On a class of reflected AR(1) processes. (September 2016)

• Record Type: Journal Article
• Authors: Boxma, Onno; Mandjes, Michel; Reed, Josh
• Is Part Of: Journal of applied probability. Volume 53:Number 3(2016)
• Page Start: 818
• Page End: 832
• Publication Date: 2016-09
• Subjects: Reflected process, -- queueing, -- scaling limit; Primary 60K25, -- Secondary 60J05; 519.2
• DOI: 10.1017/jpr.2016.42
• Languages: English
• ISSNs: 0021-9002

Abstract

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 independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Z n (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(λ) 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.