Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

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Abstract
We consider a discrete-time stochastic process {Wn,n≧0} governed by i.i.d random variables {ξn} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). GivenWn, Wn+ 1takes the valueWn+ζn+ 1if it is non-negative. OtherwiseWn+ 1takes the valueRn +1where the distribution ofRn+ 1depends only on the value ofWn+ ζn +1.This stochastic process is reduced to the ordinary Lindley process forGI/G/1 queues whenRn= 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

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