  In this paper, we provide a compositional approach for constructing finite
abstractions (a.k.a. finite Markov decision processes (MDPs)) of interconnected
discrete-time stochastic switched systems. The proposed framework is based on a
notion of stochastic simulation functions, using which one can employ an
abstract system as a substitution of the original one in the controller design
process with guaranteed error bounds on their output trajectories. To this end,
we first provide probabilistic closeness guarantees between the interconnection
of stochastic switched subsystems and that of their finite abstractions via
stochastic simulation functions. We then leverage sufficient small-gain type
conditions to show compositionality results of this work. Afterwards, we show
that under standard assumptions ensuring incremental input-to-state stability
of switched systems (i.e., existence of common incremental Lyapunov functions,
or multiple incremental Lyapunov functions with dwell-time), one can construct
finite MDPs for the general setting of nonlinear stochastic switched systems.
We also propose an approach to construct finite MDPs for a particular class of
nonlinear stochastic switched systems. To demonstrate the effectiveness of our
proposed results, we first apply our approaches to a road traffic network in a
circular cascade ring composed of 200 cells, and construct compositionally a
finite MDP of the network. We employ the constructed finite abstractions as
substitutes to compositionally synthesize policies keeping the density of the
traffic lower than 20 vehicles per cell. We then apply our proposed techniques
to a fully interconnected network of 500 nonlinear subsystems (totally 1000
dimensions), and construct their finite MDPs with guaranteed error bounds. We
compare our proposed results with those available in the literature.
