Int. J. Communications, Network and System Sciences, 2009, 7, 619-626
doi:10.4236/ijcns.2009.27069 Published Online October 2009 (http://www.SciRP.org/journal/ijcns/).
A Hybrid ARQ System with Erasures Correction
and Parity Retransmission
L. GOLDFELD, A. HOFMAN, V. LYANDRES
Department of Electrical and Computer Engineering, Communications Laboratory,
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Received June 30, 2009; revised August 4, 2009; accepted September 8, 2009
ABSTRACT
A modified type of Hybrid ARQ system with erasures correction and parity bits retransmission is considered.
Performance of the system is analyzed under assumption that the forward channel suffers from Nakagami
common fading and additive white Gaussian noise. A good agreement between theoretical results and simu-
lation is achieved. The proposed ARQ protocol is compared with other known Hybrid ARQ algorithms. It
demonstrates significantly higher throughput efficiency in a range of SNR.
Keywords: Hybrid ARQ Scheme, Erasures Correction Decoding, Fading Channel
1. Introduction
Automatic Repeat ReQuest (ARQ) systems with error
control are widely used for data transmission over noisy
channels. Their performance is usually characterized by
two parameters: Throughput Efficiency (TE) and Bit
Error Rate (BER). So-called hybrid ARQ (HARQ) sys-
tems use two codes (inner and outer) [1–8]. In a HARQ-I
system Forward Error Correction (FEC) is performed
prior to error detection [2]. In more effective HARQ-II
[3–5] system, parity-check digits for error correction are
sent to the receiver only when they are needed. For ex-
ample [3], at the initial step data blocks D with par-
ity-check bits of the outer error detection code are trans-
mitted. If errors in D are detected, the system begins not
simple repetitions of D, but repetitions of a parity block
P(D) of the inner code. P(D) as well as D itself are al-
ternately stored in the receiver buffer for error correction
until D will be recovered. As shown in [4], application of
HARQ schemes can significantly improve TE in com-
parison with a pure ARQ scheme.
It is well known [1] that error correcting capability of
block codes may be enhanced when soft decision ap-
proach is realized, for example, in the form of Soft Deci-
sion Erasures Correcting (SDEC) decoding [7–9]. In this
case the number of erroneous bits that may be corrected
is not less than dH –1. It is assumed also that all error
symbols are erased, while for FEC decoding the number
of erroneous corrected bits is not less than 0.5(dH –1),
where dH is the minimum Hamming distance of the block
code.
In this paper, a modified type of HARQ-II is consid-
ered. Two linear block codes are used in the system: an
outer systematic (n,k) block code C2 and an inner half-
rate invertible (2k,k) code C1. At the receiver FEC and
SDEC decoding are used alternately, according to the
procedure described below. The system is named HARQ
with Erasures Correction (HARQ-EC). Theoretical ana-
lysis and computer simulation of the proposed system are
performed for the case of noiseless feedback, Nakagami
common fading and Additive White Gaussian Noise
(AWGN) in the forward channel1. Moreover, we con-
sider the forward channel as memoryless, i.e. interleav-
ing/de-interleaving assumed to be ideal. The obtained
results show that HARQ-EC provides gain in BER, or
gain in TE in comparison with parameters of HARQ-II
for the same outer and inner codes [2,4–5].
The paper is organized as follows. In Section 2, we
describe the HARQ-EC algorithm. In Section 3 the rele-
vant BER and TE are analyzed. The comparison of
HARQ-EC and HARQ-II characteristics is given in Sec-
tion 4. Section 5 presents discussion of results and some
conclusions.
2. Description of the HARQ-EC System
1The assumption of noiseless feedback does not reduce the generality o
f
the analysis, as we are interested in the performance comparison o
f
HARQ-EC and HARQ-II system in the same conditions.
HARQ-EC scheme uses the outer (n,k) code C2 with

