The life time of a random binary sequence (coherent system)

We introduce and investigate a probabilistic model of a coherent system which consists of $m$ elements and is intended for fulfilment of homogeneous tasks. On the first stage, one task out of the total number $n$ of tasks enters each element of the system. The result of work of an element is either a fulfilment of the task or a failure of the element. In the case of the failure of an element, it is excluded from the system, and the task which is not fulfilled returns to the queue of those waiting for fulfilment. On the second stage, $m_1$ tasks are sent to the system, where $m_1$ is the number of elements of the system remained operable after the first stage, and so on. In the paper, a detailed analysis of the suggested model is realised in the case where each element of the system fulfils tasks with the same probability independently of the rest of the elements.