Relay races along a pair of selectable routes

Case of two teams competition, which should overcome the distance divided onto stages, is considered. In the case under consideration, every stage has its own number of routes, which the participants of the team may select to overcome. It is shown, that competition bears the character of the relay race, and two-parallel semi-Markov process is the natural approach to modelling of the situation. From all possible routes two were selected. The conception of switching space, which display all possible switching trajectories is proposed. The formula for calculation of switching trajectories number is acquired. It is shown, that ordinary semi-Markov process with the use of the recursive procedure may be obtained from the complex two-parallel semi-Markov process, which describes the wandering through selected routes. The formulae for realization of the recursion are proposed. Conception of distributed forfeit is proposed. It is shown, that forfeit depends on difference of stages, teams overcome at current time, and routes, on which participants solved to overcome stage. The formula for estimation of total forfeit, which one team pays to other team is obtained. It is shown, that the sum of forfeit may be used as the optimization criterion in the game strategy optimization task.