Motivated　by　various　applications　in　queuing　theory,　this　article　is　devoted　to　the　monotonicity　and　convexity　of　some　functions　associated　with　discrete-time　or　continuous-time　denumerable　Markov　chains.　For　the　discrete-time　case,　conditions　for　the　monotonicity　and　convexity　of　the　functions　are　obtained　by　using　the　properties　of　stochastic　dominance　and　monotone　matrix.　For　the　continuous-time　case,　by　using　the　uniformization　technique,　similar　results　are　obtained.　As　an　application,　the　results　are　applied　to　analyze　the　monotonicity　and　convexity　of　functions　associated　with　the　queue　length　of　some　queuing　systems.