Nonlinear Analysis, Theog. Methods & Applicarionr, Vol. 30. No. 6, pp. 3345-3348, 1997 Proc. 2nd World Congress of Nonlinear Aml~m 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII:SO362-546X(97)00254-X 0 ~---- -- 0362-546X/97 $17.00 + 0.00 A RELIABILITY BOUND FOR 2-DIMENSIONAL CONSECUTIVE k-OUT-OF-n : F SYSTEMS M. V. KOUTRAS,] G. K. PAPADOPGULOSt and S. G. PAPASTAVRIDIS] IDepartment of Mathematics. University of Athcns. P:mepiatemiopoIis. 15710 Athens. GREECE IGreek Pedagogical Institute, 396 Mcsogeion St., Agia Pnraskevi, 15341 Athens. GREECE Key words andphrases 2-Dimcnsionnl ~ons~c~ti~e k-out-of-n:F system. rcliahility hound. Poisson approximatioq Compound Poisson approximation, ChewStein mcth<aJ. minimal cut wt. 1. INTRODUCTION The 2-dimensional consecutive-k-out-of-,?:F system was introduced by Salvia & Lasher [I] by generalizing the notion of the consecutive-k -out-of-11 :F system [53]. It consists of a square grid of size n (containing )I2 components) and fails if and only if there is at least one square grid of size k (1 < k I W) whose components arc failed. This system has rcccntly rcccivcd extensive research interest, a fact mainly due to its applicability in various areas such as safety monitoring systems, design of electronic devices, discasc diagnosis and pattern detection. For a coherent system whose minimal cut sets have been specified, the classical lower bound is the one obtained by Esary & Proschan [4] (see also Barlow & Proshan 151). In a 2-dimensional consecutive-k -out- of-n:F system the minimal cut sets consist of k2 components placed on a rectangular kxk grid. Therefore, denoting by p, (c/ii = l-p,,) the survival (failure) probability of system’s components we deduce the following lower bound ,1-1-l,, P-l tic- 1,+&l ~,~,,P = n n (1 - n n$,J ,:I ,I 1’ -I L”, Salvia & Lasher [l] suggcstcd an alternative lower bound (obtained by employing a “binomial type” argument) whereas Koutras, Papadopoulos & Papastavridis [h] used the celcbratcd Chcn-Stein method to approximate a 2-dimensional consecutive-k-out-of-n :F system’s reliability by an exponential type lower bound. Finally, Barbour, Chryssaphinou & Roos [7] gave a compound Poisson local Chen-Stein lower bound for the same quantity. Extensive numerical experimentation revealed that the classical Esary & Proschan [4] lower bound is in general performing quite well and in view of its simplicity is certainly preferable over all other bounds. The first upper bound for a 2-dimensional conxccutivc-~-out-of-n :F system’s reliability was the one given by Salvia & Lasher [l]. As pointed out by Ksir ]X] later, this bound was incorrect and although the error for high reliability systems is negligible, it can only serve as an approximation and not as an upper 3345