Multitype branching processes with immigration in random environment, and polling systems

Abstract

For multitype branching processes with immigration evolving in a random environment and producing a final product we find the tail distribution of the size of the final product accumulated in the system for a life period. Using this result we investigate the tail distribution of the busy periods of the branching type polling systems with random service disciplines and random positive switch-over times.

Recurrence and transience of contractive autoregressive processes and related Markov chains

“…Following e.g. [Key87], [Roi07], and [Vat11], we consider a combination of these three generalizations, namely multitype Galton-Watson branching processes Z = (Z n ) n≥0 in random environment with immigration.…”

Multitype branching processes evolving in a Markovian environment

“…A multitype subcritical branching process with final product was investigated in [14] for the environment generated by a sequence of independent identically distributed random variables. A multitype subcritical branching process with immigration has been studied in [15] and [22]. The survival probability of the critical and subcritical single-type branching processes in a stationary random environment has been analysed in [21].…”

Queueing for an infinite bus line and aging branching process

“…[25]. Using multitype branching process with final product allows to describe the busy period time and could provide an efficient way to extend this work to random time service, see [31] with zero switch-over time and [33] with nonzero switch-over time, both in random environment. The stable case corresponds to subcritical branching process, and the heavy traffic limit (when the load tends to 1) has also been investigated using near critical branching processes [29].…”