ALTERNATIVE ROUTS OF GAMES WITH RIGID SCHEDULE

E.V. Larkin1, A.N. Privalov2 
1 Tula State University, Tula, Russian Federation 
2 Tula State Lev Tolstoy Pedagogical University, Tula, Russian Federation 
E-mail: privalov.61@mail.ru. Е.В. Ларкин 
1, А.Н. Привалов 
2 
1 Тульский государственный университет, г. Тула, Российская Федерация 
2 Тульский государственный педагогический университет им. Л.Н. Толстого, г. Тула, 
Российская Федерация 
E-mail: privalov.61@mail.ru


Introduction
Relay-races, as the basic conception of corporative-concurrent system description, may be applied to modeling of such fields of human activity, as industry, economics, politics, defense, sport, etc [1][2][3][4]. Due to conception announced teams, participating in relay-race, should to pass the distance, which is divided onto stages by relay-points, and team participants should to pass the stage in real physical time. Common case of random time relay-race simulation was considered in [2,5], where for description of teams behavior such abstraction, as semi-Markov process was used. Semi-Markov process is quite universal mathematical apparatus, and when instead of random time emerges rigid schedule, it can be used too. Rigidity of stage passing time permit substantially simplify model of the system and calculation of forfeit, but also leads to substantial restrictions of results obtained.
On practice teams, participated the relay race, may vary their schedules, and for an external observer such variations are the stochastic ones. This permits to consider different combinations of schedules and to improve results obtained. Approaches to modeling of relay-races with rigid schedules and alternative routes are currently known insufficiently, that explains necessity and relevance of the investigations in this domain.

Relay-race as M-parallel semi-Markov process
The graph, which shows the alternative routes rigid schedule relay-race structure, is shown on the fig. 1.
Following assumptions, when modeling this kind of races, are made bellow [5]: in relay-race participate M-teams, every of which pass its distance in real physical time; distance of every team is divided onto stages, every of which is overcame by one participant of a team, and first participants of all teams start their stages at once; every participant may choose a route, for passing the stage, for an external observer the route selection is a random event; passing the stage route by participant lasts rigid time, which is individual for the team, the stage and the route; after completion of a current stage on the selected route next participant of the team selects route and starts the passing next stage without a lag; forfeit, which is imposed on the teams is defined as the distributed payment, value of which depends on the time and difference of stages, which currently teams pass.  The model of relay-race with M teams may be performed as M-parallel semi-Markov process [6][7][8]: where t is the physical time; m µ is the ordinary semi-Markov process, m A is the set of states; ( ) h m t is the semi-Markov matrix, which describes an activity of the m-th team; Due to rigid schedule and quasi-stochastic principle of route selection [9] where ( ) Comparison of classic competition [2] and competition with rigid schedule is shown on the fig. 2; . Formulae, which describe weighted time density and probabilities of winning the j-th stage of race by m-th team, if all participating teams start their stages simultaneously, are as follows: , when min , ..., , ..., ; is the distribution function; θ is the auxiliary argument.
Pure time density of winning the j-th stage of race by m-th team is as follows: When paired competition, formulae, which describe the time density of waiting by m-th, winner, team until n-th, loser, team, finishes the stage, are as follows: nonsense, otherwise,  It is necessary to admit, that in the case under consideration, unlike the case considered at [2], the draw effect emerges. It is caused with the infinitesimal probability of two or more teams stage passing times coincidence when time intervals are a random ones, and quite real rigid schedule case, when some time intervals are quite the same (the case is shown on the fig. 2, b, bottom line).
Due to the fact, that for all teams, participated in a race, number of the stage j at every relay point may to increment only, for external viewer sequence of switches during relays in the system as a whole has the nature of evolution [10][11][12][13], which develops from functional state, being defined with vector . Trajectory of evolution depends of routes, which every team select for passing of proper stages and schedules which develop from routes selected. Owing to random character of routes selection trajectory of evolution is the random one. Common number of routes, on which m-th team as a whole may overcome the distance is equal to where ( ) , K m j is the common number of routs of m-th team j-th stage. Common number of different variants of rigid relay-races is as follows:

Recursive procedure of relay-race evolution analysis
Let us select from all possible routes on which m-th team may overcome the distance the ( ) where arrow ⇐ indicates the direction of substitution; index means the quantity of previous switches.
win the competition among other teams, but draw the competition among themselves. Quantity of switches ( ) In such a way on the second step or recursion rigid time intervals ( ) ( ) ( ) ( ) ( ) ( ) win the competition. Quantity of switches on the r-th step of recursion is equal to cardinal number of subset ( ) In such a way on the (r+1)-th step or recursion rigid time intervals , ..., min , , .
For evaluation of common forfeit, which the m-th team receives from the n-th team, one can to use the recursive procedure, described above. At initial two-elements functional state Value of forfeit is equal as follows: where insertion indices ( ) ( )  Let us assume, that on the last step of recursion only m-th team stays in race, and time, it spend from a previous switch till finishing J)-th stage, obtained on previous stage of recursion, is ( ) Value of forfeit on the last stage is is equal as follows: