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.

翻译：客户根据 Poisson 程序到达服务设施 。 到达后, 他们被迫在一个独特的队列中等待, 直至轮到客户。 服务时间假定他们离开系统 。 服务时间假定为根据指数法独立且分布相同的随机变量序列 。 此外, 假设根据 Poisson 过程灾难发生的时间, 系统将空闲。 上面提到的所有随机天体都是独立的 。 显示系统中每次客户数目的随机变量的收集, 被称为 M/ M/ /1 美元 。 在这项工作中, 我们研究这个模型的分数版本, 这个模型是通过考虑 Kolmogorov 原Markov 进程前方的分数衍生物而成的 。 关于分数 $ M/ M/ 1 美元与灾难一起的队列, 我们获取了每次客户数目的概率、 平均值和差异 。 此外, 我们讨论参数的估计 。