;

y Nreq(j): a random variable representing the number of

redundant packets required to receive for recovering

all of the k data packets in one block for the j-th

receiver in the first retransmission round, where

0≤Nreq(j)≤k.

Based on above definitions, we now begin to analyze

the PLR performance for one receiver (without loss of

generality, it is assumed to be the j-th receiver) with the

AHEC scheme. To derive the PLR performance of the

AHEC scheme for the j-th receiver, we need to calculate

the expected value of the number of missing data

packets in one encoding block of k source data packets

after the retransmission packets experiencing w

(0≤w≤Nrr,max) retransmission rounds. The PLR

performance of the AHEC scheme for the j-th receiver

then can be calculated as: E(Ik(j,w))/k, which is the final

PLR at the j-th receiver after all of the retransmission

packets experienced w retransmission rounds. Note that

if the w is set to zero, the AHEC scheme acts as the

AFEC scheme. In the following, it is always assumed

that the w is more than zero for the AHEC scheme.

First of all, in order to recover all of the missing data

packets for each receiver that received fewer than k

packets for one block, at least Nreq,max redundant packets

need to be retransmitted at the sender:

))(),...,2(),1(max(

max, recvreqreqreqreq N

Ν

=

Ν

(6)

Obviously, Nreq,max is also a random variable. Since it

is assumed that the feedback channel is error-free, the

random variable Nreq,max always reflects the true

maximum number of lost packets in one block for the

worst receiver. Before calculating the average number of

lost data packets in one block for the j-th receiver, here

we define two useful probabilities: one is the PDF of

Nreq,max (i.e. Pr(Nreq,max=i)), which is denoted by Pi

Nreq,max;

the other is the probability of Nreq,max of i in the condition

of Nreq (j) of c (i.e. Pr(Nreq,max=i|Nreq(j)=c)), which is

denoted by Preq (i,c,j). The detail derivation on these two

important probabilities is attached in the appendix.

Secondly, according to the definitions above, the

parameter Ncc is always constant for any random value

of Nreq,max in each retransmission round. Now let symbol

“r” denote the number of received redundant packets

within w retransmission rounds. Note that multiple

copies of a retransmission packet received are counted

as one redundant packet received. We then define the

probability of r different redundant packets received

after all of the m=Nreq,max+Ncc different redundant

packets experiencing w retransmission rounds for the j-

th receiver as Precv(r,m,w,Pe(j)). Note that the loss

probably of each retransmission packet within w

retransmission rounds will be∑

=

w

q

rt

N

ejP1

))(( . We thus have:

192 G.P. TAN ET AL.

rm

N

e

r

N

eerecv

w

q

rt

w

q

rt

jPjP

r

m

jPwmrP

−

⎟

⎠

⎞

⎜

⎝

⎛∑

⎟

⎠

⎞

⎜

⎝

⎛∑

−

⎟

⎠

⎞

⎜

⎝

⎛

=== 11 ))(())((1))(,,,( (7)

For the convenience of description, in the following,

we define some temp symbols as follows: symbol ‘i’

denotes the number of data packets lost in a group of

Nblk packets for the j-th receiver in the first transmission;

symbol ‘b’ denotes the total number of packets lost in a

group of Nblk packets for the j-th receiver in the first

transmission; symbol ‘s’ denotes the number of

redundant packets sent at the sender in the first

retransmission round; symbol ‘m’ denotes the number of

redundant packets received during the retransmission

round for the j-th receiver. Now, as introduced in

Section 2, we also adopt Pd(i,b) to denote the probability

of i data packets lost under the condition of b packets

lost in a group of n packets. The conditional probability

Pd(i,b) also can be calculated by (2).

