The efficiency of several interchangeable devices

The problem to be discussed is the result of a generalization of the problem from the book ‘`The course of the theory of probability" by B. V. Gnedenko. He investigated a work of the system consisting of two interchangeable devices. These devices work in the following order: the first one works until it falls out and should be repaired; then, the device is replaced by the second one which, in the own turn, also falls out and should be repaired, as well. If a repair time of the first device is less than a time operation of the second device, then the former connects to the system and switches on. The system is suggested to fall down otherwise. In this case, the full working time is assumed to equal the time operation of the both devices. If the first case holds, then one should look at the repair time of the second device and the time operation of the first one and should follow the similar scheme described above, etc. As a result, the characteristic function of the system’s working time is built. Moreover, the expected value of the working time is calculated using the function just mentioned and the methods of its augmentation are offered.
In the present work the task in question is summarized on the unlimited amount of interchangeable devices. We are interested in how, using a certain algorithm of devices' connection, to construct a characteristic function, as it is done above, so that an appropriate expectation would tend to infinity much faster than $n$. For the problem about a convergence rate of the expectation to be solved, two cases dealing with different distribution functions of the time operation and the repair time are considered, namely: 1) exponential and 2) Gaussian for both the operation time and the repair time simultaneously. It is proved that in both cases the average system's working time is equivalent to the exponential function at infinity. In order to improve the convergence rate of the expectation under consideration, in the case of exponential law, one should make the average operation time of each device as bigger than that for repair as possible. In addition, the recommendations for increasing the whole system expected working time are received.