  This paper is concerned with a compositional approach for constructing
abstractions of interconnected discrete-time stochastic control systems. The
abstraction framework is based on new notions of so-called stochastic
simulation functions, using which one can quantify the distance between
original interconnected stochastic control systems and their abstractions in
the probabilistic setting. Accordingly, one can leverage the proposed results
to perform analysis and synthesis over abstract interconnected systems, and
then carry the results over concrete ones. In the first part of the paper, we
derive sufficient small-gain type conditions for the compositional
quantification of the distance in probability between the interconnection of
stochastic control subsystems and that of their abstractions. In the second
part of the paper, we focus on the class of discrete-time linear stochastic
control systems with independent noises in the abstract and concrete
subsystems. For this class of systems, we propose a computational scheme to
construct abstractions together with their corresponding stochastic simulation
functions. We demonstrate the effectiveness of the proposed results by
constructing an abstraction (totally 4 dimensions) of the interconnection of
four discrete-time linear stochastic control subsystems (together 100
dimensions) in a compositional fashion.
