Problems of Information Transmission, Vol. 37, No. 4, 2001, pp. 380–396. Translated from Problemy Peredachi Informatsii, No. 4, 2001, pp. 112–129.
Original Russian Text Copyright
2001 by Krivulets, P olesskii.
On Bounds for the Monotone-Structure Reliability
V. G. Krivulets and V. P. Polesskii
Received March 6, 2001
Abstract—At present, two types of attainable bounds on the reliability of a general monotone
structure are known. Packing bounds are trivial. Untying bounds were obtained by Polesskii
in 1997. Their special case, the classical Esary–Proschan bounds, are known since 1963. In the
present paper, we introduce the third type of attainable bounds on the reliability of a general
monotone structure. We call them diﬀerence-untying bounds. They are generalization and im-
provement of the Oxley–Welsh bounds on the reliability of a homogeneous monotone structure
obtained in 1979. An example demonstrating the high quality of diﬀerence-untying bounds is
given. As a consequence, we obtain new bounds on the number of members of an arbitrary
A random monotone structure is the simplest model for reliability of a compound engineering
system (in particular, a data transmission network). In the monotone-structure reliability theory,
much attention is paid to constructing bounds on this reliability. There is a variety of such bounds.
Among them, of special interest are bounds on the reliability of a general monotone structure
with an arbitrary clutter and arbitrary component reliabilities.
In turn, among bounds for a general monotone structure, of special interest are bounds that
are analytical functions of some parameters of a monotone structure and are tight on some classes
of monotone structures. In other words, these are attainable “best possible” (in terms of the
parameters used) bounds.
Such bounds cannot be improved in terms of their parameters; therefore, bounds that are closer
to the actual value should necessarily employ another set of parameters.
The key role of attainable bounds in the whole class of bounds on the monotone-structure relia-
bility is as follows. By roughening them, we can construct new (generally speaking, unattainable)
bounds. Furthermore, analysis of attainable bounds indicates the parameters of the monotone
structure that should be involved to construct good indeed (close to the actual value) reliability
bounds. Therefore, constructing new attainable bounds on the monotone-structure reliability is of
At present, two types of attainable bounds on the reliability of a general monotone structure are
known, namely, packing and untying bounds. We describe them in Sections 3 and 4 respectively.
Due to triviality of packing bounds, nobody claims the authorship.
Untying bounds were obtained by Polesskii  in 1997. Their special case, the classical Esary–
Proschan bounds , are known since 1963. The idea of untying bounds is related to McDiarmid’s
results in percolation theory.
It is known that the monotone-structure reliability is studied in percolation theory  as “clutter
percolation probability.” It was introduced by Oxley and Welsh in [4,5]. Profound properties of this
reliability, which are described by the clutter percolation theorem, were shown by McDiarmid [6,7],
who noted  that his results could be used in the monotone-structure reliability theory. However,
before , no consequences from the McDiarmid theorem were used to construct bounds on the
2001 MAIK “Nauka/Interperiodica”