It is shown that multistage concurrent games, or relay-races, are widely used in practice. It is proposed to model relay-races in the state space, in which discrete co-ordinates are the mathematical analogue of stages, which participants pass in the current time, and basic principle of modelling of residence of participant in space states is the M -parallel semiMarkov process. With use of the proposed formalisms formulae for evaluation of stochastic and time characteristics of relay-races evolution are obtained. For arbitrary realization of switching trajectory the recurrent procedure of evolution with evaluation of stochastic and time characteristics of realization under investigation is worked out. Conception of distributed forfeit, which depends on dierence of stages of participants compete in pairs is introduced. Dependence for evaluation of total forfeit for every participant is obtained.

Concurrent games, in which participants run some distance in real physical time, are investigated. PetriMarkov models of paired and multiple competitions are formed. For paired competition formula for density function of time of waiting by winner the moment of completion of distance by loser is obtained. A concept of distributed forfeit, which amount is dened as a share of sum, which the winner gets from the loser in current moment of time is introduced. With use of concepts of distributed forfeit and waiting time the formula for common forfeit, which winner gets from loser, is obtained. The result, received for a paired competition, was spread out onto multiple concurrent games. Evaluation of common wins and loses in multiple concurrent game is presented as a recursive procedure, in which participants complete the distance one after another, and winners, who had nished the distance get forfeits from participants, who still did not nish it. The formula for evaluation of common winning in concurrent game with given composition of participants is obtained. The result is illustrated with numerical example.

Abstract. An ergodic semi-Markov process with the structure represented by the full graph with loops, which simulates a digital control algorithm that generated transactions onto an object, is investigated. Elementary simplifications for reduction of semi-Markov processes are defined. Recursive procedure for reduction of initial semi-Markov process structure till the model, which includes selected states with its links only, is proposed. Formulae for recalculation of probabilities, weighted densities and expectations of time of switching to linked states are obtained. It is shown that recursive procedure may be used also for calculation of time expectation of return the process to one of selected states that simplified the task of evaluation of time intervals between transactions in polling procedure.

It is shown that common method of units failure compensation, based on an introduction to the system a structural redundancy, leads to the increase of weight/size factor and energy consumption, but sometime does not prolongs its lifetime. The new approach to fault/recovery process modeling, based on use of fundamental apparatus of parallel semi-Markov process, in which ordinary processes simulate the life-cycle of individual units, and the complex process, assembled from ordinary processes, simulates reliability system as a whole, is proposed. Dependences for calculation of time intervals and probabilities of wandering through ordinary semi-Markov processes, with use of the recursive method are obtained. It is shown, that when there is rather complex model of unit life-cycle, semi-Markov process would be replaced with more coarse Markov process. Notions of complex semi-Markov process, such as functional states and semi-Markov matrices Cartesian product are introduced. Theoretical results obtained are confi rmed by the practical calculation of the reliability indicators of the system with passive redundancy.

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A semi-Markov process, as a model of interpretation of an algorithm by embedded controller is considered. Generation of transactions in digital controllers as a result of random wandering on states of semi-Markov process is described. Mathematical expressions for evaluation of stochastic characteristics and time lag densities between transactions are obtained. Latencies between transactions is considered as an important parameter for proper synthesis of control algorithms in real-time embedded digital control systems. Digital control system; real-time; algorithm; latency; semi-Markov process; ergodic processI.

Abstract. Specific problems arising, when Von Neumann type computer is used as feedback element, are considered. It is shown, that due to specifics of operation this element introduce pure lag into control loop, and lag time depends on complexity of algorithm of control. Method of evaluation of runtime between reading data from sensors of object under control and write out data to actuator based on the theory of semiMarkov process is proposed. Formulae for time characteristics estimation are obtained. Lag time characteristics are used for investigation of stability of linear systems. Digital PID controller is divided onto linear part, which is realized with a soft and pure lag unit, which is realized with both hardware and software. With use notions amplitude and phase margins, condition for stability of system functioning are obtained. Theoretical results are confirm with computer experiment carried out on the third-order system.

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