Finally, we can derive the PDF of Ik(j,w) based on

those probabilities introduced above. Now we assume

that the value of Ik(j,w) is i (where 1≤i≤k), which means

that there are i data packets lost after experiencing w

retransmission rounds. Obviously, it indicates that there

are b (where max(Np+1, i)≤b≤Np+i) packets lost in the

block of Nblk packets for the j-th receiver in the first

transmission. According to the AHEC scheme, this

receiver will require

b

–Np redundant packets for

retransmission at the sender for recovering the missing i

data packets. However, at the same time, the sender will

possibly send s (where b–Np+Ncc≤s≤k+Ncc) parity

packets due to combining all of the NACKs from overall

receivers in the multicast scenario. Finally, note that the

receiver obtained k+Np–b+m packets at the end of the w

retransmission rounds. Note that the data packets lost

only happen under the condition of k+Np–b+m being

less than k, which means the value of m will be less than

b–Np. Based on above analysis, using (2), (7) and the

probability Preq(m,c,j) (See Appendix), the PDF of Ik(j,w)

then can be expressed as this form:

⎟

⎠

⎞

⎜

⎝

⎛

−−

×==

∑

∑∑

−−

=

+

+=

+

+−=

1

0

),1max(

))(,,,(),,(

),()),(I(Pr

p

cc

ccp

Nb

m

erecvpccreq

iN

iNb

Nk

NNbs

dk

jPwsmPjNbNsP

biPiwj

(8)

where i=1,2,…,k.

Following (8), we then obtain the expected value of

Ik(j,w):

()

∑

=

=Ι×=

k

i

kk iwjiwjE

1

),(Pr)),(I( (9)

Relying on (9) and substituting (7) and (8) into (9),

we then obtain the PLR performance of the AHEC

scheme for the j-th receiver with Nrr,max retransmission

rounds immediately:

⎟

⎠

⎞

⎜

⎝

⎛−

⎟

⎠

⎞

⎜

⎝

⎛

×−−

×=

⎟

⎠

⎞

⎜

⎝

⎛∑∑

−

⎟

⎠

⎞

⎜

⎝

⎛

×−−

×=

⎟

⎠

⎞

⎜

⎝

⎛

−−

×=

=

∑

∑∑ ∑

∑

∑∑ ∑

∑

∑∑ ∑

−−

=

−

=

+

+=

+

+−=

−−

=

−

=

+

+=

+

+−=

−−

=

+

+=

+

+−=

==

1

0

1),1max(

1

0

1),1max(

1

0

max,

1),1max(

max,

)))((()))((1(

),,(),(

1

)))((()))((1(

),,(),(

1

))(,,,(),,(),(

1

)),(I(

),(

max,max,

max,

1

max,

1

p

rtrt

p

cc

ccp

p

rr

N

q

rt

rr

N

q

rt

p

cc

ccp

p

cc

ccp

Nb

m

ms

N

e

m

N

e

pccreqd

k

i

iN

iNb

Nk

NNbs

Nb

m

ms

N

e

m

N

e

pccreqd

k

i

iN

iNb

Nk

NNbs

Nb

m

errrecvpccreqd

k

i

iN

iNb

Nk

NNbs

rrk

rrAHEC

jPjP

m

s

jNbNsPbiP

i

k

jPjP

m

s

jNbNsPbiP

i

k

jPNsmPjNbNsPbiP

i

k

NjE

NjPLR

(10)

For simplifying the description, we define a vector as

follows: j

e

P

v

=[Pe(1), P

e (2),…, P

e (j-1), Pe (j+1),…, Pe

(Nrecv)]. By observing (10), we can find that it is actually

a function with these parameters: k, Np, Ncc, rt

N

v

, Pe(j)

and j

e

P

v

, which is denoted by

)),(,,,,,( max,

,

j

eerecvrtccp

AHECPLR PjPNNNNkf

v

in this paper.

In the following, let’s consider the total needed RI

with the AHEC scheme, which includes two parts: one is

the common part for all of the receivers in the first

transmission, which is Np/k; the other is the part in the

retransmissions, which is caused by the retransmissions

of redundant packets for all of the receivers. For the

convenience of calculation, we divide the second part

into two subparts: one is the needed RI in the first

retransmission round (denotes by RIAHEC-I); the other part

is the needed RI in the retransmission rounds (denotes

by RIAHEC-II) of from the second retransmission round to

the Nrr,max retransmission round. First, considering the

calculation of RIAHEC-I, note that the value i (where 0≤i≤k)

of Nreq,max means that )(

1

ccrt NiN+× redundant packets

will be retransmitted in the first retransmission round at

the sender. Using the PDF of Nreq,max (See Appendix),

thus, the RIAHEC-I is given by:

