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IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 1, 1995 MARCH
120
Survey of Reliability Studies of
Consecutive-k-out-of-n:F & Related Systems
M. T. Chao
Academia Sinica, Taipei
J. C. Fu
National Donghwa University, Hualien
M. V. Koutras
University of Athens, Athens
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The reliability of C( k,n:F) was first studied by Kontoleon
[48 - 511 but the name, consecutive-k-out-of-n:F,was first coined by Chiang & Niu [ 181. In order to give a clear picture about
C(k,n:F) to the readers, we provide the following 2 historical
examples from Chiang & Niu [18] and Chao & Lin [14].
Example 1
Key Words - Series system, Consecutive-k-out-of-n:Fsystem,
Linearly connected system, Cut, Path, Poisson convergence
Reader Aids General purpose: Widen state of art
Special math needed for explanations: Theoretical statistics
Special math needed to use results: Statistics
Results useful to: Reliability theoreticians, statisticians
Abstract - The consecutive-k-out-of-n:F & related systems
have caught the attention of many researchers since the early 1980s.
The studies of these systems lead to better understanding of the
reliability of general series systems, in computation & structure.
This manuscript is mainly a chronological survey of computing the
reliability of these systems.
1. INTRODUCTION
Today the public requires that all engineering systems, such
as atomic power plants, aircraft, automobiles and computers,
be highly reliable. Reliability evaluation is an important, integral feature of planning, design, and operation of all engineering systems. It is an undesirable fact that the reliability of a
series' system is low (especially of a large series system) and,
on the other hand, the parallel system has high reliability but
tends to be very expensive. A new system, consecutive-k-outof-n:F, and its related systems, have caught the attention of many
engineers & researchers because of their high reliability and
low cost.
Acronyms & Abbreviations
C (k,n:F) consecutive-k-out-of-n:F (system).
Let n components be linearly connected in such a way that
the system fails iff at least k consecutive components fail. This
type of structure was named consecutive-k-out-of-n:F system.
There are 2 main advantages of using C(k,n:F):
it usually has much higher reliability than the series system
it is often less expensive than the parallel system.
'The terms, series & parallel are used in their logic-diagram sense,
irrespective of the schematic-diagram o r physical-layout.
A sequence of n microwave stations transmit information
from place A to place B. The microwave stations are
spaced between places A and B. Each microwave station is able
to transmit information a distance up to k microwave stations.
This system fails iff at least k consecutive microwave stations
fail.
4
Example 2
A system for transporting oil by pipes from point A to point
B has n pump stations. Pump stations are equally spaced between points A & B. Each pump station can transport the oil
a distance of k pump stations. If one pump station is down, the
flow of oil could not be interrupted because the next station could
carry the load. However, when at least k consecutive pump sta4
tions fail, the oil flow stops and the system fails.
Since 1980 many papers were published on the reliability
of C( k,n:F) and related systems, under various assumptions.
Most early papers were published in this Transactions. Since
1990, this area has been expanded very fast and connected to
many other promising areas, eg, Poisson convergence and pattern occurrences. Thus, recent results associated with this field
are not only in reliability journals, but in many other applied
probability, operational research, and statistics journals. There
are many ways to write a survey paper, and they are always
biased. This manuscript is no exception and has its own biases.
This paper mainly chronologically reviews this fast developing
area. We do not intend to cover all published research on
C ( k , n : F )and its related systems; the evaluation of reliability
of C(k,n:F) under various models by using the Markov chain
approach is emphasized, rather than the design of a reliable
system. We apologize that some interesting articles might be
omitted from this short survey.
Notation
C (k,n:F) consecutive-k-out-of-n:F (system)
R (k,n;p) reliability of C (k,n:F)
R,( k,n;p) reliability of circular C (k,n:F)
R(m,k,n) reliability of m-C( k,n:F)
pi, qi [success, failure] probability of component i ;
p i + q i E l , pi€ (091)
implies that pi = p , qi = q
p, q
pij
transition probabilities
r0
initial probability vector
0018-95291951$4.00 01995 IEEE
.
.
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1 1
transition probability matrix
N(n,k), N,(n,k) [Fibonacci, Lucas] number of order k
Nn,k
number of non-overlapping failure runs of length k in
n s-independent Bernoulli trials
N ( n j , k ) number of binary vectors in n-space containing exactly j l's, without any of them being consecutive2
failure time; t E (0, 03).
t
A4
Other, standard notation is given in "Information for Readers
& Authors" at the rear of each issue.
