The authors 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,•). Given Wn, WnÅ+1 takes the value wn+ξnÅ+1 if it is non-negative. Otherwise WnÅ+1 takes the value RnÅ+1 where the distribution of RnÅ+1 depends only on the value of Wn+ξnÅ+1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn=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 and Keilson and Servi as special cases, thereby providing a unified view.
