Birth and Death Processes in Random Environments

Doctoral thesis, 1993

This thesis treats birth and death processes in random environments. They are modelled by Markov processes in Z+2, where the first component represents the object of interest, the "birth and death process", and the second component represents the random environment which is assumed to be a time homogeneous Markov process in its own right. In particular, we let the random environment be a birth and death process which means that the total process may be considered as a two-dimensional birth and death process. In order to investigate such processes the coupling method is used, which results in conditions for stochastic domination and monotonicity. In addition, conditions for convergence in total variation are found by using a specific coupling which turns out to be useful for several problems. For instance, the question whether one process stochastically dominates another process can be answered in terms of the intensities of the processes. The rate of convergence is estimated and found to be exponential under certain conditions. The meaning of recurrence for a birth and death process in a random environment is discussed and conditions for recurrence are given under various assumptions. When the random environment is positive recurrent, simple conditions are found under which the total process is recurrent or positive recurrent. The results are applied to queuing models, i.e. the M/M/k and M/M/* queues in random environments. When the random environment is stationary, these queues are stochastically increasing if they start empty.