∑

=

−×+×=

k

i

NccrtIAHECreq

PNiN

k

RI

1

max,

)(

1 (11)

Then, considering the calculation of RIAHEC-II, to

simplify the analysis, here it is assumed that there is only

one Equivalent Receiver (ER) in the multicast scenario

with Nrecv receivers and the loss probability of each

A NOVEL ADAPTIVE HYBRID ERROR CORRECTION SCHEME FOR 193

WIRELESS DVB SERVICES

retransmission packet for the ER ise

P, where e

Pis the

average link PLR and defined as:

recv

N

j

e

eN

jP

P

recv

∑

=

=1

)(

(12)

To derive RIAHEC-II, let’s note the following fact: if the

ER requires i (1 ≤ i ≤ k) redundant packets for repairing

missing data packets in the q-th (2 ≤ i ≤ Nrr,max)

retransmission round, it indicates that the ER required j

(i ≤ j ≤ k) redundant packets in the first retransmission

round and received only j–i redundant packets in the

group of Ncc+j retransmission redundant packets in all of

the previous q–1 retransmission rounds. Therefore, using

the PDF of Nreq,max and (7), we can obtain the probability

of the ER requiring i redundant packets in the q-th

retransmission round, i.e.:

),1,,(

max, eccrecv

k

ij

j

NPqjNijPP req −+−

∑

=

. Then the calculation

of RIAHEC-II can be written as this form:

⎟

⎠

⎞

⎜

⎝

⎛−+−

×+×=

∑

∑∑

=

==

−

),1,,(

)(

1

max,

21

eccrecv

k

ij

j

N

q

k

i

cc

q

rtIIAHEC

PqjNijPP

NiN

k

RI

req

rr

(13)

As a result, by substituting (7) into (13) and then

combining (11) and (13), the total needed RI for the

AHEC scheme with Nrr,max retransmission rounds is

given by:

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛∑

−

⎟

⎠

⎞

⎜

⎝

⎛∑

⎟

⎠

⎞

⎜

⎝

⎛

−

+

×+×

+×+×+=

++=

−+

=

==

=

−−

−

=

−

=

∑

∑∑

∑

ij

N

e

iN

N

e

cc

k

ij

j

N

q

k

i

cc

q

rt

k

i

Nccrt

p

IIAHECIAHEC

p

AHEC

q

g

rt

cc

q

g

rt

req

rr

req

PP

ij

jN

P

NiN

k

PNiN

kk

N

RIRI

k

N

RI

1

max,

)(1)(

)(

1

)(

1

21

1

(14)

Now we define two vectors e

P

v

=[Pe(1), Pe (2),…, P

e

(Nrecv)] and=

rt

N

v

[max,

,...,, 21 rr

N

rtrtrt NNN ]. By observing (14),

we can also find that the RI performance of the AHEC

scheme actually is a function with such these parameters:

k, N

p, Ncc, rt

N

v

, e

Pand e

P

v

, which is denoted by:

),,,,,,(

,eerecvrtccp

AHECRIPPNNNNkf

v

in this paper.

3.3. Optimization of the AHEC Scheme

In this section, we will propose a method to design

suitable parameters for the AHEC scheme. Note that all

of the receivers share identical parameters for this

scheme. Therefore, if the AHEC scheme can guarantee

the QoS requirements for the worst receiver, it can also

guarantee the same QoS requirements for every receiver

in the multicasting scenario. Without loss of generality,

it is assumed the first receiver is the one with the worst

situation in a multicasting scenario. In other words, the

first receiver has the largest link PLR and the largest

RTT. Our remaining task is to design suitable

parameters of the AHEC scheme, which will satisfy a

certain PLR requirement for the first receiver under strict

delay constraint with minimum total needed RI.

