# Article

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Keywords:
residual lifetime; inactivity time; stochastic order; dependence; reliability
Summary:
In this paper, we consider the linear and circular consecutive $k$-out-of-$n$ systems consisting of arbitrarily dependent components. Under the condition that at least $n-r+1$ components ($r\leq n$) of the system are working at time $t$, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive $k$-out-of-$n$ systems. In the following, we investigate the inactivity time of the component with lifetime $T_{r:n}$ at the system level for the consecutive $k$-out-of-$n$ systems under the condition that the system is not working at time $t>0$, and obtain some stochastic properties of this conditional random variable.
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