Calculation and assessment of measures of the residual operating life of non-restorable items

The widespread use of cutting-edge components in the composition of non-restorable items has made it difficult to assess their reliability at the stages of designing and production. Therefore, items with underestimated values of the assigned (warranted) operating life are put into operation. By now, the service life of such items has reached the assigned limits, the items still maintaining a fairly high dependability rate. Thus, there has been a problem of extending the service life beyond the originally assigned rates. As originally assigned rates are determined by such measures as the mean operating life and gamma-percentile operating life, the non-restorable item’s life margin is determined as related to these rates by means of measures of the residual operating life, namely the mean residual operating life and gamma-percentile residual operating life. The mean residual operating life and gamma-percentile residual operating life are determined in this paper, to calculate the extendable service life. For these measures, we have derived calculation formulas and assessments which make it possible to determine the extended service life. In addition, for these measures, calculation formulas and assessments are obtained for non-restorable items whose margin allocation is distributed exponentially and uniformly. The paper examines the issues of attainability of the obtained assessments and the conditions of their attainability; also, the influence of failure rate monotonic change on the assessments is analyzed.


Mean residual operating life
Currently, there are different interpretations of the term "operating life", which is specially discussed in papers [1 -4]. In this paper, according to GOST 27.002-2015, the term 'operating life' is only meant as the operating time of the item since the beginning of its operation to its transition to the limit state. Henceforth, the non-restorable item under study is assumed as operating without interruptions, i.e., its operating time is continuous.
Under the residual operating life over time  is assumed here as the item's operating time from the moment of time  until its transition to the limit state, under the established application modes and operating conditions. While solving the task of extending the service life in practice, the value  is usually understood as the assigned operating life [5].
If  is the operating time of the item from the beginning of operation to its transition to the limit state, then the residual operating life   after the time  is determined by formula here and further on, the slash will denote the condition for the event enclosed within the first parenthesis.
where ( ) E  is the mathematical expectation of a random variable which is inside the parentheses.
The measure ( ) R  should not be confused with the mean residual operating life, which is equal to r  − , where r is the mean (non-residual) operating life, since the value (1) and the value     =− are related as follows: Hence, taking into account (2), we obtain is the mean (non-residual) operating life. For measure (2), the following formula is valid [8]: where ( ) P  is the probability of failure during the time specified inside the parentheses. For example, for the exponential law of item operating life allocation: where 0   is constant value, the measure calculated by the formula (4), is equal to ( ) , here 1  = is mean residual operating life.
In the case of the item's operating life being of a limited time interval ( ) 0,l , the formula (4) uns as follows: So, for an item whose operating life allocation is uniformly distributed on the segment ( ) 0,l : according to the formula (7), we obtain Since the mean operating life of the item r is equal to [8]: For example, for the law (8), from (9) we obtain 2 l r = .
In the tasks of extending the items' service life, information about the originally assigned values of dependability measure is important as a starting point of the residual operating life. Proceeding from this, let us prove the following. Theorem 1. For the mean residual operating life of a non-restorable item over time  , the following formula is valid: is the failure rate of the item.
Proof. While differentiating (4), we obtain is the failure rate of the item.
While integrating the expression (12) in the range from zero to  , and taking into account (10), we get the required formula (11).
The meaning of formula (11) is that it determines the presence of the item's mean operating life margin at the time  with regard to the initially assigned rate r (or its absence). For example, for the allocation law of the item's operating life (5)

Gamma-percentile residual operating life
For some items, high rates of failure-free operating are required for the extendable period of operation. In this case, the extendable period has to be determined depending on the specified rate of failure-free operating. It is obvious that the extendable service life, determined on the basis of the calculation and assessment of the mean residual operating life, does not have such properties. Let us determine one more measure of the residual operating life that has such properties [9].
Let the value be set To find a solution for extending the item's service life, it is necessary to determine the operating life margin at the time  with regard to the initial rate. Thereby, let us prove the following statement. Theorem 2. For the gamma-percentile residual operating life of a non-restorable item over time  the following formula is valid: Tt    hence it follows that such an item does not have a margin of gamma-percentile residual operating life at a time  with regard to the rate t  .

Conclusion
The calculation formulas and assessment of the residual operating life measures of non-restorable items are proved that make it possible to determine the extendable service life.