We consider smooth mappings acting from one Banach space to another and depending on a parameter belonging to a topological space. Under various regularity assumptions, sufficient conditions for the existence of global and semilocal continuous inverse and implicit functions are obtained. We consider applications of these results to the problem of continuous extension of implicit functions and to the problem of coincidence points of smooth and continuous compact mappings.

We consider the problem of optimizing the operation of a queuing system in which the number of working service channels can be changed in a controlled manner at control times separated from each other by a fixed time step. It is assumed that, when passing from step to step, the simplest arrival flow intensity changes in accordance with some homogeneous Markov chain. The criterion for choosing a strategy for switching service channels is the minimum total average costs over a multistep planning horizon. The parametric structure of the optimal strategy for switching service channels is revealed.

A new model of inventory control with returns is considered, when it is possible for consumers to return (under certain conditions) the products they have pur-chased. It proved to be that the optimal inventory control strategy in such a sys-tem turns out to be four-level.

The article presents a novel approach to solving complex problems of queuing theory in general and to the analysis of closed queuing networks, in particular. The main idea is to combine a simulation on a limited set of input parameters with data mining methods. As an example of applying this approach, the paper evaluates the main performance parameters of a closed exponential queuing network, for which, as is known, methods of analytical and numerical analysis have been developed with some restrictions on its structure. The specified network was chosen in order to qualitatively test the proposed approach, as well as to demonstrate its effectiveness on a very large amount of input data, which is an essential distinguishing feature of this work. The maximum approximation error of the estimates of the mean sojourn times of customers obtained in the study, as well as their average number in the network nodes, does not exceed 5%. The new approach is universal in nature, and its field of application is not limited to studies of only exponential networks, therefore, it can be used to study more complex queuing networks.

In this paper we consider an unreliable retrial queueing system consisting of the main unreliable server, the reliable backup server and the orbit of infinite size. Customers and breakdowns arrive to the main server according to Markovian arrival processes (MAPs). If the main server is busy at a customer arrival moment, then the customer goes to the orbit. If at the moment of a primary or an orbital customer arrival the main server is under repair and the backup server is idle, then the customer goes to the backup server. If during the service by the backup server the main server restores, the customer goes to this server and is serviced as a new. If at the time of the arrival of a primary or an orbital customer the main server is repaired and the backup one is busy, then the customer goes to orbit. If a breakdown arrives at the busy main server, then the under-served customer is transferred to the backup server and is serviced as a new. Service times and repair times have phase type (PH) distributions. The system under consideration can be used when modeling the hybrid communication system presented by the main optic channel and the reserve radio channel. We investigate the stable system operation, calculate the stationary distribution and a number of performance measures. We also present illustrative numerical examples to show the behavior of the performance measures of the system.