Some remarks on uniformly bounded markov chains: multimodality analysis

Abstract

Since 1975 when Carleial and Hellman [ IEEE Trans. Commun. COM-23, 401–410 (1975)] 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. In this paper, we generalize the problem and ask under what conditions a one-dimensional Markov chain possesses a multimodal steady-state distribution. We restrict our analysis to uniformly bounded Markov chains. In this class we distinguish so called near birth and death processes and prove 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, we establish conditions under which some Markov chains can be approximated by a birth and death process.