L. GOLDFELD ET AL. 620
minimal Hamming distance and the inner (2k,k)
half-rate invertible code [2] C1 with minimal Hamming
distance . It is called invertible since with the help
of inversion of k parity-check bits k information bits can
be uniquely determined. We assume that each transmit-
ted message D consists of k information bits and that
each encoded message, called a code word CW, consists
of n bits. When D is ready for transmission, it is encoded
by the outer encoder into the transmitted n-length code
word CW2(D,Q), where Q denotes the vector of n–k par-
ity-check bits. In parallel the transmitter computes k bits
of the parity-check block P(D) of the half rate invertible
(2k,k) code C1. The block P(D) is not transmitted and is
stored in the buffer for later use.
 2
H
d
 1
H
d
Let CW2(Dr ,Qr) denotes the received vector if
CW2(Dr ,Qr) was transmitted. The received data block
Dr is passed to the forward channel receiver of the
HARQ-EC. Its key elements are Soft Decision Maxi-
mum Likelihood Demodulator (SDMLD), FEC and Er-
rors Erasure Correction (EEC) decoders for the outer and
inner codes respectively. In SDMLD the decision

 r
l
A
about symbol Al is obtained according to the following
rule [7]:

if max
where,1, 2,..,
Erasureif max
ik
k
ii r
l
k
k
ii
A
andthri kM
A
and thr



 



 


(1)
where Λi is the log-likelihood ratio calculated for the i-th
symbol, M is the dimension of the used constellation, and
thr is the threshold level determining the width of the
erasure zone in the decision space of the SDMLD. If the
number of the erased bits in the code word is zero,
the received vector feeds the outer code of the FEC de-
coder and after error correction the restored message Dr
is sent to the user. If t, the received vector is
passed to the outer code of the EEC decoder. If this
combination is identified by the EECD with only one of
the codebook C2, it is considered as the transmitted
codeword and the message Dr is sent to the user. Other-
wise, the ReQuest signal (RQ) is sent to the transmitter
via the feedback channel. Simultaneously, the message
Dr (with erased elements) is saved in the buffer of the
receiver. Upon receiving this request, the transmitter
encodes the k-th parity bits block P(D) of the inner code
C1 into the n-length codeword CW2(P(D),Q(p)) of the
code C2 where Q(p)denotes the n–k parity-check digits for
P(D).
 2
er
t

0
2
er
Let CW2(Pr(D),Qr
(p)) denotes the received vector cor-
responding to CW2(P(D),Q(p)). The SDMLD, according
to (1), erases its unreliable symbols. If the number of the
erasures in CW2(Pr(D),Qr
(p)) is equal to zero, the
received vector is passed to the FEC decoder of the outer
code. After error correction procedure, the message D is
recovered from Pr(D). by inversion and is sent to the user.
Otherwise, the received vector is passed to the EECD of
the outer code. If vector CW2(Pr(D),Qr
(p))is identified by
the EECD, the message D is recovered with the help of
inversion of Pr(D). The message Dr that is stored in the
receiver memory (after recovery of D from Pr(D)) is then
discarded. If the combination CW2(Pr(D),Qr
(p)) is not
identified by the EECD of the outer code, the received
parity block Pr(D) is integrated with Dr kept in the buffer.
The code word CW1(Dr,Pr(D)) of the code C1 with
 2
er
t

 1
er
t
erased symbols is formed and passed to EECD of the
inner code. If this combination is identified by EECD
with certain vector of the code book C1 then it is consid-
ered to be correct and the recovered message D is passed
to the user. Otherwise, the request signal is generated and
transmitted via the backward channel. Simultaneously,
the message D is discarded from the receiver buffer, and
the parity block Pr(D) (with erased symbols) is saved in
the receiver buffer. Upon receiving the second request
for the message D, the transmitter resends the code word
CW2(D,Q) and the procedure described above is repeated.
The block diagram in Figure 1 illustrates transmission
and retransmission procedures in the proposed HARQ-
EC.
3. Analysis of the HARQ-EC Performance in
Fading Channel
The main characteristics of any ARQ systems are BER
and TE, defined as




,
11 k
NC
EN k
TE EV n
BER P

 
(2)
where V is the total number of transmitted code words
and N is the number of information messages sent during
the transmission interval, E[V], E[N] denote the expecta-
tions of V and N respectively, and PNC is the probability
of an undetected word error. As follows from [2], for
selective-repeat ARQ scheme with noiseless feedback
channel, unlimited receiver buffer and maximal number
of retransmissions the values of TE and PNC may be
written as
 



 
ll
crd erd
l
erd
NC ll
erd crd
k
TEP P
n
P
P
PP


(3)
where

 l
crd
P and

 l
erd
P are probabilities of correct and
rroneous reception of the data block, respectively, at the e

