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Modified Lindley process with replacement: Dynamic behavior, asymptotic decomposition and applications. (English) Zbl 0698.60076
The authors consider a discrete-time stochastic process $$\{W_ n$$, $$n\in {\mathbb{N}}_ 0\}$$ with $$W_{n+1}=W_ n+\xi_{n+1}$$ if $$W_ n+\xi_{n+1}=:A_ n\geq 0$$ and $$W_{n+1}=R_{n+1}$$ if $$A_ n<0$$ for given $$W_ 0$$. The $$\xi_ n$$, $$n\in {\mathbb{N}}$$, are real i.i.d. random variables, the $$R_ n$$, $$n\in {\mathbb{N}}$$, are non-negative random variables, and the distribution of $$R_{n+1}$$ depends only on the value of $$W_ n+\xi_{n+1}.$$
It is shown that $$\{W_ n$$, $$n\in {\mathbb{N}}_ 0\}$$ is ergodic if $$E(\xi_ n<0)$$ and $$R_ n\to^{d}R_{\infty}$$ as $$n\to \infty$$ and $$R_{\infty}$$ is honest. Under these conditions $$\{W_ n\}$$ converges to $$W_{\infty}$$ in distribution as $$n\to \infty$$. Furthermore, as a central result is shown the asymptotic decomposition $$W_{\infty}=X_{\infty}+W^ L_{\infty}$$ with two independent random variables $$X_{\infty}$$ and $$W^ L_{\infty}$$ defined as follows. $$W^ L_{\infty}$$ is the limit of $$\{W^ L_ n\}$$ for $$n\to \infty$$ with $W^ L_{n+1}=\max [W^ L_ n+\xi_{n+1},0]\quad (ordinary\quad Lindley\quad process),$ and $$X_{\infty}$$ is the limit of $$\{X_ n\}$$ for $$n\to \infty$$ with $X_{n+1}=X_ n+\min [W^ L_ n+\xi_{n+1},0]=:B_ n\quad if\quad B_ n\geq 0\quad and\quad X_{n+1}=R_{n+1}\quad if\quad B_ n<0$ starting with $$X_ 0=W_ 0$$. This decomposition theorem contains recent results of B. T. Doshi [ibid. 22, 419-428 (1985; Zbl 0566.60090)] and J. Keilson and L. D. Servi [see ibid. 23, 790-802 (1986; Zbl 0612.60087)] as special cases. The authors show that these general results have useful applications in a variety of single server queueing models with several types of vacation or priority policies.
Reviewer: H.Schellhaas

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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