TCSET'2010, February 23-27, 2010, Lviv-Slavske, Ukraine 107 Estimation of Reliability Indices for Symmetric Ramified Systems Andriy Sydor Abstract - Main reliability indices for unrestorable symmetric systems ramified to level 3 and with ageing output elements are examined in this paper. Models for the failure probability, the failure frequency and the failure rate are worked out in the case when the lifetime of ageing output elements is circumscribed by the Weibull distribution. Keywords – ramified systems, symmetric systems, reliability indices, ageing elements. I. INTRODUCTION Reliability theory is a subset of quality control; in it the characteristic studied is the length of life of the item. Reliability deals with products in service [1]. On output level ramified systems have sensors, printers, keyboards, disk drivers, which are exposed to aging. Lifetime of such devices is often circumscribed by the Weibull distribution. Elements of upper levels have lifetime circumscribed by the exponential distribution [2]. II. ESTIMATION OF RELIABILITY INDICES Let us consider a symmetric complicated system, ramified to level 3, with Weibull distributed output elements , where a 1 elements of level 1 are subordinate to the element of level 0, a 2 elements of level 2 are subordinate to every element of level 1, a 3 elements of level 3 are subordinate to every element of level 2. It was proved that lifetime of elements on upper levels (levels 0, 1 and 2) is circumscribed by the exponential distribution and lifetime of elements on the output level (level 3) is circumscribed by the Weibull or the Rayleigh distribution [4]. Main traditional reliability indices of unrestorable systems in case of ramified systems are the failure probability, the failure frequency and the failure rate in the prescribed availability condition [5]. By analogy with [6] we obtain the following generation function of the system: 0 a 1 a 2 a 3 3 2 1 0 3 q ) q ) q ) q z p ( p ( p ( p ) z ( S 1 2 3      (1) Let O 3W (k,t) be the failure rate of the system in the prescribed availability condition which is determined as a result of division of the failure frequency by the availability function. We obtain: u  ｦ ｦ ｦ ｸ ｸ ｹ ｷ ｨ ｨ ｩ ｧ   ｸ ｸ ｸ ｸ ｸ ｸ ｹ ｷ ｨ ｨ ｨ ｨ ｨ ｨ ｩ ｧ ｸ ｸ ｹ ｷ ｨ ｨ ｩ ｧ  1 2 3 3 2 3 2 3 2 1 2 3 2 1 3 1 2 3 3 1 1 1 ( ) , ( 3 x a a x ceil x x x a x x a a a a k x a a a x ceil ceil x x a W C C C t k O u    u ｦ ｦ ｦ          3 2 3 3 3 3 1 2 3 2 1 2 2 2 2 2 1 2 1 1 1 1 1 1 1 0 0 0 ) 1 ( ) 1 ( ) 1 ( x x a j j j x x a x x a j j j x x a x a j j j x a C C C  u       u  ) ( ) ( ) ( 2 2 2 1 1 1 0 1 3 3 3 3 3 j x j x t j x O O O E O E   3 3 3 3 2 2 2 1 1 1 ) ( ) ( ) ( E O O O t j x j x j x e e       u )/ (2)   u  ｦ ｦ ｦ ｸ ｸ ｹ ｷ ｨ ｨ ｩ ｧ   ｸ ｸ ｸ ｸ ｸ ｸ ｹ ｷ ｨ ｨ ｨ ｨ ｨ ｨ ｩ ｧ ｸ ｸ ｹ ｷ ｨ ｨ ｩ ｧ     1 2 3 3 2 2 1 2 3 2 1 3 1 2 3 3 1 1 1 1 1 1 1 1 1 /( x a a x ceil x x x a a a a k x a a a x ceil ceil x x a t t x x a C e e C O O   3 2 3 3 3 3 3 3 3 2 3 2 1 2 2 2 2 1 1 x x a t t x x x a x x a t t x e e C e e       ｸ ｹ ｷ ｨ ｩ ｧ   u E E O O O O ). III. CONCLUSIONS The paper deals with mathematical models of main indices for unrestorable ramified systems with ageing output elements. The main thrust of this paper is to reduce the computational time and complexity when evaluating reliability indices of complicated symmetric systems. REFERENCES [1] D. L. Grosh. A Primer of Reliability Theory. N.Y.: John Wiley & Sons, 1989. [2] M. Z. Raqab. “Optimal prediction-intervals for the exponential distribution, based on generalized order statistics”, IEEE Transactions on Reliability, vol. 50, pp. 112-115, 2001. [3] J.-W. Wu, T.-R. Tsai, L.-Y. Ouyang. “Limited failure- censored life test for Weibull distribution”, IEEE Transactions on Reliability, vol. 50, ʋ 1, pp. 107-111, 2001. [4] A. Sydor . “A choice of lifetime distribution functions for elements of hierarchical ramified systems”, Proceedings of the VII th International Conference CADSM 2003 “The Experience of Designing and Application of CAD Systems in Microelectronics”. Lviv – Slavske, 2003, pp. 262-263. [5] D. Marunchak, A. Sydor. “Reliability indices of ramified systems”, Proceedings of the VI-th International Conference “The Experience of Designing and Application of CAD Systems in Microelectronics” (CADSM2001). Lviv – Slavsko, 2001, pp. 130-131. [6] A. Sydor. “Evaluation of reliability indices for ramified computing systems”, Proceedings of the Second IEEE International Workshop on Intelligent Data Acquisition and Advanced Computing Systems. Lviv, 2003, pp. 80-83. [7] D. Marunchak, A. Sydor. “Estimation of reliability for hierarchical symmetric systems with Weibull distributed output elements”, Proceedings of International Conference on Modern Problems of Telecommunications, Computer Science and Engineers Training. Lviv – Slavsko, 2000, pp. 1-2. Andriy Sydor - Lviv Polytechnic National University, 12 S.Bandery Str., Lviv, 79013, UKRAINE, E-mail: sydor@polynet.lviv.ua