Example 6.28. A radio system, having 1000 elements (with a failure rate $\lambda_{i}=10^{-6}$ failures/hour), has passed testing and has been accepted by the customer. It is required to determine the probability of failure-free operation of the system in the interval $t_{1}<\left(t=t_{1}+\Delta t\right)<t_{2}$, where $\Delta t=1000$ hours.
Solution. The failure intensity of a system consisting of $n=1000$ non-redundant elements, each with $\lambda_{i}=10^{-6}$ failures/hour, is $\lambda_{s}=n \cdot \lambda_{i}=10^{-3}$ failures/hour. Given these initial conditions, the probability of system failure over a time $t=1000$ hours is:

$$
P(X>t)=e^{-\lambda t}=e^{-10^{-3} \cdot 1000}=e^{-1}=0.37
$$