Tight Bounds on the Coeffcients of Consecutive kk-out-of-nn:FF Systems

In this paper we compute the coefficients of the reliability polynomial of a consecutive-$k$-out-of-$n$:$F$ system, in Bernstein basis, using the generalized Pascal coefficients. Based on well-known combinatorial properties of the generalized Pascal triangle we determine simple closed formulae for the reliability polynomial of a consecutive system for particular ranges of $k$. Moreover, for the remaining ranges of $k$ (where we were not able to determine simple closed formulae), we establish easy to calculate sharp bounds for the reliability polynomial of a consecutive system.Comment: 10 pages, 3 figures, accepted for presentation at the International Conference on Computers Communications and Control (ICCCC), May 202