First of all, it is known that the delay requirement will

limit the number of data packets in one block and the

number of retransmissions. In the following, we will

derive the boundary of the two parameters Nrr,max and k

based on the strict delay constraint. For those

retransmission packets in the first receiver, the

maximum possible end-to-end delay includes four parts:

the one-way delay in the first transmission (which is

RTT(1)/2); Nrr,max of two-way delays in the

retransmission rounds (which is Nrr,max×tlp(1)); the

decoding delay caused by the length of encoding block

(which is (k×ts)) and total number of intervals for the

copies of retransmission packets (which is (Nrt,max×ts)).

Thus, for those retransmission packets of the first

receiver, the maximum possible end-to-end delay must

satisfy:

targetmax,max, D)()1(

2

)1( ≤×++×+ srtlprrtNktN

RTT

(15)

Because the value of (k + Nrt,max) is at least 2 for the

AHEC scheme, the maximum allowable number of

retransmission rounds is constrained by:

⎥

⎦

⎥

⎢

⎣

⎢×−−

=)1(

2

)1(

D

ˆtarget

max,

lp

s

rr t

t

RTT

N (16)

where

⎣

⎦

x denotes the largest integer not greater than x.

Therefore, for the AHEC scheme the parameter Nrr,max

will be limited in the range of between one andmax,

ˆrr

N.

Then, we define the length of k with w retransmission

rounds and maximum v copies of retransmission packets

as k(w,v), which is given by:

194 G.P. TAN ET AL.

⎥

⎦

⎥

⎢

⎣

⎢×−−×−

=

s

slp

t

tv

RTT

tw

vwk2

)1(

)1(D

),(

target (17)

where 1≤w≤max,

ˆrr

N and v≥w.

Note that in (17) the parameter k will only rely on the

parameters w and v if tlp(1), ts, RTT(1) and Dtarget are

fixed. To simplify the design, we assume that the

parameters tlp(1), ts and RTT(1) is always fixed in a

scenario and Dtarget is also constant for the given QoS

requirements. Therefore, the length of k will only

depend on the parameters w and Nrt,max..

Considering practical implementation, actually, we

can set an upper band of the maximum possible value

for Nrt,max (denotes bymax,

~

rt

N, note that it is not less

than max,

ˆrr

Ndue to the practical consideration).

Theoretically, the value of max,

~

rt

N can be set to infinite.

However, as shown in (14), the total needed RI increase

with the increasing of max,

~

rt

N significantly. Therefore,

the minimum total needed RI usually can be acquired

under a small value ofmax,

~

rt

N. Now the parameter rt

N

v

can be limited in a small finite space max,rr

N

Φ(i.e.

max,rr

N

rt

NΦ∈

v), which is:

[]

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

≥≥≥

≤

∑

=Φ=

1,1

~

,...,,

max,

1

21

max,

iNr

Nr

rrr

rri

rt

N

kk

N

rr

(18)

where: max,max, ˆ

1rrrr NN ≤≤

Note that the parameter Nrt,max of the AHEC scheme

is actually the sum of all of the elements in the vector rt

N

v

,

which is denoted by sum(rt

N

v) here. Depending on the

PLR and RI performance analyzed for the AHEC

scheme in Subsection 3.2, our optimization problem thus

can be written as the following form:

()

eerecvrtccprtrrAHECRI

N

optAHEC

PPNNNNNNkf

RI

rr

N

rt vvv

v

,,,,,)),sum(,(

minarg

max,,

,

max,

Φ∈

=

Subject to:

max,max, ˆ

1rrrr NN ≤≤

(

)

target

1

max,,

PLR

),1(,),sum(,,)),sum(,(

≤

eerecvrtccprtrrAHECPLR PPNNNNNNkf

v

vv

(19)

By solving (19) with traversing the full space, we

thus can obtain the optimal parameters for the AHEC

scheme: k, Np, Ncc, Nrr,max and rt

N

v.

Remarks: If k is set to one, the AHEC scheme acts as

a pure HEC-PR scheme; If k is set to more than one and

Np is set to more than zero, the AHEC scheme acts as the

traditional Type I HARQ scheme; If k is set to more than

one and Np is set to zero, the AHEC scheme acts as the

traditional Type II HARQ scheme. As a result, the

AHEC scheme can choose the best scheme automatically

among the HEC-PR scheme, traditional Type I and Type

II HARQ scheme by solving (19).

