On the fractional queueing model with catastrophes

Customers arrive at a service facility according to a Poisson process. Upon arrival, they are made to wait in a unique queue until it is their turn to be served. After being served it is assumed that they leave the system. Services times are assumed to be a sequence of independent and identically distributed random variables following an exponential law. Besides, it is assumed that according to the times of a Poisson process catastrophes occur leaving the system empty. All the random objects mentioned above are independent. The collection of random variables describing the number of customers in the system at each time is what is called of $M/M/1$ queue with catastrophes. In this work, we study the fractional version of this model, which is formulated by considering fractional derivatives in the Kolmogorov's Forward Equations of the original Markov process. For the fractional $M/M/1$ queue with catastrophes, 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.