Notation
U
I
M;
(1,O,...,0), l x ( k + l ) vector
(1,1, ..., l,O), l x ( k + l ) vector
implies the transpose
Mi =
pi qi 0
.... 0 0
p. 0
. 9.
0 0
p; 0 0
1 J
If all the components have the same failure probability, then
(2-4) reduces to:
Many formulae, such as (2-l), (2-3), and those in Papastavridis
& Hadjichristos [75] are special cases of (2-4).
The same Markov structure is also important in quality
inspection-systems such as continuous sampling plans Blackwell [3] and Chao [l 11.
In the early 1980s, evaluating the reliability of a C(k,n:F)
system through equations such as (2-2) & (2-3) was tedious.
Hwang [39], Derman et aZ[22], Shanthikumar [87], Lambiris
& Papastavridis [59], and Fu [31] developed recursive equations to evaluate the reliability. However, with the simplicity
.
of (2-5), these recursive equations have lost their appeal. Eq
(2-5) can be easily turned into a recursive equation. For k=2,
(2-5) yields the recursive equation:
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j=O
i
0 q;
1 0 0 0 - 0
2. RELIABILITY O F C(k,n:F) SYSTEM
Kontoleon [48-511 studied the reliability of C(k,n:F) where
all the components are operating s-independently and have the
same failure probability - i.i.d. Bernoulli trials with the same
success probability for all components. Chiang & Niu [ 181provided (2-1) and recursive (2-2):
**.*
(k+l)x(k+l)
transition matrix.
Derman, Lieberman, Ross [22] expressed R(k,n;p) in the
form:
n
R(k,n;p) =
N(n,j,k) .p"-J-q'.
1=0
A similar approach was used by Bollinger & Salvia [7]. The
N ( n,j , k ) have special combinatorial meaning. Their computation can use either recurrence relations [5, 7, 221 or explicit
sums [36, 40, 591.
An important observation for the general C ( k,n:F) is that
it can be imbedded in a Markov chain - Chao & Lin [14].
However, their Markov chain has 2k states and it can be conveniently manipulated only for small k. Fu [31] successfully
introduced a (k+ 1)-state Markov chain which simplified the
probability structure of C(k,n:F) considerably. Subsequently,
Fu & Hu [32] and Chao & Fu [12, 131 developed a simple formula for the general case of s-independent but not necessarily
identically distributed components:
Let p & q be functions o f t : p ( r ) = Pr{T 2 t}. Then,
1 - R (k,n;p( t )) is the failure time distribution of C( k,n:F).
It has been studied by Derman et al [22], Griffith & Govindarajulu [38], Shanthikumar [88], Bollinger [6], Griffith [37],
Papastavridis [65], and Iyer [47]. Papastavridis & Hadjichristos
[74] obtained E { T } & Var { T }.
For n
03, let q ( t )
[ h ( t ) / r ~ ] Papastavridis
''~.
[67],
Chao & Fu [12], and Papastavridis & Chrysaphinou [73] proved that the failure time of the system follows the Weibull
distribution:
-
n--m
lim
[R(k,n;p(t))]
-
=
exp(-h(t)).
(2-7)
An interesting question about C(k,n:F) is: Let all pi,
i = I , ... ,n be different (without loss of generality, assume pI
'Number of ways in which j identical balls can be placed in n-j + 1
distinct urns, subject to: at most k-1 balls are placed in any 1 urn.
< p2 < ... < p n - I < p"). Then what is the best arrangement
of the components so that the system has the highest possible
reliability? Derman et a1 [22] proved that for n =4 & k = 2, the
arrangement (1,4,3,2) is the solution. Even for k=2, the problem is non-trivial. Recently, several manuscripts have addressed
more general problems in this direction, eg, Wei, Hwang, SOS
[97], Malon [62], Tong [95,96], Du & Hwang [23 - 261, Hwang
~
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122
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IEEE TRANSACTIONS ON RELIABILITY, VOL. 44. NO. I , 1995 MARCH
.
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1401, Papastavridis [66], and Papastavridis & Sfakianakis [79].
The concept of optimal arrangement is rather restrictive. It exists only in a very special case when all the components are
functionally interchangeable.