4. Analysis Results

In this section, we firstly analyze the performances of

three schemes over an erasure error channel: the HEC-

PR scheme, the AFEC scheme and the AHEC scheme,

respectively; and then compare them with each other.

Then, we study the effect of the parameter k in the

AHEC scheme. For the convenience of comparing

different schemes fairly, we make some assumptions as

follows: the entire receivers experience the i.i.d erasure

channel with the same level of original link PLR; the

three schemes use identical system parameters with the

same QoS requirements for the PLR and the same

latency. In this case, we consider DVB services over

wireless home networks with a group size of less than 7,

RTT of less than 15ms and a wireless link PLR of up to

10% when the video multicast data rate is more than

500Kbps [7]. The target PLR requirement is set to 10-6

under the strict delay constraint of 100ms (refers to [5]).

However, it should be clear that the proposed AHEC

scheme is suitable for any wireless multicasting scenario

under strict delay boundary. Now we apply the three

schemes in a typical scenario with the common system

parameters, which are summarized in Table 2.

Table 2. System parameters

PLR Requirement: PLRtarget 10-6

Latency Constraint: Dtarget 100ms

Multimedia Data Rate: 4Mbps

Packet Loss Model GE Model

RTT 15ms

trw+tsw: 0ms

Encoding Packet Length: Nsymb 1250bytes

Original Average Link PLR: Pe 10-3~10-1

Actually, the following theoretical analysis results

have been accompanied by simulations with ns-2 [19].

However, those simulation results are not further

explained in this paper due to the fact theory matches

simulation very well.

4.1. Optimization Results

In the following, we will design the optimum parameters

for the AHEC scheme for this typical scenario and then

A NOVEL ADAPTIVE HYBRID ERROR CORRECTION SCHEME FOR 195

WIRELESS DVB SERVICES

compare it with the HEC-PR scheme and the AFEC

scheme. First, to meet the latency requirement we can

obtain the maximum allowable number of retransmission

rounds max,

ˆrr

N with the AHEC scheme by solving (16),

which are at most five. Then, since it is found that the

optimum value for the parameter Nrt,max is usually much

less than five by our numerical calculations for this case,

the maximum value of Nrt,max is set to five (i.e. 5

~

max, =

rt

N)

to make sure that the optimum results of the AHEC

scheme can be acquired by the searching algorithm.

Note that in the traditional Type I and Type II HARQ

schemes, multiple copies of a parity packet only can be

counted as one redundant packet at the receivers. For the

consideration of efficiency, we should only adopt new

parity packets instead of multiple copies of redundant

packets at each retransmission round. For this reason,

the parameters of the AHEC scheme will always satisfy

Nrt,max =Nrt,max (i.e. max,

1,1rr

q

rt NqN ≤≤≡) in the case of

k>1. Based on these boundaries, the optimum

parameters of AHEC with different link PLR and small

group size can be obtained by solving (19). Parts of the

optimum parameters are shown in Table 3 and 4.

Table 3. Optimum parameters of the AHEC scheme with

Nrecv=2

Optimum Parameters of the AHEC scheme

rt

N

v

Average

Link PLR

k

Np

Ncc

1

rt

N 2

rt

N 3

rt

N4

rt

N

0.001 23 0 0 1 1 - -

0.01 16 0 0 1 1 1 -

0.03 9 0 0 1 1 1 1

0.05 9 0 0 1 1 1 1

0.07 1 0 0 1 1 1 2

0.09 1 0 0 1 1 1 2

0.10 16 2 1 1 1 1 -

Table 4. Optimum parameters of the AHEC scheme with

average link PLR of 0.06

Optimum Parameters of the AHEC scheme

rt

N

v

Nrecv

k

Np

Ncc

1

rt

N 2

rt

N 3

rt

N4

rt

N

1 1 0 0 1 1 1 1

2 1 0 0 1 1 1 1

3 23 2 1 1 1 - -

4 23 2 1 1 1 - -

5 23 3 1 1 1 - -

6 23 3 1 1 1 - -

7 23 3 1 1 1 - -

From these two tables, we can see that the AHEC

scheme can automatically choose the most suitable

scheme according to current group size and average link

PLR among the HEC-PR scheme, the Type I HARQ

scheme and the Type II HARQ scheme. For example, as

shown in Table 3, the AHEC scheme will act as the

Type II HARQ scheme if the real-time multicast

scenario is with Nrecv=2 and average link PLR of 0.05.