L. GOLDFELD ET AL. 621
Data
Source Inner Encoding
Control
Symbol
Memorize
Outer
Encoding
Erasure of
"unreliable"
symbols Ner>0 FEC of
CW2
Recovery of
CW2
Information
Symbols
Memorize
Control
Symbol
Memorize
Outer
Encoding
Erasure of
"unreliable"
symbols Ner>0 FEC of CW2
Recovery of
CW2
Creating of
CW1 Ner>0 FEC of CW1
Recovery of
CW1
Output
Retransmission of
Data Block
Transmission of
following Data Block
Step I of
Transmission
Step I of Reception
No
CW2 (D, Q)
D
P(D)
Dr
Successful
Recovery
Yes
Yes
No
NAK 1
Step II of
Transmission
P(D) CW2
(P(D), Q)No
Yes
Successful
Recovery
Yes
No
P(D)
b
CW1
(D, P(D))No
Successful
Recovery
Yes
No
NAK2
Step II of
Reception
Step III of Reception
Figure 1. Transmission and retransmission procedures in HARQ-EC.
l-th2 start of the procedure described above (see Figure
1).
symbol time duration T is represented by



 


Re exp
mm c
s
tAgtmTj   t (4)
We analyze performance of HARQ-EC for the case of
a binary modulation. The transmitted signal within one where Am is the information symbol, g(t) is the impulse
response of the transmitter filter and wc is the carrier
frequency3. As was mentioned earlier, we consider the
case of the forward channel with additive white Gaussian
noise and flat fading with Nakagami distribution [8]
2The index l will be omitted as the statistics of errors and erases in the
received codeword do not depend on l.
3The kind of modulation and alphabet dimension M does not reduce the
generality of the analysis, as we are interested in a comparison of the
HARQ-EC performance to HARQ-II systems in the same conditions.

L. GOLDFELD ET AL. 622


2
21
2
00
2exp
m
m
ch
mm
fm2




 

 
 
 





(5)
where is the gamma function,
 m


22
0
 E, and m
≥0.5 is the fading depth parameter4. Moreover, it is as-
sumed that fading is slow, which means that μch may be
considered constant, at least for one symbol interval T.
The signal xm(t) is demodulated by SDMLD which in-
cludes a Log-Likelihood Ratio (LLR) estimator followed
by a Decision Device with Eraser (DDE) [1]. The output
of LLR is
2112 qq ,
where
 

T
imi idttstxq
0
2,1,
for coherent SDMLD and
2,1,
22  iYXq iii
for noncoherent SDMLD.
Here
 
dttstxX
T
imi 

0
,
  ,
ˆ
0


T
imidttstxY

 ts
ˆi is the
Hilbert transform of si(t), and T is the symbol duration.
The vector Λcw=[Λ1, Λ2,…Λl…, Λn] is passed to DDE
which produces a version of the outer code word. The
elements of the received vector are obtained ac-
cording decision rule (1), which in our case can be writ-
ten as
 r
C2

1if
0if
erase if 1
l
r
l l
l
thr
Athr
thr thr





 

(6 )
Using (6) and results of BER analysis for binary or-
thogonal set of signals [1] probabilities of symbol error
Pe and symbol erasure Pers are written as (see Appendix
A)


0
11 exp1
111
m
em
mthr thr
p
thr mthr 
 



m
(7)




0
0
1
11 11
11 exp1
111
m
m
ers m
m
m
mthr
p
thr mthr
mthrthr m
thr mthr










(8)
where


0
2
0
2
0NEE s
  and Es is the energy of the
transmitted signal element. With the help of (7) and (8)
we estimate performance of the considered system.
Taking into account transmission and retransmission
procedures in HARQ-EC, probabilities of correct and
erroneous decoding of the code word can be evaluated
with the help of the following expressions


 

 

    
123 11
12 11
crdrq rq
crd crdcrd
erdrq rq erd erderd
PPPPPP
PPPPPP
 
 3
(9)
where







 



 

22
22
1
22
1
22
2 12
,,
,,
FEC RC
crr rcrr r
crd
FEC RC
err rerr r
erd
rqer H
PPCWDQ PCWDQ
PPCWDQ PCWDQ
PPt d
,
,

 





 




(10)
are probabilities of correct decoding, erroneous decoding
and request of the code word CW2(Dr,Qr) respectively, at
the first stage of transmission,
 







 







2
2
2
2
2
2
2
2
2
2
,
,
,
,
FEC p
crr r crd
RC p
crr r
FEC p
err r
erd
RC p
err r
PP CWPDQ
PCWPDQ
PP CWPDQ
PCWPDQ




























(11)
are probabilities of correct decoding and erroneous de-
coding of the codeword CW2(Dr,Qr) respectively, at the
second stage of transmission,
 



 
 



 
1
1
1
1
3
1
1
3
1
1
,
,
,
,
FEC
crr r crd
RC
crr r
FEC
err r erd
RC
err r
PP CWDPD
PCWDPD
PP CWDPD
PCWDPD




















(12)
are probabilities of correct and erroneous decoding, re-
spectively, of the code word CW1(Dr,Pr(D)), which is
created from the block of data Dr extracted from the re-
ceiver memory and the parity block Pr(D) of the inner
code received at the second stage of transmission. Here


j
FEC
cr
P,


j
FEC
er
P are probabilities of correct and errone-
ous reception of code words at the output of the FEC
decoders, and


j
RC
cr
P,


j are probabilities of cor-
rect and erroneous reception of code words at the output
of the errors erasure correction decoders, for the inner
(j=1) and outer (j=2) codes respective
RC
er
P
ly.
Bearing in mind the assumptions made above on the
statistical properties of the channel, we consider that er-
rors, as well as erasures in the received stream of code
4The Nakagami pdf includes, as a special case, the Rayleigh pdf fo
r
1m, and can approximate both the Rice and log-normal pdf’s [8].

L. GOLDFELD ET AL. 623
symbols, are independent. In this case probabilities


j
FEC
cr
P,


j
FEC
er
P,


j
RC
cr
P,


j
RC
er
P and for the
outer and inner codes can be determined (Appendix B)
with the help of the following expressions:

j
rq
P

 

 




0
1
1
11,
11,
1,
1
crjj
jjj
jj
jjj
crj
jj
j
Hj
Hjjer
jjer jj
erj
erj
erj
erd j
tn
FEC nni
i
crerse e
i
i
nn
FEC nni
i
ererse e
i
it
nnni
ji
rqers ers
i
id
dnt
RC nt
crers ers
t
t
t
t
ppp
Pp pp
Ppp
Ppp









 




 
























0
0
1
1
1
1
1,
erj
erd er erd
jjj
jj
erd j
Hjjer
jjer jj
erj
erj
er er
jjerd ererd
jjj
jj
erd j
erd j
t
ttt
erserser erd
t
dnt
RC nt
erers ers
t
t
ttttt
erserser erd
t
t
pp Pcrtt
Ppp
pp Pertt









 













 








,





(13)
where






0.5 1
0
0.5 1
,1
,1
Herd
jj jer
j
jer
j
jj
jer
jjerj
jer
j
jj
Herd
jj
dt nt
nt i
i
er erdee
i
i
nt nt nt i
i
er erdee
i
idt
Pcrttpp
Perttpp
 

















(14)
nj is the length of code word, is the minimum
Hamming distance of the used code,
j
H
d
erj
t
is a number of
codeword symbols erased in the SDMLD, and j erd
t
is
the number of the erased symbols taken from those that
determine the Hamming distance of the original code


jjererd tt0. Substituting (7), (8) into (13), (14), and
the results into (9-12) and afterward into (2) and (3), we
obtain values of TE and BER. For the given inner and
outer codes they depend on the average SNR γ0 and on
the threshold thr.
4. Comparison of HARQ-EC and HARQ-II
Performance
In this Section, we compare the theoretical results ob-
tained above for HARQ-EC with those obtained by
computer simulation. Then we will compare the per-
formance of HARQ-EC to that of HARQ-II schemes [2,
4,5] for the same inner and outer codes5. The systematic
linear block code with parameters n2=15, k=11 and
is used as the outer code C2, and the half in-
vertible code with n1=22, k=11 and is used as
the inner code C1.
3
2
H
d
7
1
H
d
Figures 2, 3 show for HARQ-EC dependence of BER
and TE (for the threshold values thr =1.5, thr=2, and
thr=3), obtained as a result of analytical calculation and
computer simulation respectively. Inspection of Figures
2 and 3 demonstrates good agreement between theory
and simulation results. It follows that increase of the
threshold level thr in HARQ-EC leads to decrease of
both BER and TE.
Figure 4 shows BER for HARQ-EC (for the threshold
values thr=1.5, thr=2, and thr=3), HARQ-II and FEC
systems with the same code redundancy as a function of
the average SNR while at Figure 5 dependence of TE of
the compared systems from average SNR is presented.
Figure 2. Average BER of HARQ-EC2 as a function of the
average SNR (for the threshold values thr=1.5, thr=2, and
thr=3); analytical calculation and computer simulation.
Figure 3. Throughput efficiency of HARQ-EC2 as a func-
tion of average SNR (for the threshold values thr=1.5, thr=2,
and thr=3); analytical calculation and computer simulation.
5The kind of modulation and type of codes do not reduce the generality
of the analysis, as we are interested in comparison of the HARQ-EC
p
erformance to HAR
Q
-II s
y
stems in the same conditions.

