Some remarks on uniformly bounded Markov chains: Multimodality analysis

Since 1975 when Caleial and Hellman published their paper, it has been known that the bistable behavior of the ALOHA system is associated with a bimodal shape of the backlog steady-state distribution. This paper generalizes the problem and asks under what conditions a one-dimensional Markov chain possesses a multimodal steady-state distribution. It restricts analysis to uniformly bounded Markov chains. In this class the author distinguishes so called near birth and death processes and proves that under some additional assumptions a shape of the distribution is determined by the transition probabilities located on the principal diagonal, subdiagonal and supdiagonal of the transition matrix. This provides a theoretical explanation for the bistable behavior of the ALOHA system. In addition, the author establishes conditions under which some Markov chains can be approximated by a birth and death process.