Similarly, as shown in Table 4, the AHEC scheme acts

as the HEC-PR scheme if the scenario is with Nrecv=2

and average link PLR of 0.06; and it will act as the Type

I HARQ scheme if the scenario is with Nrecv=7 and

average link PLR of 0.06.

Figure 3. The total needed RI with different schemes

4.2. Performance Comparisons

Upon those optimum results shown in Table 3 and Table

4, now we obtain the total needed RI by (14) of the

AHEC scheme with different group size, which are

shown in Figure 3.

For comparing the performances among different

schemes, Figure 3 also show the total needed RI with the

HEC-PR scheme and the AFEC scheme. Note that the

AFEC scheme is actually a special case of the AHEC

scheme with Np=0 and Nrr,max=0. From this figure, we

can see that the total needed RI of the HEC-PR scheme

increases with the increase of the number of receivers

significantly but not strict linearly. The reason is clear:

although all of the receivers are independent, they also

can recover some common missing packets by

retransmitting a small part of identical packets with the

HEC-PR scheme. As a matter of fact, the probability of

sharing identical packets among different receivers will

increase with the increase of the number of receivers.

This leads to the total needed RI of the HEC-PR scheme

increase with the number of receivers significantly but

not strict linearly. In other words, the speed of the

increase of the total needed RI will slow down with the

increase of the number of receivers.

Additionally, as shown in this figure, the AHEC

scheme always outperform the HEC-PR scheme and the

AFEC scheme, because it can choose the best scheme

automatically among different HEC schemes. However,

from this figure, we also can see that the performance of

196 G.P. TAN ET AL.

the HEC-PR scheme is very close to the AHEC scheme

when the number of receivers is no more than two or the

average link PLR is less than 10-2. Because the

implementation of the HEC-PR scheme is very simple

due to without any encoding and decoding algorithm, the

HEC-PR scheme should be considered for such as those

real-time multicast services with small group size and

small average link PLR.

4.3. The Effect of the Parameter k

Finally, we study the effect of the parameter k in the

AHEC scheme in this section. For the convenience of

analysis, we searched for the optimum parameters for

the AHEC scheme with the fixed parameters Nrecv=5 and

Nrr,max=2 under different average link PLR of 0.01,0.05

and 0.10. Actually, on the effect of the k, the tendency is

similar for any average link PLR. Here we only take

three typical examples to demonstrate the tendency of its

effect. Part of those optimum parameters is listed in the

Table 5.

Table 5. Optimum Parameters of the AHEC Scheme with

Nrecv=5 and Nrr,max=2

k 20 40 80 120 160200

Np 1 1 1 1 2 2 PLR=0.01

Ncc 0 0 0 0 0 0

Np 2 3 6 8 10 12 PLR=0.05

Ncc 1 1 1 1 1 1

Np 4 7 12 16 21 25 PLR=0.10

Ncc 2 2 2 2 2 2

Note that although here we only show the results for

the AHEC scheme with the length of k being less than

200, it should be clear that the AHEC scheme is suitable

for any length of k upon requirements; moreover, the

higher k is employed by the ideal erasure error code, the

more efficient code rate can be adopted. From this table,

we can see that the parameter Ncc is always invariable

under certain average link PLR. That is, we only need to

change the parameter Np for the AHEC scheme

according to different value of k under certain average

link PLR. Note that the variable value of k means

different multimedia data rate under certain delay

constraints if the packet size is constant. Obviously, this

feature of the AHEC scheme can simplify its

implementation in real-time multicast scenarios with

variable source data rate.

Upon those optimum parameters with different length

of k, the total needed RI of the AHEC scheme is

obtained and shown in Figure 4.

