Solution of the integral equation for the average cost of restoration in the theory of reliability of technical systems
- Authors: Vainshtein V.I.1, Vainshtein I.I.1, Safonov K.V.2
-
Affiliations:
- Siberian Federal University
- Reshetnev Siberian State University of Science and Technology
- Issue: Vol 24, No 4 (2023)
- Pages: 628-638
- Section: Section 1. Computer Science, Computer Engineering and Management
- URL: https://journals.eco-vector.com/2712-8970/article/view/625628
- DOI: https://doi.org/10.31772/2712-8970-2023-24-4-628-638
- ID: 625628
Cite item
Abstract
Failures of elements during the operation of technical and many other systems are, as a rule, random in nature. This leads to various models of the recovery process, studied in probability theory and mathematical reliability theory. During the restoration process, failed elements are restored or replaced with new ones, and there is often a change in the costs and quality of the restored elements (time-to-failure distribution functions).
The work examines the cost function (average cost of restoration) in the process of restoration of order in which, according to a certain rule, the costs of each restoration and the distribution functions of operating time change.
Considering, that the recovery function (average number of failures) is well studied in reliability theory, a solution to the integral equation for the cost function is obtained through the recovery function of the model under consideration.
For the order restoration process a formula is obtained for calculating the cost function through the restoration function of a simple process formed by the convolution of all distribution functions of the periodic part. For practical application, explicit formulas are obtained for the cost function during the restoration process, in which the periodic part is distributed according to an exponential law or
an Erlang law of order with the same exponent α.
The resulting formulas can be used to study the properties of the cost function and solve optimization problems in strategies for carrying out the restoration process in terms of price, quality, risk, if, for example, the average number of failures is taken as quality, the average cost of restorations as price, the dispersion of the number of failures as the risk, or cost of restoration.
Full Text
Introduction
In the mathematical theory of reliability, when studying recovery processes, the numerical characteristics associated with the random number of failures and the random cost of recovery are first considered, for example, the average and dispersion of the number of failures and the cost of recovery, through which various criteria for the optimality of recovery strategies are determined.
The paper discusses models of the recovery process , taking into account the cost of restoration. Where Хi , random operation time with distribution functions elements from до failure; cost of recovery; – element cost, set at the initial time , for a case for a case [1–4].
Let – be the number of failures (recoveries), – be the cost of recovery for the time from 0 to :
–-multiple convolution of distribution functions
For [1; 2]: recovery function (average number of failures)
cost function (average cost of restorations)
During operation, the quality of the restored elements and the cost of restoration may differ. This leads to different models of the recovery process [1, 3, 5–9].
The work considers the restoration process taking into account the cost of restoration of the order in which the distribution functions and cost of restoration satisfy the condition: if the indices when divided by give the same excess [1, 3, 8, 9].
In the process under consideration, after the first restorations , a periodic process of the order begins.
Note, that in the case we have a periodic process of order restoration , and if , process of restoring order .
If coincide or coincide starting from number , we have simple (ordinary) and general (delayed) recovery processes, well studied in reliability theory.
Note that for the model under consideration, the recovery function has been well studied. Numerical methods for finding it have been developed, and for many distribution laws characteristic of reliability theory, there are its explicit representations [1, 6].
To find the cost function there are integral equations [1, 2, 10].
The purpose of the work is to obtain a solution to the integral equation for the cost function in the form of an integral representation through the restoration function Such a representation will be convenient for its study and calculation in various theoretical and applied problems of reliability theory.
Representation of the cost function through the recovery function
Let us write the integral equation for the cost function [1, 2]
(1)
for a case
for a case ,
– convolution of all distribution functions The functions define the periodic part of the recovery process.
Let be the restoration function of a simple process, let be the restoration function of the general process formed by the first distribution function and the following
(2)
In equation (1) we make the replacement
(3)
We obtain
Hence, to find the function we obtain the integral equation
(4)
Let us make a replacement
(5)
Equation (4) will be rewritten as
(6)
(7)
Note, that and are distribution functions, as a convolution of work distribution functions, and – mixture of distribution functions.
