Models for estimating quality and reliability indicators, Models...

Models for assessing quality and reliability

As a result of studying this chapter, the student must:


• models of distribution functions, general quality model, reliability model models

be able to

• use the methods of mathematical statistics in the construction of models;


• Methods for assessing quality and reliability indicators.

Models of distribution functions used in reliability theory

The distribution function (FD) models are used in reliability theory to describe the distribution of failure characteristics [2]. failure characteristics means the time of appearance of failures, the rate of change of parameters (characteristics) of the product, etc. They can be represented as sets (disordered, ordered).

Poisson distribution

Poisson distribution plays an important role in reliability theory, it describes quite well the pattern of random failures in systems of varying degrees of complexity. This distribution law has found wide application in determining the probability of occurrence and restoration of failures.

The random variable X is distributed according to Poisson's law if the probability that this value takes a certain value is expressed by the formula


where λ is the distribution parameter (some positive quantity); m! denotes the factorial of the number m = 0, 1, 2 , ..., e = 2,71828 - the base of the natural logarithm.

The mathematical expectation and variance of the random variable X for the Poisson law are equal to the distribution parameter λ:

Example 4.1

In the workshop for the maintenance and repair of televisions from the public there are applications with an average density of 5 pcs. for a work shift of 10 hours. Assuming that the number of applications in any time interval is distributed according to Poisson's law, it is possible to find two bids for 2 hours of a work shift.


The average number of applications per 2 hours is λ = 2 • 5/10 = 1.

Applying formula (4.1), we calculate the probability of receipt of two applications

Answer: 0.184.

Exponential distribution

The exponential distribution law is perhaps the most known and commonly used in practice (Figure 4.1). It is even called the basic law of reliability, as it is often used to predict reliability during normal operation of products, when gradual failures have not yet manifested and reliability is characterized by sudden failures. These failures are most often caused by an unfavorable coincidence of one or another circumstance, and therefore they have a constant intensity. The exponential law in the theory of reliability has found wide application also because it is simple for practical use. Most of the problems solved in reliability theory using the exponential law are simple to solve, in any case easier than using other distribution laws. The main reason for this comparative simplicity is that under the exponential law the probability of failure-free operation depends only on the duration of the time interval and does not depend on the time of the previous work.

Graph of distribution of exponential distribution density

Fig. 4.1. Graph of distribution of exponential distribution density

Examples of unfavorable combination of working conditions in the operation of products are many. In particular, for a gear train, this can be the action of the maximum load on the weakest tooth when it is engaged; for electronic equipment elements - exceeding the permissible current or leaving the temperature outside the specified temperature range.

The distribution density of the exponential law is described by the relation

the distribution function of this law - the ratio

The reliability function is defined as

The mathematical expectation of a random variable x

variance of the random variable x

The exponential distribution finds quite a wide application in a wide variety of areas, including the theory of mass service, which is especially true in our time.

It describes the distribution of the operating time to failure of complex products and systems, the time of failure-free operation of the elements.

Example 4.2

Based on the results obtained during the operation of the generator, it is established that the operating time for failure is subject to an exponential law with the parameter λ = 2 • 10-5 h-1. Find the probability of failure-free operation of this generator during the time t = 1 00 hours. Define the mathematical expectation of the time between failures.


To determine the probability of failure-free operation, we use the formula (4.1), according to which

The mathematical expectation of the time between failures is equal to

Answer: 0.998; 5 • 104 hours.

thematic pictures

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