13.07.2011 г.

1. B.V.Gnedenko's model. The system consists of two elements, one of which is in working order, and another - in the non-loaded reserve. After refusal of a basic element he acts on restoration, the reserve element replaces given up basic element. If working element refuses, and reserve has not time to be restored, there is a refusal of system. If the reserve element has time to be restored during work of a basic element it becomes reserve and process further repeats.
The mathematical model of non-failure operation of system is represented by two equations [1]:

In (1) P2(t), P(t) — probabilities of non-failure operation of system and a basic element, a(t) — density of probability of time before refusal of an element, W(t) — conditional probability of non-failure operation of system provided that at the initial moment of time the reserve element has put into operation, and the basic instantly started to be restored. The control of a condition of elements continuous and ideal.
Using transformation Laplas, we shall write down (1) as:

where  . The decision of system of the equations (2) looks like:

From (3) we shall find, for example, average time before refusal of system:

In particular, at exponencials laws of distribution of time before refusal P(t) = e-λt and time of restoration G(t) = 1 — e-μt it is received:

2.  The account nonideal the control of elements of system. In [2] it is offered to take into account nonideal the control in restored system introduction of probability q. On this size « the resource of restoration » is multiplied, namely:   where   – probability no restoration an element in time t provided that his(its) refusal was found out with probability q. Size of integral by analogy to « a resource of reliability » professor N.M.Sedjakin,     where λ(t) — intensity of refusal of an element, we shall name « a resource of restoration ». In view of introduction of probability q in it further we shall designate probabilities as Gq(t), Rq(t), and size of a resource -q(t).

In the given expressions (1) - (5) with probability q should be directly connected G(t) b*(s), therefore it is finally possible to write down:

In that specific case, for exponentials distributions, the formula will be fair:

Thus, average time of non-failure operation of the duplicated system in this case in direct ratio reliability of the control of a condition of given up element.

Example 1. Let λ = 0,01 ч-1, μ = 0,1 ч-1. Then T2q = 200 ч  at q = 0, T2q = 700 ч  at q = 0,5, T2q = 1200 ч  at q = 1.

Example 2. Let the law of distribution of time before refusal normal. The density of distribution is equal . The law of distribution of time of restoration Weiboull. Function of distribution of time of restoration is equal . Then . So T = 25 ч., Θ = 6,267 ч. In figures 1 diagram b*q(q), T2q(q) the probabilities q constructed for various values are shown. From them follows, that with increase of reliability of the control about refusal of elements of system the size of conditional probability b*q(q) grows. With increase of this probability, as well as probability q, value of average time before refusal of the duplicated system T2q(q) grows, and increase is nonlinear. It testifies to importance of size of reliability of the control in the duplicated system.

3. Readiness of the duplicated system with the control. To receive the equations for research of readiness of the duplicated system at any distributions directly as it is made for definition of probability (1), inconveniently enough. It is easier to take advantage of expression (6) and to find from it image Лапласа of density of distribution of time before refusal of the duplicated system, having applied the formula:

where a* 2q (s) — the image of required density. Having executed necessary transformations, we shall receive:

Further we shall take advantage of the formula for the image of function of readiness as:

In which g*2q (s) — image of density of probability of restoration of the duplicated system after it refusal. We shall pay attention to that fact, that the given density can accept various kinds depending on discipline of restoration of system, namely restoration of both elements can be carried out by one or two brigades. We shall take into account it further at the analysis of readiness.

Expression (11) after substitution in it (10) is resulted in a kind:

Let's remind, that . Having executed limiting transition,
we shall find size of factor of readiness:

average time of restoration of system one or two brigades.

Example 3. We shall assume, that laws of distribution of time before refusal and restoration of elements exponentials. To intensity of refusal and restoration of an element are equal λ, μ. If restoration of system is made by one brigade under the formula (13) we shall receive:

If restoration of system is made by two brigades then the factor of readiness will be equal:

Correctness (14) and (15) can be checked up, having applied system of the differential equations.

Example 4. To determine factor of readiness of system, if density of probability of time before refusal of an element  , and function of distribution of its time of restoration  .

Service of two given up elements is made by one brigade. Average time of restoration of both elements equally:

Substituting these values in the formula

Let's receive dependence of factor of system on values of parameter of the control q. She is shown in figure 2. From figure follows, what even at rather small values of parameter of the control factor of readiness

becomes enough close to unit. At service by two brigades of restoration this effect will be increased.

The conclusion. Expression for function of readiness of the duplicated system in transformation Laplas and value of it stationary condition is received at any distributions of time before refusal and restoration of making elements.

The parameter of reliability of the control of a condition of elements is entered into the received expressions after their refusal. The size of the given parameter allows to take into account duration of restoration of elements after their refusal. Due to this generalization of the result received by B.V.Gnedenko, for readiness of the duplicated system with the control of a condition of elements is received.

#### The literature

1. Gnedenko B.V., Beljaev J.K., Solowiev A.D. Mathematical's methods in the theory of reliability. –  M.: the Science. –  1965. – 524 with.
2. Smagin V.A. To one probability models of the control. – AVT. – 2010. – № 6. – With. 25-33.

Smagin V.A.

Статья поступила в редакцию 14.06.2011

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