Zuo & Kuo [98] summarized the results available for the
invariant optimal design of C( k,n:F) and identified invariant
optimal designs of some C ( k , n : F ) .For those systems without
invariant optimal designs, a heuristic method and a randomization method are presented to provide suboptimal designs.
If all components of a k-out-of-n system (i.i.d. case) have
increasing failure rate (IFR), then the system also is IFR. Derman et a1 [22] conjectured that C(k,n:F) is IFR. Contrary to
their belief, Hwang & Yao [45, 461 proved:
for every fixed k there exists nk such that for every n L nk,
the C(k,n:F) does not have increasing failure rate.
for fixed d there exists nd large enough such that for all n
L nd, the C(n-d,n:F) is IFR.
that components #1 & #n become adjacent (consecutive). Since
Derman et a1 [22] brought this system into focus, quite a few
results on R,(k,n;p) of the i.i.d. circular system have appeared. Derman et a1 showed that [22]:
k- I
R,(k,n;p) =
(j+
1 ) -4'-R(k,n-j-2;p)
p2.
j=O
Lambiris & Papastavridis [59] and Hwang [40] derived direct
summation formulas. In their papers, recurrence relations for
the sequence R, ( k , n ; p ), n = 1,2,. . . are provided: those formulas are especially helpful for tabulation.
From the combinatorial point of view, R , ( k , n ; p ) can be
written:
For fixed q ( t ) ,
lim [ ~ ( k , n ; p ( t ) )=j
n-m
o for all t
E (o,oo).
They did not give the rate of convergence. The situation is rather
different if q ( t ) is a function of n. If, for example, q ( t )
X ( t ) / n " k , then the failure time of C ( k , n : F ) has a Weibull
distribution (for details see section 5 ) . Hence, it preserves the
IFR property without violating the results of Hwang & Yao [45].
Chan, Chan, Lin [9], by using the structure function of
C(k,n:F)obtained the algorithm for finding all the minimal path
sets and cut sets of the system, and the system reliability.
All the results in this section assume that the components
fail s-independently . For the Markov dependency, the reliability
& bounds of C ( k , n : F )have been studied by Fu [31], Fu & Hu
[32], Papastavridis & Lambiris [78], and Chao & Fu [131. With
a slight modification of the transition matrix Mi,eq (2-4) can
capture the reliability of the Markov dependent system. Linear
dependency of components in C( k,n:F) was studied by Boland,
Proschan, Tong [4]. s-Dependency of components in C(k,n:F)
was studied by Lau [60].
The aspect of economical design of large C ( k , n : F ) has
been studied mainly by Chao & Lin [ 141 and Chang & Hwang
[lo]. They obtained, for fixed costs of the components and
budget, the highest reliability C(k,n:F).
Let N be the number of components for the first consecutive
kcomponentsfail ( N = k , k + 1,...,n ) . Ther.v. Ncanbeviewed as the time for the first consecutive k components to fail.
The distribution of the random variable N has been obtained
via the moment generating function, f(s) = E(sN} by Chen
& Hwang [16], Chrysaphinou, Papastavridis, Sypsas [21], and
Hwang & Shi [44].
-
3. RELATED SYSTEMS
Many systems are closely related to C ( k , n : F ) . The
simplest variation of the C(k,n:F) system is the circular
C(k,n:F) wherein the n components are placed on a circle so
is the cyclic counterpart of the N ( n , j , k ) ; see Hwang & Yao
[46], Hwang & Papastavridis [43], and Koutras & Papastavridis
[56]. In the combinatorial literature,
n
N,(n,k)=
N,(n,j,k),
j=O
are referred to as Fibonacci and Lucas numbers of order k
respectively; see Philippou [80], Philippou & Makri [81], and
Charalambides [ 151.
All the results so far in this section refer to the i.i.d. case.
For circular C( k,n:F) consisting of s-independent but not
identically-distributed components, the exact reliability can be
obtained by the recursive methods of Hwang [39] and Antonopoulou & Papastavridis [ 11. For the more general case of
not s-independent and not identically-distributed components,
reliability evaluation techniques have been proposed by Kossow
& Preuss [52], Papastavridis [69], and Sfakianakis &
Papastavridis [94].
In the microwave stations and oil transportation examples
(see the Introduction), a major restriction in the models was
that all stations (pumps) have the same transmission (transporting) capability. However, in the real world, it is quite common to have telecommunication systems where some stations
can communicate to the adjacent relays only, while others have
a higher power and are able to transmit information to more
than one adjacent relay.
