Recursive Approach for Evaluation of Time Intervals between Transactions in Polling Procedure

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.


Introduction
Polling is one of the most common procedure for transactions organization in real time, embedded, swarm etc., control systems [1,2].In such a systems control computer algorithmically organizes a cycle of control of peripheral equipment, so there are rigid restrictions for time period between transactions.
For evaluation of time intervals the model of data processing must be worked out.Taking into account the number of specific features of control algorithms, such as cyclic recurrence, quasi-stochastic way of continuation in places of branching, quasi-probabilistic of time of execution of operators, there is the only formalism for modeling the computation, ergodic semi-Markov process [3][4][5][6], and interpretation of algorithm may be represented as a wandering through states of semi-Markov process [7,8].
In [9,10] matrix expressions for evaluation of time intervals were obtained, but they are cumbersome enough for practice use.So, the known approach needs high computational resources, commensurable with compilation of program and experimental evaluation of time intervals under consideration [11,12].Recursive procedure proposed below permits substantially decrease time complexity of analysis of ergodic semi-Markov processes.

Initial semi-Markov process
In most general case semi-Markov process, which simulates a polling generator of transactions, is the next [3][4][5][6]: where where mn S S r r -is the adjacency matrix of size When reducing, states of subset A S are eliminated, so every switch of semi-Markov process (3) generates one transaction.For the reducing may be used matrix method, described in [10], however similar result may be obtained with the aid of recursive procedure of simplification of (1).
For realization of recursive procedure let us to introduce three elementary operations of reducing of semi-Markov process, which are shown on figure 2 [15][16]. where where k -is the number of arcs from a m to a n ; The recursive procedure presupposes sequential elimination of states from (S+J)-th, to (S+1)-th.Let us assume that at the previous simplifications states from (S+J)-th till (S+1+1)-е were eliminated, i.e. semi-Markov process (1) was reduced till the process   Process (3) still stay the ergodic one, so density of time intervals between transactions may be defined in accordance with [3,4] the next expression:

Time intervals between transactions
Procedure must be repeated for different states, which must be placed onto the first position to achieve all parameters necessary to use the formula (10).

Experimental verification of method
For verification of proposed method direct computer experiment was executed with use the Monte-Carlo method.Under verification was homogeneous semi-Markov process  .If L l , then 3).12) End of experiment.On figure 5 the histogram of time of returning to state 1 is shown, which was constructer after 10000 realizations of the process M

Conclusion
In this article relatively simple universal procedure of recursive simplification of semi-Markov process was presented to determine time intervals between transactions at polling.Time characteristics were obtained for most common structure of semi-Markov generator of transactions (full graph with loops).Time intervals are significantly important parameter from point of view insertion the computer into control circuit as the feedback element.Optimization of time under consideration permits to improve quality characteristics of control.Further research in this area may be directed onto working out the method of optimization of time intervals with use recursive procedure.

Figure 2 .
Figure 2. Operations of reducing On the fig. 2 the union of sequential states is shown.Probability, weighted density, expectation and dispersion of time of wandering from m a to n a are deter- mined by the next dependencies:

1
density and expectation of time of switching from a m to a n on k-th arc.On the fig.2(c) elimination of loop is shown.Probability, weighted density, expectation and dispersion of time of switching from m a to n a are deter- mined by the next dependencies: Recursion procedure repeated from j = J till j = 1.As a result of such a procedure elements of matrices culated for the semi-Markov process (3).
recursive procedure, described above.Without loss of community let us describe it for determining applied to process (3) after a changing of numeration of states of the process.Procedure for decrement of m, from S to 1n,k d m -1; union of parallel arcs