Boxma, Onno; Mandjes, Michel; Reed, Josh On a class of reflected \(\mathrm{AR}(1)\) processes. (English) Zbl 1351.60121 J. Appl. Probab. 53, No. 3, 818-832 (2016). Summary: We study a reflected \(\mathrm{AR}(1)\) 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 \(\mathrm{Exp}(\lambda)\) 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. Cited in 9 Documents MSC: 60K25 Queueing theory (aspects of probability theory) 60J05 Discrete-time Markov processes on general state spaces 90B22 Queues and service in operations research Keywords:reflected process; queueing; scaling limit × Cite Format Result Cite Review PDF Full Text: DOI Link