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