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M. T. Chao. J. C. Fu. M. V. Koutras. The reliability of C( k,n:F) was first studied by Kontoleon. [48 - 511 ... examples from Chiang & Niu [18] and Chao & Lin [14].

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

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

.

.

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 =

1

pi qi 0

.... 0 0

p. 0 . 9.

0 0

p; 0 0

j=O

i Derman, Lieberman, Ross [22] expressed R(k,n;p) in the form:

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):

**.*

1

(k+l)x(k+l) transition matrix.

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:

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

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 ) .

+

D

~

123

CHAO ET AL: SURVEY OF RELIABILITY STUDIES OF CONSECUTIVE-K-OUT-OF-N:F SYSTEMS

Notation

kj

positive integers such that j

+ kj 5 n + 1.

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{X,%k-l).Pr{X2~k-1(Xl~k-l)

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].

-

Pr {X35 k - 1 ( X 2Ik - 1,XIIk - 1} .. . 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)

IEEE TRANSACTIONS ON RELIABILITY, VOL. 44, NO. 1, 1995 MARCH

124

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. , the C(k,n:F) reliability exp(-hk) case: if p - h / n l / k then ca. They proved that result for k I4 and conjectured as n 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:

-

-

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:

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

Another interesting inequality, valid for the non-i.i.d. case as well, is:

i

~ ( k , n)

R ( m , k , n ) = Pr{Nn,k Im - l } .

(5-4)

qj.....qn.....qk+n+j-l IR , ( k , n )

j=n-k+2

5 R(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:

(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.

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.

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CHAO E r A L SURVEY OF RELIABILITY STUDIES OF CONSECUTIVE-K-OUT-OF-N:F SYSTEMS

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AUTHORS Dr. M.T. Chao; Inst. of Statistical Science: Academia Sinica: Taipei 115 TAIWAN - R.O.C. Internet (e-mail): [email protected] 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-

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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): [email protected] 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. Manuscript received 1994 March 20 IEEE Log Number 94-08109

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