Test method of distribution functions of the semi-markovian system dwell times in states

The article presents and proves the theorem on the stay times in the trajectory states taking the repeated returns into account. A method for determining the distribution functions of the stay times of the semi-Markov system in the states, called the method of trajectories, is introduced. An example of semiMarkovian modeling systems proving the correctness of the proposed method is examined.


Introduction
In many cases the hierarchical approach is employed for modelling of the complex semi-Markovian (SM) systems for their model development.In this case it is necessary to abut separate elements of the system with each other, for what it is necessary to know the distribution functions (DF) of the operation times of these elements.
In the case of SM systems the integral equations of Markov renewal are solved (MRE).The article proposes new method of SM systems modelling and its application to the study of functioning of the production element (PE).The model allows to assess the reliability impact of PE on its performance under the following hypothesis: in case of PE failure the production service is interrupted, and after the performance restoration of the element the production service continues in consideration for the time of the interrupted service (non-devaluate failures) [1,2].
The proposed method is based on the theorem presented below.

Problem formulation
Consider the SM system with the common phase space of states M .Select the two subsets + M and − M in the phase space of states M of the SM processes, such that M M M = ∪ − + .In the future, we will only talk about the subset + M , as all the above will be true for the subset − M .The times of single stay in the states

Problem Solution
The theorem on the times stay in the states of trajectory in view of the repeated returns.
If the discrete ergodic SM system with the known stationary distribution for the selected k-th trajectory with the states and in the original domain, they look like: where -is coefficient of increasing the stay time of the system in the states by re-entering it; i ρ -is fixed rate of the system stay in the state i S for the selected trajectory; e ρ -is fixed rate of the system stay in the state e S (a final state of the current trajectory from which there is direct transition to the subset − M ) for the selected trajectory; ej P -is the probability of transition from state e S to j S belonging to the subset − M .
Theorem proof.Stay times of the system in the states ( ) increase by the system re-entering with some probability in these states during the stay of the entire system in the subset k M + , and there is nothing else due to what they can increase which is obvious.But this means that the number of the system entering follows the geometrical law of distribution with the i P probability that the system will come out of this state, and the probability that it will remain is ( Iterating the resulting equation (3), we have [3]: The density of distribution ) (t The unknown that is to be determined here is i P .It is known that the geometric distribution law leads and this leads to the following: Let us find i mθ .According to the known theorem [1] + T can be accurately determined.For the discrete states the formula is: Then Let us introduce the coefficient k i c of the increase of the system stay time in the states i S equal to Since there is only one way out for the current trajectory of the state e S -the final state of the trajectory from which there is direct transition to the state j S of the subset − M , the coefficient k i c of time increase spent by the system in the states i S would be: Then: , and the probability i P is determined by the condition of the increase i m up to the value i mθ that is of equality , which implies that the sought probability i P on the basis of (5) equals to: Substituting ( 7) into (4) we obtain (2).
Applying the Laplace transformation to the formula (4), we obtain The resulting expression is an infinitely decreasing geometric progression.Let us take the limit of the Then the expression (8) will look like: ( ) ( ) Accordingly, the image of DD ) (t f k i will look like: Substituting ( 7) into ( 9) and (10) we will obtain The theorem is proved.Corollary 1.The time T θ θ θ , where n -is the number of states included in the trajectory.
Based on the formula of total probability the expression for the distribution function Σ θ of the system stay time in the subset + M is: The stay times in the states of the entire system are determined by the following expression: Corollary 4. DF of the stay times in the particular trajectories of renewal correspond to EMC for the distribution function of system stay times in the subset + M with given initial state.Thus, the exact solution of the Markov renewal equation is obtained for the discrete SM system.
Note that in general, the algorithms of phase consolidation (for example, the one proposed in [2]) used for the transition from the systems with continuous set of states to the systems with discrete states are rough, that is why it is incorrect to speak about the exact solution of the Markov renewal equation in the continuous phase space.
The Method Of Trajectories Step One.The transition from system with continuous states to system with discrete states . For that the DF i F of the system stay times in the new discrete states, ij P transition probabilities of these states to other states (transitional probabilities), the specific frequencies i ρ of entering the states (stationary distribution of EMC) and fixed probabilities of stay in the states (the stationary distribution of the SM process) are determined.The procedure is performed with the help of known methods of SM systems modelling.
