A classical problem in computer/network reliability is that of identifying simple, regular and repetitive building blocks (motifs) which yield reliability enhancements at the system-level. Over time, this apparently simple problem has been addressed by various increasingly complex methods. The earliest and simplest solutions are series and parallel structures. These were followed by majority voting and related schemes. For the most recent solutions, which are also the most involved (e.g., those based on Harary and circulant graphs), optimal reliability has been proven under particular conditions. Here, we propose an alternate approach for designing reliable systems as repetitive compositions of the simplest possible structures. More precisely, our two motifs (basic building blocks) are: two devices in series, and two devices in parallel. Therefore, for a given number of devices (which is a power of two) we build all the possible compositions of series and parallel networks of two devices. For all of the resulting twoterminal networks, we compute exactly the reliability polynomials, and then compare them with those of size-equivalent hammock networks. The results show that compositions of the two simplest motifs are not able to surpass size-equivalent hammock networks in terms of reliability. Still, the algorithm for computing the reliability polynomials of such compositions is linear (extremely effcient), as opposed to the one for the size-equivalent hammock networks, which is exponential. Interestingly, a few of the compositions come extremely close to size-equivalent hammock networks with respect to reliability, while having fewer wires.a

No abstract

The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operatorvalued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.

As a continuation of [3] this paper treats the stationary and stationarily cross-correlated multivariate stochastic mappings. Moreover for the case of multivariate random distribution fields, a particular form for the operator cross covariance distribution is given, from which a Kolmogorov type isomorphism theorem and a spectral representation of a stationary multivariate random distribution field are derived.

No abstract