From this figure, we can see that the total needed RI

decreases significantly when the parameter k increases

from 10 to 60. When k is more than 60, however, this

parameter has only a little effect on the performance of

the AHEC scheme. Note that different k values mean

different delay constraints or different source data rates

if the packet size is fixed. Therefore, under certain delay

constraints with fixed packet size, the higher the

multicast source data rate is, the better performance can

be achieved in the AHEC scheme. Moreover, since the

stable good performance can be obtained if the

parameter k is more than 60, a suitable fixed short length

of k (≥60) can be always adopted when the data rate is

high enough to provide good delay performance. On the

other hand, the parameter k is only associated with the

delay constraints if both the source data rate and the

packet size are fixed: to guarantee a certain PLR

requirement, shorter delay constraints the system has,

shorter length of the parameter k in the AHEC scheme

has to be adopted so that more RI needed. Therefore, the

AHEC scheme also provides a good way for the tradeoff

between the total needed RI and delay constraints by

choosing different k.

Figure 4. The total needed RI of the AHEC scheme with

Nrecv=5 and Nrr,max=2

5. Conclusions

In this paper, we propose a novel Adaptive Hybrid Error

Correction (AHEC) scheme by choosing the most

suitable HEC scheme among HEC-PR scheme, Type I

HARQ scheme and Type II HARQ scheme under strict

delay constraints. Using the proposed mathematical

framework for the AHEC scheme, we can design the

most suitable parameters of the AHEC scheme for

guaranteeing a certain PLR requirement under strict

delay constraints with minimum needed RI. By applying

the proposed AHEC scheme in a typical Wireless DVB

scenario for performance analysis and comparisons, we

have found:

1) The AHEC scheme outperforms the HEC-PR scheme

and AFEC scheme in all cases. However, when either

the group size or the average link PLR is small

enough, the performance of the HEC-PR scheme is

very close and even equal to the best performance of

the AHEC scheme. Due to the simplicity of the HEC-

A NOVEL ADAPTIVE HYBRID ERROR CORRECTION SCHEME FOR 197

WIRELESS DVB SERVICES

PR scheme without any encoding or decoding

algorithm, this scheme should be considered in the

real-time multicast scenarios with small group size or

small average link PLR.

2) In most cases, the best performance of the AHEC can

be obtained with variable network and channel

conditions by only varying the parameter Np. Thus,

the AHEC scheme is usually robust and simple to

implement for practical systems.

3) The length of k has a great effect on the performance

of the AHEC scheme. The performance increases in

case the scenario allows for the choice of a bigger k.

It indicates that the AHEC scheme is very suitable for

the real-time multicast systems with high data rate.

Also, it provides a good way for the tradeoff between

the total needed RI and the strict delay constraints.

In this paper, for simplifying the analysis, we have

made a strong assumption: all of the receivers are

independent and experience i.i.d channel with uniform

distribution. That is, we do not consider the effect of

temporal and spatial correlation in real wireless channels.

For future works, we will analyze the performance of the

AHEC scheme based on accurate Gilbert-Elliot [20,21]

channel model for practical wireless channels. First

results, however, do confirm that all conclusions made

in this paper remain valid and that spatial and temporal

correlation shifts the architectural choice in the

parameter space but do not change the conclusions.

6. Appendix: PDF of Nreq,max and Derivation

of Preq(i,c,j)

First, to derive the PDF of Nreq,max, we define two basic

probabilities: one is the probability of Nreq(j) being i,

which is denoted by )( jP i

req

=; the other is the probability

of Nreq(j) being less than i, which is denoted by)( jP i

req

<.

Using (1), these two probabilities can be calculated as

follows, respectively:

))(,,())(Pr()( jPNiNPijjP eblkpreq

i

req +==Ν=

=

∑−+

=

<=<Ν=

1

0

))(,,())(Pr()(

iN

g

eblkreq

i

req

p

jPNgPijjP

(20)

where 1 ≤ i ≤k and 1≤ j ≤ Nrecv

To simplify the analysis in this paper, it is assumed

that all of the receivers have the same PLR level of e

Pas

defined by (12). Thus, following (20) we define:

),,()( eblkp

i

req

i

req PNiNPjPP +====

∑−+

=

<< ==

1

0

),,()(

iN

g

eblk

i

req

i

req

p

PNgPjPP (21)