The mathematical modeling of these networks is implemented
by the consecutively-connected systems which were first introduced by Shanthikumar [89]. These systems consist of a source (0),
a sink ( n 1) ,and n components { 1,2,. ..,n} . The source is directly connected by arcs to components { 1,2,. ..,b},
and component
j (1 s j s n ) to (i+1, j + 2 ,..., j + k j ) .
+
zy
D
~
Notation
kj
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CHAO ET AL: SURVEY OF RELIABILITY STUDIES OF CONSECUTIVE-K-OUT-OF-N:F SYSTEMS
+ kj 5 n + 1.
123
Its reliability has been studied by Tong [95] and Kuo, Zhang,
Zuo [58].
The strict C(k,n:F) was first proposed by Bollinger [6].
Its reliability has been studied by Papastavridis [64], Philippou
& Makri [81], Rushdi [83], and Kossow & Preuss [52].
However, the concept of this system is dubious, and probably
is unrealistic. Several articles have debated this subject, eg,
Papastavridis [7 11, Hwang [42], and Rushdi [58].
Salvia & Lasher [86] introduced the 2-dimensional
C(k,n:F) which reasonably extends the C ( k , n : F )structure to
the plane (instead of working on a line). This system consists
of n x n components placed on a grid; it fails iff there exists a
square subgrid of size at least k x k with all its components
failed. There is an error in their system-reliability algorithm.
Using the Stein-Chen method, Koutras, Papadopoulos,
Papastavridis [57] studied the bounds and limit theorems for
the reliability of 2-dimensional C ( k,n:F) .
Another interesting variation, consecutive k-out-of-m-fromn:F system, applies in radar detection problems (sliding window detection probabilities) and quality control (sampling acceptance procedures). It consists of n components arranged in
a line (linear system) or on a cycle (circular system) and fails
iff there are m consecutive components which include among
them at least k failed components. For m = k the system
becomes C(k,n:F) while form = n it reduces to the k-out-ofn:F system. The consecutive-k-out-of-m-from-n:F system was
first mentioned by Tong [95]. Since then, many publications
have appeared on the reliability evaluation and optimal arrangement of this structure, and its connection to certain combinatorial
problem; see Kounias & Sfakianakis [53, 541, Koutras &
Papastavridis [55], Papastavridis & Sfakianakis [79],
Sfakianakis [91, 921, and Sfakianakis, Kounias, Hillaris [93].
A Weibull limit theorem for large (n w) consecutive-k-outof-m-from-n systems was given by Papastavridis [67] and
Papastavridis & Koutras [77].
zyxwvutsrq
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positive integers such that j
The source, sink, and arcs are perfect while the n components
{ 1,2,...,n} are failure prone. The system is functioning iff there
is a connection from source to sink through working components. In particular, the special case kj = min(k,n - j 1),
0 rjIn is the C(k,n:F).
Shanthikumar [89, 901 developed reliability bounds and
a recursive algorithm for the exact reliability evaluation of a
more general consecutively connected system (system with consecutive minimal cutsets). Hwang & Yao [45] extended the
Shanthikumar algorithm [89, 901 to circular consecutively connected systems. Koutras & Papastavridis [55] studied the limit
behavior of system reliability as n- m.
An engineering system containing n components is linearly
connected if it can be imbedded into a finite Markov chain {X,:
t E rn = (1,2, ...,n)} defined on a finite state space Cl =
(0,1,... , k } with ( k 1) x ( k 1) transition matrices:
+
+
+
(3-1)
l
o
....
1
1
State k is absorbing (the system breaks down and cannot be used
anymore).
Since {X,} is a Markov chain and state k is absorbing,
R ( k , n ) = P r { X l s k - l ,...,X n s k-1)
=
(3-2)
Pr {X35 k - 1 ( X 2Ik - 1,XIIk - 1} .. .
-
zyxwvuts
Pr{X,%k-l).Pr{X2~k-1(Xl~k-l)
Pr{X,sk- 1 lXn.-l~ k 1 -, X n - 2 s k - 1,...,XI ~ k 1)
-
4. UPPER & LOWER BOUNDS
For fixed n, eq (3-2) yields a simple formula for the reliability
of a linearly connected system:
(3-3)
This general system was introduced in papers by Fu [31] and
Chao & Fu [12, 131. The structure of the linearly connected
system is so general that it covers almost all important systems:
series, standby, k-out-of-n:F, C ( k , n : F ) , deteriorating,
deteriorating & repairable, and m-consecutive-C(k,n:F) . Their
reliabilities can be evaluated by the above simple formula (3-3).
There is a dual to C (k,n:F) : consecutive-k-out-of-n:G.This
system is good iff at least k consecutive components are good.
Several upper & lower bounds have been proposed for approximating the C(k,n:F) reliability, eg, Derman et a1 [22],
Fu [29, 301, Papastavridis [63], and Papastavridis & Koutras
[76]. The Fu method is probabilistic while the Papastavridis
method is based on analyzing the roots of the denominator of
a generating function. For the simple i.i.d. case, the lower &
upper bounds for C (k,n:F) reliability are given by Fu [29, 3 11
and Papastavridis & Koutras [77]:
-
, C(k,n:F) reliability can
If q is small ( q X / ~ Z ” ~ )the
be approximated by exp(-Xk). Using the Stein-Chen method,
Chrysaphinou & Papastavridis [ 191, proved:
( R ( k , n ; p ) - exp(-X,)J
I
(2k-l)-qk
+
2(k-l).q,
(4-2)
124
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IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 1, 1995 MARCH
a result from which, upper & lower bounds for R ( k , n ; p ) can
be easily obtained.
Barbour, Holst, Janson [2], also using the Stein-Chen
method (along with a certain coupling), gave the improved
inequality:
The Stein-Chen method has become popular for studying
Poisson convergence for sequences of s-dependent r.v. It
shocked us that, as the following numerical results indicate, the
bounds (4-2) & (4-3) generated by the Stein-Chen method performed poorly, especially for small n.
Chrysaphinou & Papastavridis [20] and Papastavridis & Koutras
[76] used (4-4) to extend the result into the case of a circular
system. They also showed that certain limit theorems were valid
for the linear non-maintained & maintained C (k,n:F) systems.
5 . LARGE C(k,n:F) SYSTEMS
AND POISSON CONVERGENCE
Chao & Lin [14] were the first to observe that in the i.i.d.
case: if p - h / n l / k then
,
the C(k,n:F) reliability
exp(-hk)
as n
ca. They proved that result for k I4 and conjectured
that it is also true for k > 4 . Their proof was rather tedious.
Fu [29, 301 gave a direct, simple proof for the conjecture, based
on a large deviation type inequality:
-
-
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Notation
L, U
[lower, upper] bound of (4-1)
Lcp, U,, [lower, upper] bound for (4-2)
LE, U, [lower, upper] bound for (4-3).
Notation (i.i.d. case)
ai,n
bi,n
10
10
10
10
2
2
4
4
0.05
0.20
0.10
0.20
0.8703
0.1777
0.3986
-0.2223
0.9669
0.5818
0.9986
0.9792
0.9777
0.6925
0.9993
0.9889
0.9788
0.7462
0.9994
0.991 1
0.9909
0.9180
1.0002
1.0029
1.0853
1.2177
1.6000
2.2001
50
50
50
50
2
2
4
4
0.05
0.10
0.05
0.10
0.1772
0.3826
0.6997
0.3946
0.8781
0.5674
0.9997
0.9950
0.8846
0.61 11
0.9997
0.9953
0.8900
0.6421
0.9997
0.9958
0.9021
0.6894
0.9998
0.9966
0.9922
0.8426
1.2998
1.5960
100
100
2
2
0.05
0.10
0.6733
0.1416
0.7785
0.3642
0.7805
0.3697
0.7903
0.4086
0.8025
0.4562
0.8883
0.6016
qk
q k *( l + q ) .
These are the lower & upper bounds in (4-1).
-
Take q
X / n 1 l k ;then q n
limiting expression:
- bi,n -
( A k + o ( l))/n. The
lim [R(k,n:F)] = exp(-Ak),
(5-2)
n-m
is an immediate consequence of (5-1).Further, if A=A(t), then
the failure time of a large C ( k , n : F )has a Weibull distribution.
This result was first presented by Papastavridis [67].
For the non identically-distributed case, if qi h i / n l l k ,
i = 1,... ,n using the inequality (5-l), the limiting reliability of
the system is:
-
At present, we don't know exactly why the upper & lower
bounds from the Stein-Chen method perform so poorly. We
speculate that the Stein-Chen method uses only the first two
moments of the process, but ignores the Markov structure of
the reliability system.
