  We have provided a fractional generalization of the Poisson renewal processes
by replacing the first time derivative in the relaxation equation of the
survival probability by a fractional derivative of order $\alpha ~(0 < \alpha
\leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler
distribution is examined. We also discuss some properties of this new model and
its relevance to time series. Distribution of gliding sums, regression
behaviors and sample path properties are studied. Finally we introduce the
$q$-Mittag-Leffler process associated with the $q$-Mittag-Leffler distribution.
