A multiplicative version of the Lindley recursion

Onno Boxma, Andreas Löpker, Michel Mandjes, Zbigniew Palmowski (Corresponding author)

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

This paper presents an analysis of the stochastic recursion Wi+1=[ViWi+Yi]+ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing Yi= Bi- Ai, for independent sequences of nonnegative i.i.d. random variables {Ai}i∈N0 and {Bi}i∈N0, and assuming {Vi}i∈N0 is an i.i.d. sequence as well (independent of {Ai}i∈N0 and {Bi}i∈N0), we then consider three special cases (i) Vi equals a positive value a with certain probability p∈ (0 , 1) and is negative otherwise, and both Ai and Bi have a rational LST, (ii) Vi attains negative values only and Bi has a rational LST, (iii) Vi is uniformly distributed on [0, 1], and Ai is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.

Original languageEnglish
JournalQueueing Systems
VolumeXX
Issue numberXX
DOIs
Publication statusAccepted/In press - 2021

Keywords

  • Autoregressive models
  • Laplace transform
  • Lindley recursion
  • Wiener–Hopf boundary value problem

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