Comparing equations (6) and (2), we find that equation (6) defines the restoration function of the general process specified by the first distribution function of the second and subsequent ones
Thus,
(8)
and taking into account (3), (5), (7), (8)
(9)
Taking into account (2),
(10)
formula (9) will be written in the form
or taking account of (10)
We found that calculating the cost function comes down to calculating the finite number of convolutions of distribution functions and finding the restoration function of a simple restoration process formed by the distribution function , or restoration function
In the practical implementation of the obtained formulas (9), (10), (11), one can use numerical and analytical methods for calculating convolutions and restoration functions, discussed in [1, 11]. We also note that the resulting formulas make it possible to study the properties of the cost function and consider various optimization problems based on strategies for carrying out the restoration process in terms of price, quality, and risk. If, for example, we take the average number of failures as quality, the average cost of restorations as price, and the dispersion of the number of failures or the cost of restorations as the risk [1, 12–15].
This work is a continuation of work [11] and it can be noted that the theorems on the asymptotic behavior of the cost function obtained in [11] are much easier to obtain using the resulting formula for representing the cost function (9).
The cost function for a simple restoration process with exponential distribution
We consider a restoration process in which only the restoration costs change according to the law if the indices when divided by give the same excess. This corresponds to the common case where failures result in full restorations, but the costs of restorations change, for example, only the price of the element changes.
Let the operating time of the elements be distributed according to the exponential law For this case, we obtain calculation formulas for calculating the cost function.
Taking into account, that n-multiple convolution of the distribution functions of independent random variables is a function of the distribution of their sum, and that the Erlang order distribution n is the distribution of the sum of random variables n distributed according to the exponential law, we conclude that for the case under consideration
Erlang order distribution and [1, 6]
(12)
Now, according to (11)
. (13)
Taking into account (12), when calculating the integrals included in formula (13) are calculated explicitly. For example [16]
When substituting
in (13), we obtain
(14)
We select the real part in (14):
Therefore
Let us write down the formula for the cost function
We also consider the cost function during the process of restoring order , when the operating time of the periodic part of the process is distributed according to the Erlang law of orde with a parameter
Let We find Let us write down the integral equations for
(15)
(16)
Let there be given
Laplace-Stieltjes transform function [1; 6]. Considering from (15),(16) we obtain
(17)
(18)
Comparing (17), (18), we conclude that
We found that the restoration function of a simple restoration process formed by multiple convolution of Erlang order distributions with the parameter , is the restoration function of a simple restoration process formed by an Erlang order distribution with the same parameter .
We have
Now in accordance with (11)
(19)
Integral
in (19) it is calculated similarly to the previous example with replacing by and by
We also note that if additionally = , i = 1, 2, …,, то = , n = 1, 2, …, and in accordance with (19)
Conclusion
The most important performance indicators of technical and many other systems are random variables [17]. These are, for example, the operating time of the restored elements before failure, the number of failures and the cost of restoration during the restoration process. In the mathematical theory of reliability, when studying restoration processes, the numerical characteristics of these quantities are first considered, for example, the average and dispersion of the number of failures and the cost of restoration, through which various criteria for the optimality of restoration strategies are determined.
Considering that the recovery function for the model under consideration is well studied, the work obtained a solution to the integral equation for the cost function through the recovery function of a simple process specified by the convolution of all distribution functions of the periodic part. As a practical example, explicit formulas for the cost function are obtained for the restoration process, in which the periodic part is distributed according to an exponential law or Erlang law of order with the same property
Note that the resulting formulas make it possible to study the properties of the cost function and consider various optimization problems in strategies for carrying out the restoration process in terms of price, quality, and risk. If, for example, we take the average number of failures as quality, the average cost of restorations as price, and the variance of the number of failures or the cost of restorations as risk.
We also note that, along with the obtained formulas for calculating the cost function, limit theorems for the cost of restorations (as a random variable), similar to those for the number of failures [3], as well as finding the dispersion of the cost of restorations in the models under consideration will also be important.
About the authors
Vitaly I. Vainshtein
Siberian Federal University
Author for correspondence.
Email: vvaynshtyayn@sfu-kras.ru
Cand. Sc., Associate Professor, Head of the Department of Information Security
Russian Federation, 79, Svobodny Av., Krasnoyarsk, 660041Isaak I. Vainshtein
Siberian Federal University
Email: isvain@mail.ru
Cand. Sc., Professor, Department of Applied Mathematics and Computer Security
Russian Federation, 79, Svobodny Av., Krasnoyarsk, 660041Konstantin V. Safonov
Reshetnev Siberian State University of Science and Technology
Email: safonovkv@rambler.ru
Dr. Sc., Associate Professor, Head of the Department of Applied Mathematics
Russian Federation, 31, Krasnoyarskii Rabochii prospekt, Krasnoyarsk, 660037References
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