{

}

recv

Nj ,...,2,1:where

∈

∀

Now let )(

max, hPi

Nreq be the probability of h receivers

lost Np+i packets and the other Nrecv–h receivers lost less

than Np+i packets. Based on the definitions above and

using (21), this probability can be obtained:

hN

i

req

hi

req

recv

i

Nrecv

req PP

h

N

hP −

<=

⎟

⎠

⎞

⎜

⎝

⎛

=)()()(

max, (22)

Then, Let i

Nreq

Pmax, denotes the probability of

Pr(Nreq,max=i). Upon (22), we can obtain the PDF

of max,req

Ν:

kihPiP

recv

reqreq

N

h

i

Nreq

i

N≤≤==Ν= ∑

=

1,)()Pr(

1

max, max,max,

(23)

Secondly, let’s consider the probability P

req(i,c,j) for

the j-th receiver in the following. Similarly, to simplify

the analysis in this paper, we assume that all of the other

Nrecv–1 receivers except for the j-th receiver have the

same PLR level of j

e

Pas defined as follows:

1

)()(

1

−

+

=

∑∑ +=

−

=

recv

N

ji

e

j

i

e

j

eN

iPiP

P

recv

(24)

Based on this assumption for those Nrecv–1 receivers,

we define:

),,()(

ˆj

eblkp

i

req

i

req PNiNPdPP +== ==

∑−+

=

<< ==

1

0

),,()(

ˆ

iN

g

j

eblk

i

req

i

req

p

PNgPdPP (25)

where:

{

}

}{,...,2,1jNdrecv

−

∈

∀

Now concerning those Nrecv–1 receivers except for the

j-th receiver, let ),(

max, jhPi

Nreq be the probability of h

receivers lost Np+i packets and the other Nrecv–h–1

receivers lost less than Np+i packets. Using (25), then,

the probability can be calculated by:

1

)

ˆ

()

ˆ

(

1

),(

max,

−−

<=

⎟

⎠

⎞

⎜

⎝

⎛−

=hN

i

req

hi

req

recv

i

Nrecv

req PP

h

N

jhP (26)

Actually, the calculation of Preq(i,c,j) should be

divided into two parts according to two different cases:

1) One part is the probability of Pr(Nreq,max =i, Nreq(j) = c)

with i=c, in which case the number of missing

packets in one block are no more than Np+c for any

receiver among those Nrecv–1 receivers except for the

j-th receiver. Using (26), Preq(i,c,j) can be expressed

as:

198 G.P. TAN ET AL.

⎟

⎠

⎞

⎜

⎝

⎛

+=

=Ν=Ν=

∑−

=

1

0

max,

),())(,,(

))(,Pr(),,(

max,

recv

req

N

h

c

Neblkp

reqreqreq

jhPjPNcNP

cjcjccP

(27)

2) The other part is the probability of Pr(Nreq,max =i,

Nreq(j) =c) with i> c, in which case at least one

receiver among those Nrecv–1 receivers except for the

j-th receiver lose Np+ i packets in one block and all of

the other receivers lose less than Np+ i packets in the

block. Similarly, in this case, P

req(i,c,j) should be

calculated by:

⎟

⎠

⎞

⎜

⎝

⎛

+=

=Ν=Ν=

∑−

=

1

max,

),())(,,(

))(,Pr(),,(

max,

recv

req

N

h

i

Neblkp

reqreqreq

jhPjPNcNP

cjijciP

(28)

To integrate (27) and (28) for the expression of

Preq(i,c,j), we define a function fcmr(x1,x2) (where x1≥x2)

as follows:

⎩

⎨

⎧

>

=

=21,1

21,0

)2,1( xx

xx

xxfcmr (29)

As a result, based on (27), (28) and (29), the

calculation of Preq(i,c,j) can be expressed as the

following form:

⎟

⎠

⎞

⎜

⎝

⎛

×+=

∑−

=

1

),(

),max( ),(

))(,,(),,(

max,

recv

cmr

req

N

cifh

ci

N

eblkpreq

jhP

jPNcNPjciP

(30)

7. References

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[19] http://www.isi.edu/nsnam/ns/.

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