Queuing models with Mittag-Leffler inter-event times

Abstract We study three non-equivalent queueing models in continuous time that each generalise the classical M/M/1 queue a different way. Inter-event times all are Mittag-Leffler distributed, which is heavy tail distribution with no moments. For of we answer question being at zero infinitely often (the ‘recurrence’ regime) or not transient regime). Aside from this question, analytical properties allow us to number questions such as existence and description equilibrium distributions, mixing times, asymptotic behaviour return probabilities moments functional limit theorems.