Some Remarks on Uniformly Bounded Markov Chains: Stability Analysis
نویسنده
چکیده
Since the appearance of a paper by Carleial and Hellman [2] in 1975, it is known that the bistable behavior of the ALOHA system is associated with a bimodal shape of the backlog steady-state disbibution. 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 wtiformly bounded Markov Chains. In this class of Markov Chains we distinguish so called near birth and death processes and prove that under some additional assumptions a shape of the distribution is detennined 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.
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