L. GOLDFELD ET AL. 624
Figure 4. Average BER of HARQ-EC and HARQ-II sys-
tems as functions of average SNR.
Figure 5. Throughput efficiency of HARQ-EC and HARQ–
II systems as functions of average SNR.
Examination of these figures demonstrates the fact that
by choice of thr value in SDMLD of HARQ-EC, gain in
BER or TE can be achieved. For example, TE of
HARQ-EC with thr=2 exceeds TE of HARQ-II for a
roughly equal value of BER in a quite wide range of av-
erage SNR.
5. Conclusions
We propose the Hybrid ARQ system with erasure of un-
reliable symbols and retransmission of the code words
(HARQ-EC). Its performance is considered for the case
of flat Nakagami fading and AWGN in the forward
channel. The obtained theoretical results are valid for any
memoryless channel with common slow Nakagami fad-
ing, while the calculations and simulation were per-
formed for Rayleigh fading. Good agreement between
theoretical and simulation results is obtained. It has been
shown that performance of HARQ-EC may be better
than HARQ-II over a wide range of average SNR when
the same codes are used.
6. References
[1] J. C. Proakis, “Digital communications,” McGraw–Hill,
New York, 1995.
[2] S. Lin and P. S. Yu, “A hybrid ARQ scheme with parity
retransmission for error-control in satellite channels,”
IEEE Trans. Commun., Vol. COM-30, No. 7, pp. 1701–
1719, 1982.
[3] Y. Wang and S. Lin, “A modified Selective-Repeat
Type-II Hybrid ARQ system and its performance analy-
sis,” IEEE Trans. Commun., Vol. COM-31, No. 5, pp.
593–607, 1983.
[4] K. Q. Archer and J. A. Edwards, “Effect of channel fade
rate on throughput of three GHARQ-II schemes over
Rayleigh fading channel,” Electronic Letters, Vol. 31, No.
16, pp. 1320–1322, 1995.
[5] C. Sunkyun and K. Shin, “A class of adaptive hybrid
ARQ for wireless links,” IEEE Trans. Veh. Tech., Vol.
50, No. 3, pp. 777–790, 2001.
[6] S. Kallel, R. Link, and S. Bakhtiyari, “Throughput per-
formance of memory ARQ schemes,” IEEE Trans. Veh.
Tech., Vol. 48, No. 3, pp. 891–899, 1999.
[7] S. B. Wicker, “Reed–Solomon error coding for Rayleigh
fading channels with feedback,” IEEE Trans. Veh. Tech.,
Vol. 41, No. 2, pp. 124–133, 1992.
[8] L. Goldfeld, V. Lyandres, and D. Wulich, “ARQ with
erasures correction in the frequency-nonselective fading
channel,” IEICE Trans. Commun., Vol. E80–B, No. 7, pp.
1101–1103, 1997.
[9] T. Hashimoto, “Comparison of erasure-and error thresh-
old decoding schemes,” IEICE Trans. Fundamentals, Vol.
E76-A, pp. 820–827, 1993.

L. GOLDFELD ET AL. 625
Appendix A
Let us assume that symbol “1” was transmitted. The
probability of correct reception Pcr in SDMLD can be
found as the probability that Λi>thr, and the probability
of symbol erasure Pers in SDMLD, in turn, can be ob-
tained as the probability that thr
th
r
i
1. From (6) we
obtain.



