## Abstract

In this chapter, we consider the monotonicity properties of discrete-time Markov chains where each monotonicity is characterized in terms of transition matrices. A Markov chain {*X* ^{ n } }is said to be increasing (decreasing, respectively) if *X* ^{ n+1} ≻ *X* ^{ n } (*X* ^{ n } ≻ *X* ^{ n+1}) for all n = 0,1, ..., where ≻ denotes an ordering relation in some stochastic sense, and in either case we call {*X* ^{ n } }*internally monotone*, or monotone for short. An *external monotonicity* is such that, for two Markov chains {*X* ^{ n } }and {*Y* ^{ n }, we have *X* ^{ n } ≻ *Y* ^{ n } for all *n*. Monotonicity properties are important both theoretically and practically because they lead to a variety of structural insights. In particular, they are a basic tool for deriving many useful inequalities in Markov chains for stochastic modeling.

## Keywords

Markov Chain Hazard Rate Probability Vector Discrete Random Variable Service Time Distribution## Preview

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