Step Two.Selection of all possible trajectories of transition from the subset + M to the subset − M .Moreover, each state of the system is included in one or several trajectories at once.
Step Three.The stationary distribution of EMC and transition probabilities for each trajectory are determined.
Step Four.Based on the foregoing theorem, replaced by the stay times in the states i α are replaced by i θ -for them the densities and distribution function of system stay times in the states are determined, in view of the repeated returns in accordance with the presented theorem.
Step Five.In accordance with the theorem of total probability, the probabilities T k P of each of the trajectories realization are determined on the basis of transition probabilities of the Markov chain.
Step Six.We find the distribution function of the stay time in + M , regardless of the initial state, which is defined as the weighted sum (the mixture) of DF of each trajectory emerging from the state i S .The coefficients of the mixture are found on the fifth step of the T k P probability of trajectories.Let us consider specific example implementation of the proposed method of modelling.
We will describe the operation of the element [ .In case of PE failure the production service is interrupted, and after the performance restoration of the element the production service continues in consideration for the time of the interrupted service.
It is necessary to define the DF ) (t F ε RV ε -of the cycle time service of PE production unit in view of its failures, the mean and variance of said RV, as well as the performance of PE. To describe the operation of the system we use the Markov renewal process (MRP) and the corresponding SM process (SMP) with the states: 10x -PE is operational, the service of the next production unit has begun; the remaining time before failure of PE, as well; 11x -momentary state corresponding to the end of the maintenance unit of production; the remaining time before failure of PE, equals ; 20th -was restored and continued efficiency PE interrupted service production unit; the time remaining before the end of the interrupted service as well; 21x -has failed PE, service unit of production is interrupted; the time remaining before the end of the interrupted service as well.
The transition graph of the system -in Fig. 2.
Fig. 2. The states graph of the system with continuous states.
The phase space of states is as follows: M = {10x, 11x, 20x, 21x}.We describe the densities and EMC transition probabilities: of the density of the stationary distribution for the states of 10x, 11x, 20x, 21x, respectively.The system of integral equations for the densities of stationary distributions is: In [1] it is shown that the solution of this system is determined by formulas As mentioned above, let us complete the transition to an equivalent system with discrete states.The graph of such system is similar to the graph in Fig. 3.As shown in Fig. 3, the graph states have no continuous components, that is, the system is discrete.To go to discrete system phase consolidation algorithm (PCA) is used.It is proposed in [2].It is necessary to define the transition probabilities, the stationary distribution of EMC and the stay times of distribution function in the discrete states 10 and 20 of the formulas [2]: ) For the state 20: ( ) According to the formula (11) we find the transition probabilities: ( ) ( ) We find the DF system for stay time in the discrete state 20 using (12): Let us transform the numerator: For the state 10:   We find the DF system for stay time in the discrete state 10 using (12): Let us transform the numerator: With serial convolution we determine the stay time of distribution function of each trajectory.
, where * -sign of the convolution operation.
Let us find the probability of each of trajectories Hence, it becomes possible to determine the stay time of distribution function ) (t F θ Σ of the system in the subset, regardless of the initial state as a weighted sum (a mixture) of the stay times of distribution function of the subset of each trajectory with coefficients equal to the probability of those trajectories: DF cycle time maintenance PE because of its failure, and also for the mathematical expectation and variance obtained by known method using the Markov renewal equation, given in [4] Results comparing the two methods are presented in Fig. 5.
Mathematical expectation and variance RV -time cycle service of PE production unit in view of its failures is defined in two ways: • known method using the Markov renewal equation:

Conclusions
The results of modelling prove the correctness of the proposed method.The obtained exact solution is of great interest for it allows to determine accurately the DF of time between the events in the stream of the resulting superposition of the renewal stream and alternating renewal stream.In the future it is expected to approve the proposed method for the modelling of other systems.
The research was partially supported by the Russian Foundation for Basic Research (research project No. 15-01-05840 a.).
time period in the subset + M having mathematical expectation + T .The times of the multiple stays of the system in the state+ ∈ M S idue to the repeated entering into them during the Σ that the states of the trajectory form the set -is the number of trajectory.And also, the probabilities T k P of the k-th trajectory on the basis of the transition probabilities of the system stay time in the states

Fig. 3 .
Fig. 3.The graph of the system states with discrete states.

Fig. 5 .
Fig. 5.Comparison of the modelling results with the help of the classic approximate method of successive approximations, and of the precise method proposed: 1 -with approximate method; 2 -the solution offered with the precise method.