The superposition of independent discrete Markovian packet streams

Abstract

In a finite, aperiodic, irreducible Markov chain, a visit to some states generates a one; to some others, a zero, while the remaining states are considered silent, in that neither symbol is generated. States during which either a one or a zero is generated are called active states, and sojourns in the set of active states correspond to messages. The output process is called a Markovian packet stream. Informally, a stream is called thin, if the steady-state fraction of time spent in the active states is small and messages are separated by silent periods of long durations. A limit theorem for the superposition of a large number of independent, stochastically identical and appropriately thin Markovian packet streams is obtained. The class of limit processes consists of a family of stationary, integer-valued, discrete-parameter processes of dependent Poisson random variables. Some properties of the limit processes are established.