  In this paper, we develop a probabilistic framework for analyzing coded
random access. Our framework is based on a new abstract receiver (decoder),
called a Poisson receiver, that is characterized by a success probability
function of a tagged packet subject to a Poisson offered load. We show that
various coded slotted ALOHA (CSA) systems are Poisson receivers. Moreover,
Poisson receivers have two elegant closure properties: (i) Poisson receivers
with packet routing are still Poisson receivers, and (ii) Poisson receivers
with packet coding are still Poisson receivers. These two closure properties
enable us to use smaller Poisson receivers as building blocks for analyzing a
larger Poisson receiver. As such, we can analyze complicated systems that are
not possible by the classical tree evaluation method. In particular, for CSA
systems with both spatial diversity and temporal diversity, we can use the
framework of Poisson receivers to compute the exact (asymptotic) throughput. We
demonstrate that our framework can be used to provide differentiated services
between ultra-reliable low-latency communication (URLLC) traffic and enhanced
mobile broadband (eMBB) traffic. By conducting extensive simulations, we also
verify that our theoretical results match extremely well with the simulation
results.
