346 research outputs found
Exact two-terminal reliability of some directed networks
The calculation of network reliability in a probabilistic context has long
been an issue of practical and academic importance. Conventional approaches
(determination of bounds, sums of disjoint products algorithms, Monte Carlo
evaluations, studies of the reliability polynomials, etc.) only provide
approximations when the network's size increases, even when nodes do not fail
and all edges have the same reliability p. We consider here a directed, generic
graph of arbitrary size mimicking real-life long-haul communication networks,
and give the exact, analytical solution for the two-terminal reliability. This
solution involves a product of transfer matrices, in which individual
reliabilities of edges and nodes are taken into account. The special case of
identical edge and node reliabilities (p and rho, respectively) is addressed.
We consider a case study based on a commonly-used configuration, and assess the
influence of the edges being directed (or not) on various measures of network
performance. While the two-terminal reliability, the failure frequency and the
failure rate of the connection are quite similar, the locations of complex
zeros of the two-terminal reliability polynomials exhibit strong differences,
and various structure transitions at specific values of rho. The present work
could be extended to provide a catalog of exactly solvable networks in terms of
reliability, which could be useful as building blocks for new and improved
bounds, as well as benchmarks, in the general case
Exact solutions for the two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the generalized fan
The two- and all-terminal reliabilities of the Brecht-Colbourn ladder and the
generalized fan have been calculated exactly for arbitrary size as well as
arbitrary individual edge and node reliabilities, using transfer matrices of
dimension four at most. While the all-terminal reliabilities of these graphs
are identical, the special case of identical edge () and node ()
reliabilities shows that their two-terminal reliabilities are quite distinct,
as demonstrated by their generating functions and the locations of the zeros of
the reliability polynomials, which undergo structural transitions at
Analytic properties of mirror maps
We consider a multi-parameter family of canonical coordinates and mirror maps
o\ riginally introduced by Zudilin [Math. Notes 71 (2002), 604-616]. This
family includes many of the known one-variable mirror maps as special cases, in
particular many of modular origin and the celebrated example of Candelas, de la
Ossa, Green and\
Parkes [Nucl. Phys. B359 (1991), 21-74] associated to the quintic
hypersurface in . In [Duke Math. J. 151 (2010),
175-218], we proved that all coeffi\ cients in the Taylor expansions at 0 of
these canonical coordinates (and, hence, of the corresponding mirror maps) are
integers. Here we prove that all coefficients in the Taylor expansions at 0 of
these canonical coordinates are positive. Furthermore, we provide several
results pertaining to the behaviour of the canonical coordinates and mirror
maps as complex functions. In particular, we address analytic continuation,
points of singularity, and radius of convergence of these functions. We present
several very precise conjectures on the radius of convergence of the mirror
maps and the sign pattern of the coefficients in their Taylor expansions at 0.Comment: AmS-LaTeX; 40 page
Exact Failure Frequency Calculations for Extended Systems
This paper shows how the steady-state availability and failure frequency can
be calculated in a single pass for very large systems, when the availability is
expressed as a product of matrices. We apply the general procedure to
-out-of-:G and linear consecutive -out-of-:F systems, and to a
simple ladder network in which each edge and node may fail. We also give the
associated generating functions when the components have identical
availabilities and failure rates. For large systems, the failure rate of the
whole system is asymptotically proportional to its size. This paves the way to
ready-to-use formulae for various architectures, as well as proof that the
differential operator approach to failure frequency calculations is very useful
and straightforward
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