For the reliability approximation of a circular C (k,n:F),
Derman et a1 [22] suggested an upper bound of the form:
zyxwvutsrqpo
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lim [ R ( k , n ) ] = exp(-A'),
n-m
A' = n-m
lim
(5-3)
[ i d].
i= 1
R,(k,n;p)
I1
- A/B.
Papastavridis [64],
working with generating function techniques,
proved that
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Another interesting inequality, valid for the non-i.i.d. case as
well, is:
~ ( k , n)
For the i.i.d. case, (5-3) was also proved by Fu [30] using
(5-1) and by Papastavridis [63] using a generating function
technique. Godbole [35] and Koutras & Papastavridis [55]also
obtained bounds and limiting results via the Stein-Chen method.
The m-C(k,n:F) fails iff there are m non-overlapping sequences of k failed consecutive components3. The reliability of
an m-C(k,n:F) system is:
i
(5-4)
qj.....qn.....qk+n+j-l IR , ( k , n )
j=n-k+2
5 R(k,n).
R ( m , k , n ) = Pr{Nn,k Im - l } .
(4-4)
31n the sense of the Feller [28] counting, eg, I FFF 1 FSS I FFF I FFF I
SFF has 3 non-overlapping consecutive triples of failed components.
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zyxwvutsrqpon
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CHAO ET A L SURVEY OF RELIABILITY STUDIES OF CONSECUTIVE-K-OUT-OF-N:F
SYSTEMS
Notation
Nn,k
number of non-overlapping failure runs of length k in
n s-independent Bernoulli trials.
This system is linearly connected as well. Hence, for fixed n
its reliability can also be evaluated via (3-3).For large n, roughly
speaking, if the failure probabilities of components are small,
then the r.v. Nn,kconverges to a Poisson r.v. Mathematically,
it can be stated as:
If q = X/n”k (i.i.d. case) then:
lim [ R ( m , k , n ) ]= lim [Pr{Nn,k 5 m - I } ]
n-m
n-m
(5-5)
There are several approaches to establish ( 3 - 3 , eg,
Papastavridis & Chrysaphinou 1721, Papastavridis 1701, Fu [33],
and Godbole 1351.
ACKNOWLEDGMENT
We are pleased to thank the Associate Editor and the
referees for their useful comments and constructive suggestions.
125
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AUTHORS
Dr. M.T. Chao; Inst. of Statistical Science: Academia Sinica: Taipei 115
TAIWAN - R.O.C.
Internet (e-mail): mtchao@stat. sinica.edu,tw
Min-Te Chao was born in 1938 and received a BS (1961) in Mathematics
from National Taiwan University, an MA (1965) and a PhD (1967) in Statistics
from University of California at Berkeley. He is a member & Fellow of the
Institute of Mathematical Statistics, and a member of the International Statistical
Institute. He is an associate editor of 5 journals: Statistica Sinica, J. Chinese
Statistical Assoc, Statistics & Probability Letters, J . Statistical Planning & In-
INVITATION To MEMBERSHIP
127
ference, and Qualiry Engineering. He worked as a member of technical staff
for Bell Labs (1968-78), and as a district manager for AT&T (1978-82).
Dr. James C. Fu; Inst. of Applied Mathematics; National Donghwa Univ;
Hualien, TAIWAN R.O.C.
Dr. James C. Fu was born in Nanking, P.R. China, in 1937. He received
his BS (1960) in Mathematics from the National Cheng-Kung University, the
MS (1968) in Biostatistics from Cornell University, and the PhD (1971) from
Johns Hopkins University. He is a member & Fellow of the Inst. of Mathematical
Statistics, and is a Professor and Director of the Inst. of Applied Mathematics,
National Donghwa University.
Dr. Markos V. Koutras; Dept. of Mathematics and Statistics: Univ. of Athens:
Athens, 1.5784, GREECE.
Internet (e-mail): mkoutras@atlas.uoa.ariadne-t.gr
Markos V. Koutras was born in Arta. Greece in 1957. He received his
MSc (1981) in Computer Science and Operations Research and PhD (1983)
in Statistics from the University of Athens. He is an Assoc. Professor in the
Dept. of Mathematics (Section of Statistics and Operations Research) in the
University of Athens.
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INVI7’ATION To MEMBERSHIP
Manuscript received 1994 March 20
IEEE Log Number 94-08109
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