### Abstract

In this paper, we study a reflected AR(1) process, i.e., a process $(Z_n)_n$ obeying the recursion
$Z_{n+1}$ = max\{$aZ_n + X_n, 0$\}, with $(X_n)_n$ a sequence of i.i.d. random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case $X_n$ can be written as $Y_n - B_n$, with $(B_n)_n$ being a sequence of independent random variables which are all
exp($\lambda$) distributed, and $(Y_n)_n$ i.i.d.; when $|a| <1$ we can also perform the corresponding
stationary analysis. Extensions are possible to the case that $(B_n)_n$ are of phase-type.
Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein-Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a Normal random variable conditioned on being positive.
Keywords: Reflected processes . queueing . scaling limits

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Eurandom |

Number of pages | 16 |

Publication status | Published - 2015 |

### Publication series

Name | Report Eurandom |
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Volume | 2015012 |

ISSN (Print) | 1389-2355 |

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## Cite this

Boxma, O. J., Mandjes, M. R. H., & Reed, J. (2015).

*On a class of reflected AR(1) processes*. (Report Eurandom; Vol. 2015012). Eurandom.