A complex stochastic Boolean system (CSBS) is a complex system depending on an arbitrarily large number

This paper deals with the mathematical modeling of a special kind of complex systems, namely, those depending on an arbitrary number

We call such a system a complex stochastic Boolean system (CSBS). These systems can be found in many different scientific or engineering areas like mechanical engineering, meteorology and climatology, nuclear physics, complex systems analysis, operations research, and so forth. CSBSs also arise very often when analyzing system safety in reliability engineering and risk analysis; see, for example, [

Each one of the

Throughout this paper, the

As an example of CSBS, we can consider a technical system like the accumulator system of a pressured water reactor in a nuclear power plant, taken from [

Thus, this accumulator system can be considered as a CSBS where each one of its

Real-world CSBSs arising in many different engineering areas, like the above-mentioned accumulator system, are typically analyzed in many works dealing with system safety and reliability theory. In this context, let us mention that formula (

Moreover, we must highlight that the assumption of independent failures (an essential hypothesis for formula (

The behavior of a CSBS is determined by the ordering between the current values of the

The most useful representation of a CSBS is the intrinsic order graph: a symmetric directed graph on

In this context, the main goal of this paper is to compare the occurrence probabilities of two binary strings with the same length

For this purpose, this paper has been organized as follows. Section

Let us start this section with some basic concepts and notations which will be used in the rest of the paper.

Let

The decimal equivalent of

In what follows, we indistinctly denote the

The Hamming weight (or simply the weight)

The complementary

Note that two binary

Let

According to (

Let

However, assuming a simple (but not restrictive in practice) hypothesis on the parameters

Let

In the following, we assume that the parameters

The

The term “corresponding,” used in Theorem

The matrix condition IOC, stated by Theorem

For all

For

For

For

For all

Many different properties of the intrinsic order relation can be derived from its simple matrix description IOC. For the purpose of this paper, it suffices to recall here the following necessary (but not sufficient) condition for intrinsic order.

Note that if

The intrinsic order respects the Hamming weight. More precisely, for all,

The converse of Corollary

For

since matrix

does not satisfy IOC.

For

To finish this section, we present the graphical representation of the poset

For small values of

The intrinsic order graph for

However, for large values of

Let

In Figure

The intrinsic order graphs for

The intrinsic order graphs for

Note that

On the contrary, each pair

Also, looking at any of the digraphs in Figure

Recall that two binary

The edgeless graph associated with a given graph is obtained by removing all its edges, keeping its (isolated) nodes at the same positions. In Figures

The edgeless intrinsic order graph for

The edgeless intrinsic order graph for

For further theoretical properties and practical applications of the intrinsic order and the intrinsic order graph, we refer the reader to [

In this section, we present our results about the comparison, by intrinsic order, between the occurrence probabilities of two binary

For the special case that the bitstrings

Let

Using Definition

The following lemma characterizes the intrinsic order between binary

Let

Using Lemma

Now, we introduce the following notation for binary

Let

the vector of positions of

the vector of positions of

We also use the symbol “

Let

The following theorem characterizes the intrinsic order between binary

Let

Using Lemma

Let

Given a fixed binary

Let

Using Theorem

Now, we establish the dual result of Theorem

Let

Clearly, the

Given a fixed binary

Let

It suffices to use Lemma

Let

On one hand, the vector of positions of 1-bits in

The following corollary characterizes the binary

Let

the set of binary

the set of binary

(i) Let

(ii) Using again (

Finally, we establish the dual result of Corollary

The following corollary characterizes the binary

Let

the set of binary

the set of binary

Using Lemma

Given a fixed binary

The author declares that there is no conflict of interests regarding the publication of this paper.

The author sincerely thanks the anonymous referee for useful comments and suggestions. This work was partially supported by the “Ministerio de Economía y Competitividad” (Spanish Government) and FEDER, through Grant Contract CGL2011-29396-C03-01.