Semi-Markov model of a technical system with the component-wise instantly replenished time reserve

. Time redundancy is one of the methods to increase the reliability and efficiency of technical systems. When it is used, the system is given additional time (a time reserve) for restoring characteristics. In this paper we construct a semi-Markov model of a two-component system with a component-wise instantly replenished time reserve. In this paper we construct a semi-Markov model of a two-component system with a component-wise instantaneous replenishment of the time reserve. For an approximate determination of the stationary characteristics of the reliability of the system, the phase merging scheme algorithm is used.


Introduction
When designing and operating technical systems, great attention is paid to the reliability and efficiency of both the system as a whole and its individual components. There are various ways to improve the reliability and efficiency of technical systems, one of which is time redundancy.
About the time redundancy [1][2][3][4][5][6][7][8] say in cases where the system during its operation is given additional time (time reserve) for the restoration of its technical characteristics. In production systems, the time reserve sources can be different: warehouses, various types of inter operational storage devices, a stock of productivity, etc. Systems with functional inertia have a reserve of time.
For systems with a time reserve, a malfunction (failure of an object) does not yet mean a failure of the system itself, if the restoration of the operability of the object ends before the use of the time reserve. At present, semi-Markov processes are often used to model systems for various purposes [9][10][11][12][13].
In this paper, using the theory of semi-Markov processes with a common phase space of states [9][10][11][12][13], a model of a two-component system with a component-wise instantly replenished time reserve is constructed. The reliability characteristics of the system under consideration are determined; the effect of the time reserve on the characteristics obtained is analyzed.

The system description
The system S, consisting of two components 1 K and 2 K , and the component 1 K has an instantly replenished time reserve equal to h const = . Components 1 K , 2 K time to failure are random variables (RVs) 1 The case of parallel connection (in reliability sense) of components without their disconnection is considered. The failure of system S occurs if both components are restored and the time reserve is fully used up.

Semi-Markov model building and stationary characteristics definition
To describe the system S operation let us introduce the following set E of system semi-Markov states: 1  K continues restoration, the time reserve is fully used up, 2 K begins restoration, time 0 x > is left till 1 K begins operate, system failure; 200xz -1 K continues restoration and continues to function due to the time reserve, 2 K begins restoration, time 0 x > is left till 1 K begins operate, z is the value of the remaining time reserve; 1 2 101x x -1 K continues restoration, the time reserve is fully used up, time 1 x is left till 1 K begins operate, time 2 x is left till 2 K failure; 201xz -1 K continues restoration and continues to function due to the time reserve, 2 K begins operate, time 0 x > is left till 1 K begins operate, z is the value of the remaining time reserve; 1 2 100x x -both components continue restoration, the time reserve is fully used up, time 1 x is left till 1 K begins operate, time 2 x is left till 2 K begins operate, system failure. Time diagram of the functioning of the initial system is shown in Figure 1: 1(2) corresponds to the functioning of the 1 K ( 2 K ), 1р corresponds to the state of the time reserve. We proceed to determine stationary characteristics of the system reliability. For this, we use phase merging algorithms developed in the works [10][11][12].
Suppose that the stochastic kernel of the embedded Markov chain (EMC) { ; 0} n n ξ ≥ of a semi-Markov process ( ) t ξ [11,13] is close to the stochastic kernel of the EMC { ; 0} n n ξ ≥ of supporting system S0 with a unique stationary distribution ( ) dx ρ . Then for an approximate calculation of the mean stationary operating time of the system to failure T + , the mean stationary restoration time T − and stationary availability factor a K of the initial system S we can use the following approximate formulas [14]: Let's assume that the initial system's uptime is significantly longer than the restoration time. Then, the reference system will be the system S0, in which the components are recovered instantaneously, that is, the superposition of the two renewal processes.
Time diagram of the functioning of the support system is shown in Figure 2.
As shown in [11], the system of equations (2) has the following solution: where the value of 0 ρ can be obtained from the normalization requirement.
The mean values of sojourn times in the states of the initial system are represented by formulas: For the considered system r = 2, because initial system can in two steps go into a subset of failure states E − from a subset of up-states entering the ergodic class 0 E .
We find stationary characteristics of the system T + , T − , a K by using the formulas (1), (2) 0 ( , 1 ) P ρ = 0 1 (2) 0 ( , ) P m As an example of the use of formulas (1,(9)(10), let us consider a system in which 1 K operating time Eα1 = 8 h, 2 K operating time Eα2 = 6 h , 1 K recovery time Eβ1 = 0.71 h, time T − and stationary availability factor a K of the system for the specified distribution were calculated. The results are presented in Figure 3.

Conclusion
Time redundancy is widely used to ensure the reliability and efficiency of the operation of technical systems. Unlike other kinds of redundancy, the reserve here is the time reserve that occurs during the system operation. The time reserve can be used to switch the structural reserve, detect and eliminate failures, repetition works impaired failures, standby load in working condition. Time redundancy is widely used in technical and information systems, computer networks, communication systems.
In this paper, we construct a semi-Markov model of a two-component technical system with a component-wise time reserve in the case when the reserve has only one component. Using the phase merging scheme algorithm, stationary reliability characteristics of the system under consideration are approximately found. The analysis of the effect of the instantly replenished time reserve value on the obtained reliability characteristics is carried out.
In the future, it is planned to build two-component and multi-component models of systems with a random component-wise time reserve and various strategies for its use.
The results of this work can be used to build models of technical systems with different types and strategies for using the time reserve, engineering calculations and solving optimization problems.
Work was supported by the Ministry of Education and Science of the Russian Federation within the framework of the main part of the state order (№ 1.10513.2018/11.12) and the Russian Foundation for Basic Research (No. 18-01-00392a).