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We consider continuous-time Markov chains on integers which allow transitionsto adjacent states only, with alternating rates. We give explicit formulas forprobability generating functions, and also for means, variances and stateprobabilities of the random variables of the process. Moreover we studyindependent random time-changes with the inverse of the stable subordinator,the stable subordinator and the tempered stable subodinator. We also presentsome asymptotic results in the fashion of large deviations. These results givesome generalizations of those presented in Di Crescenzo A., Macci C.,Martinucci B. (2014).

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates