  A $M/M/1$ queue with catastrophes is a modified $M/M/1$ queue model for
which, according to the times of a Poisson process, catastrophes occur leaving
the system empty. In this work, we study a fractional $M/M/1$ queue with
catastrophes, which is formulated by considering fractional derivatives in the
Kolmogorov's Forward Equations of the original Markov process. For the
resulting fractional process, we obtain the state probabilities, the mean and
the variance for the number of customers at any time. In addition, we discuss
the estimation of parameters.
