  Integrated satellite-terrestrial networks (ISTNs) can significantly expand
network coverage while diminishing reliance on terrestrial infrastructure.
Despite the enticing potential of ISTNs, there is no comprehensive mathematical
performance analysis framework for these emerging networks. In this paper, we
introduce a tractable approach to analyze the downlink coverage performance of
multi-tier ISTNs, where each network tier operates with orthogonal frequency
bands. The proposed approach is to model the spatial distribution of cellular
and satellite base stations using homogeneous Poisson point processes arranged
on concentric spheres with varying radii. Central to our analysis is a
displacement principle that transforms base station locations on different
spheres into projected rings while preserving the distance distribution to the
typical user. By incorporating the effects of Shadowed-Rician fading on
satellite channels and employing orthogonal frequency bands, we derive
analytical expressions for coverage in the integrated networks while keeping
full generality. Our primary discovery is that network performance reaches its
maximum when selecting the optimal density ratio of users associated with the
network according to the density and the channel parameters of each network.
Through simulations, we validate the precision of our derived expressions.
