We consider the multipath communication between a client and a server that is established at the transport and session layers of an SDN/NFV protocol stack in a fog computing scenario. We analyze the reliability function of an associated redundant transport system comprising two logical channels that are susceptible to failures. The failure processes of both channels are described by Markov-modulated failure times that are driven by the transitions of a common Markovian environment. We model the error-prone system with repair and independent phase-type distributed repair times by a continuous-time Markov chain. We identify its generator matrix in terms of Kronecker products of the underlying parameter matrices that are determined by the interarrival times driven by Markov-modulated failure processes and the independent phase-type distributed repair times. We show that the steady-state distribution of the restoration model can be eectively calculated by an iterative aggregation-disaggregation method for block matrices and compute the associated reliability function of the transport system by a uniformization method.

In this paper we consider an optimal control problem for the MAP(t)/M/2 queueing system with heterogeneous servers is introduced. The Markov arrival process (MAP) has time-dependent and periodic rates for phase transitions.We built a continuous time finite-horizon Markov decision process (MDP) with the aim to minimize a cost function. We solve a Bellman equation as a system of ordinary differential equations with time-dependent coefficients. We show that the optimal policy is of threshold type with threshold levels depending on the phases of arrival process. Moreover, the periodic variation of arrival attributes makes a threshold control policy piecewise constant time-dependent and periodic. We study numerically the speed of convergence of the policy to a periodic pattern. For the fixed control policy we calculate a transient solution. and provide a sensitivity analysis to determine how sensitive the performance measures are to changes in parameter values and in inter-arrival time correlation.

The paper describes approach to reliability analysis of systems with cyclic mode of operation, in which recovery actions after failures become possible only in offline mode. Theoretical basis of this investigation is continuous-time discrete state Markov processes. Complex reliability measure which is both cumulative and instantaneous is created for investigation of operative availability of such systems. Step by step schema for calculating operative availability index is suggested. Results of analysis of main redundant configurations of these systems with imperfect fault coverage are presented.

Рассмотрена математическая модель восстанавливаемой системы с теплым резервом, в случае, когда все рабочие и ремонтные времена ограничены экспоненциальной случайной величиной (сверху и снизу), а рабочие и ремонтные времена могут быть зависимыми. Показано, что оценка сверху границы скорости сходимости распределения этой системы экспоненциальны. Оценены границы скорости сходимости коэффициента готовности.

В этой статье исследованы отображения, действующие в частично упорядоченных пространствах. Для этих отображений введено условие, аналогичное условию Каристи, используемое для функций, определенных в метрических пространствах. Доказано утверждение о достижении минимальной точки отображением частично упорядоченных пространств. Показано, что из полученного предложения следует известный результат о существовании минимума полунепрерывной снизу функции, определенной на полном метрическом пространстве. Получены новые результаты о точках совпадений отображений частично упорядоченных пространств.