Modified Lindley process with replacement: Dynamic behavior, asymptotic decomposition and applications. (English) Zbl 0698.60076

Shanthikumar, J. George; Sumita, Ushio

J. Appl. Probab. 26, No. 3, 552-565 (1989).

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

Keywords:

Lindley process; single server queueing models with vacation or priority policies; asymptotic decomposition; types of vacation or priority policies