^{1}

^{*}

^{1}

^{*}

In the present work, it is assumed that the n-components are arranged in the hierarchial order. The
*n*-cascade system surviving with loss of m components by
*k* number of attacks is studied; the general equation for the reliability is obtained for the above said system; and the system reliability is computed numerically for 6-cascade system for 2-number of attacks.

Reliability of a system is the probability that a system will adequately perform its intended purpose for a given period of time under stated environmental conditions [

Stochastic process is a mathematical model that evolves over time in a probabilistic manner. A special kind of stochastic process is called Markov process, where the outcome of an experiment depends only on the outcome of the previous experiment, i.e., the next state of the system depends only on the present state, not on preceding states. Cascade redundancy is the provision of alternative means or parallel paths in a system for accomplishing a given task such that all means must fail before causing a system failure. The reliability model is being studied here. The probabilities of component failure depend on the relative positions of the particular components along the hierarchy.

It is assumed that the n-components are arranged in the hierarchical order as

In the present, probability distribution of the number of attacks required for failure of a n-component hierarchical cascade system, as defined above, is investigated and the probability of the system surviving k attacks, sustained a loss of the first m components, is studied.

consists of zeros except for unity at the

Obviously,

Let a system consist of three components

Here consider the probability that the system survives “k” attacks with a loss of the first “m” components. Let

The Two Component System:

For m = 0, we get

For m = 1, we have the corresponding event

The corresponding probability

here ^{nd} attack.

It is obvious that when m = 2

The Three Component System:

When m = 0, we get

When m = 1, we have the corresponding event

Hence

here ^{nd} attack.

When m = 2, the corresponding event

Hence

It is obvious that

The Four Component System:

When m = 0, we get

Similarly, for m = 1

For m = 2

For m = 3

For m = 4

The Five Component System:

For m = 0, we get

For m = 1, the corresponding probability

For m = 2,

For m = 3,

For m = 4,

For m = 5,

The Six Component System:

For m = 0, we get

For m = 1, the corresponding probability

For m = 2,

For m = 3,

For m = 4,

For m = 5,

For m = 6,

The general equation for probability of n-component system fails in the kth attack

The general equation for reliability of n-component system for kth attack

Let us consider a transition probability matrix of order 7 for 6-component system

The first element in the ith column matrix represents the probability of failure of the system at the end of ith attack i.e.,

Let us find the reliability of above system in 2 number of attacks

The present work deals with the cascade reliability model represented as Markovian model. Reliability of stress strength model is derived with Markovian Approach. In this paper, the reliability of n-cascade system for k attacks with loss of m components has been derived for n > 4 and the general formula for reliability of n cascade system for k number of attacks has been derived. Using above general equation reliability has been calculated numerically for 6-cascade system for 2-number of attacks.

NalaboluSwathi,Tallapureddy Sumathi UmaMaheswari, (2015) Reliability Analysis of a Redundant Cascade System by Using Markovian Approach. Journal of Applied Mathematics and Physics,03,911-920. doi: 10.4236/jamp.2015.37111