Dynamic behavior of kk-out-of-nn:G systems

In this paper, we study the distribution and expected value of the number of working components at time tt in usual and weighted kk-out-of-nn:G systems under the condition that they are working at time tt. We evaluate the distribution of the corresponding conditional random variable and compute its expected value for the systems consisting of independent but nonidentical components. Illustrative examples are presented and an optimization problem which makes use of the conditional random variable is also formulated and solved numerically.