,
1
cr i
ers i
ppthr
p
pt
thr







hr

(A1)
where (see [1])
1
2
1
22
;
2
i
j
S
U
U
UEeN
UN






1
(A2 )
N1 and N2 are complex-valued Gaussian random vari-
ables with zero mean and variance σ2=2μEs, U1 and U2
are mutually independent variables with distributions [1]


222
11
1
00
2
22
2
00
4
exp ,
24
pUexp.
24
S
SS
SS
UUEU
pU I
ENEN N
UU
EN EN















1
0
0
(A3)
Taking into account (A1), (A2) we obtain
 
 

 
1
1
1
12
122
00
2
12
122
0
;
cr
Uthr
ers
Uthr
Uthr
ppUthrU
1
1
p
UpUdUd
U
ppUthrU
thr
pUpUdU dU







 












U
(A4)
With the help of elementary algebra and tabulated in-
tegrals
 
 1
1exp exp
1
cr
thr thr
pthrthr thr





 






1
(A5)
 



1
exp exp
11
11
expexp
11
ers
thr thr
pthr thr thr
thr
thr
thr thr

























( A6)
where

2
0
S
E
N

  (A7)
Taking into account that
erscre ppp 


1
we have
 
 11
exp exp
11
e
thr
thr
pthr thr





 








(A8)
Probabilities (A6) and (A8) are conditional probabilities
of erasure and error reception in SDMLD respectively,
given μ and therefore Pe and Per are


0
0
,
,
er ers
ee
Pp fd
Pp fd








(A9 )
where f(μ) is defined by (5).
Appendix B
Let us determine probability of request Prq, probabilities
of correct and erroneous reception of a code word at the
output of the ECE decoder (

and ), and also
at the output of the FEC decoder ( and
 RC
cr
P
 RC
er
P
 FEC

cr
P

 FEC
er
P)
under the following conditions:
1) The code used is a linear block code (n,k) with the
minimal Hamming distance dH;
2) The encoded bit stream is represented by a code-
word CW with length n, supplied from the SDMLD out-
put to the FEC decoder, if CW does not contain the
erased symbols. Otherwise, a codeword CW feeds the
ECE decoder.
3) Errors and erasers in a sequence of code symbols
CW are independent (the channel is memoryless). For
Nakagami frequency -nonselective fading in the forward
channel, the probabilities of symbol error Pe and symbol
erasure Pers are defined by (7), (8).
First, we find probabilities

and
 FEC
cr
P

 FEC
er
P. Since
symbol errors are independent events, the binomial law
determines probabilities of correct and erroneous recep-
tion of a code word at the output of the FEC decoder.
Keeping in mind that FEC decoding is used when the
number of erased symbols in the received codeword
ter=0, we thus obtain

 

 
0
1
11
11
cr
cr
tn
nn
FEC i
crerse e
i
i
nn
nn
FEC i
erersee
i
it
Pppp
Pp pp


i
i



 



 




(B1)

L. GOLDFELD ET AL. 626
where tcr is the number of correct errors per codeword. Probabilities of correct and erroneous reception of a
code word at the output of ECE decoder depend on the
random variables ter and terd, i.e. they have to be consid-
ered as conditional probabilities P(cr/ter,terd) and
P(er/ter ,terd) written as
Probabilities rq
P
,


RC
cr
P and


RC
er
P may be ob-
tained as follows. The ECE decoding is used when ter>0,
where ter is a number of the erased symbols in the re-
ceived code word. The erasure of ter symbols from n cre-
ates a new shorter code with code word length n(sh)=n–ter
and the Hamming distance


erdH
sh
Htdd  , where terd is
the number of erased symbols taken from those ones that
determine dH of the original code (0≤terd≤ter). Since
symbol erasures are independent, the probability of the
erasure of ter symbols from n as well as probability of the
erasure of terd symbols from dH are determined by the
binomial law. Taking this into account, we obtain Prq as







0.5 1
0
0.51
,1
,1
erd er er
er erer
erd
dt nt nt i
i
er erdee
i
i
nt nt nt i
i
er erdee
i
idt
Pcrt tpp
Pert tpp
 













(B3)
Twice averaging (B3) by the binomially distributed ter
and terd, we obtain the unconditional probabilities

 RC
cr
P
and

 RC
er
P (see 13).


1
er
er
er
er H
nnnt
t
rqer Hersers
t
td
PPtdpp 


 


 (B2)