In this paper a stochastic model of a large distributed system where users'
files are duplicated on unreliable data servers is investigated. Due to a
server breakdown, a copy of a file can be lost, it can be retrieved if another
copy of the same file is stored on other servers. In the case where no other
copy of a given file is present in the network, it is definitively lost. In
order to have multiple copies of a given file, it is assumed that each server
can devote a fraction of its processing capacity to duplicate files on other
servers to enhance the durability of the system.
A simplified stochastic model of this network is analyzed. It is assumed that
a copy of a given file is lost at some fixed rate and that the initial state is
optimal: each file has the maximum number d of copies located on the servers
of the network. Due to random losses, the state of the network is transient and
all files will be eventually lost. As a consequence, a transient
d-dimensional Markov process (X(t)) with a unique absorbing state describes
the evolution this network. By taking a scaling parameter N related to the
number of nodes of the network. a scaling analysis of this process is
developed. The asymptotic behavior of (X(t)) is analyzed on time scales of
the type t↦Npt for 0≤p≤d−1. The paper derives asymptotic
results on the decay of the network: Under a stability assumption, the main
results state that the critical time scale for the decay of the system is given
by t↦Nd−1t. When the stability condition is not satisfied, it is
shown that the state of the network converges to an interesting local
equilibrium which is investigated. As a consequence it sheds some light on the
role of the key parameters λ, the duplication rate and d, the maximal
number of copies, in the design of these systems