Fractional Erlang queues

Abstract

We introduce a fractional generalization of the Erlang Queues M/E k /1. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward equation for such process, then we use such equation to obtain an interpretation of this process in the queuing theory context. Then we also exploit the transient state probabilities and some features of this fractional queue model, such as the mean queue length, the distribution of the busy periods and some conditional distributions of the waiting times. Finally, we provide some algorithms to simulate their sample paths.

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

“…Comparing (3) with (10), it is clear that σ α (1) ∼ S α, 1, cos 1 α π 2 α , 0 . The latter observation can be used to state the following lemma (see also [23] Proof. Let σ α be an α-stable subordinator independent of Y.…”

“…Together with a purely mathematical interest, let us stress that this procedure leads also to some interesting applications (see, for instance, [18] in economics, [19] in computing and [20] in computational neurosciences). In the context of queueing theory, this has been first done in [4], in the case of the M/M/1 queue, and then extended in [21,22] to the case of M/M/1 queues with catastrophes, [23] in the case of Erlang queues and [24] for the M/M/∞ queues.…”

Fractional immigration-death processes

“…With the same approach, some classes of fractional birth-death processes have been introduced and studied [37][38][39]: in these papers, properties of these processes are deduced from a fractional version of their Kolmogorov forward equation. Let us recall that fractional processes are shown to be interesting in different application contexts, as, for instance, queueing theory ( [4,5,11]). Here, following the approach of [25], we show the existence of strong solutions for the time-fractional counterpart of the Kolmogorov backward and forward equations of immigration-death processes with the aid of Charlier polynomials and link them to a time-changed immigration-death process.…”

Non-Local Solvable Birth-Death Processes

“…The same approach has been used to study the case of fractional immigration-death processes in [5]. Let us also stress out that this approach can be used to study a fractional M/M/∞ queue or a fractional M/M/1 queue with acceleration of service (for some models of fractional queues, we refer to [4,6,15]). However, one could consider a time-change with a different inverse subordinator.…”