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Scaling limits for some Mittag-Leffler queues

Giacomo Ascione, Luigia Caputo

Abstract

In this paper, we consider five models of heavy-tailed queues involving Mittag-Leffler distributions that generalize the classical $M/M/1$ queues. These models are suitable modifications of previously defined models in such a way that the classical $M/M/1$ queue can be recovered by a suitable selection of parameters. We provide the distribution of inter-arrival and service times of both the original and modified queueing models. We then study the scaling limits of all the proposed models and we argue that the behaviour of the limiting processes can be used to characterise the traffic regime of the queues.

Scaling limits for some Mittag-Leffler queues

Abstract

In this paper, we consider five models of heavy-tailed queues involving Mittag-Leffler distributions that generalize the classical queues. These models are suitable modifications of previously defined models in such a way that the classical queue can be recovered by a suitable selection of parameters. We provide the distribution of inter-arrival and service times of both the original and modified queueing models. We then study the scaling limits of all the proposed models and we argue that the behaviour of the limiting processes can be used to characterise the traffic regime of the queues.
Paper Structure (40 sections, 42 theorems, 184 equations)

This paper contains 40 sections, 42 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. The Queueing System Models
  3. Single server queueing systems
  4. The $M/M/1$ queue
  5. The $M/M/1$ as a single server queueing system
  6. The $M/M/1$ as a Markov-renewal process
  7. The $M/M/1$ as the reflection of a continuous-time random walk
  8. The $M/M/1$ as a reflected difference of competing Poisson processes
  9. Model 1: the fractional M/M/1 queue
  10. Model 2: the fast renewal Mittag-Leffler queue
  11. Model 3: the renewal Mittag-Leffler queue
  12. Model 4: the restless Mittag-Leffler queue
  13. Model 5: the Mittag-Leffler GI/GI/1 queue
  14. Scaling limits of Mittag-Leffler queues: Main Results
  15. The scaling limits of the classical $M/M/1$ queue: a summary
  16. ...and 25 more sections

Key Result

Proposition 2.1

For a fractional $M/M/1$ queue of order $\alpha \in (0,1]$ and inter-arrival and service rates $\lambda,\mu>0$, the sequences $(T_k)_{k \ge 1}$ and $(S_k)_{k \ge 1}$ are constituted by i.i.d. random variables and it holds The two sequences, however, are not independent of each other unless $\alpha=1$.

Theorems & Definitions (55)

  • Proposition 2.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 45 more