Higher-order Lindley equations
This article is devoted to an analysis of the generalized Lindley equation \[ w \cong \max \bigl[ w^{(1)} + s^{(1)} - \tau^{(1)}, \dots, w^{(m)} + s^{(m)} - \tau^{(m)} \bigr] \] where the \(w\) all have the same unknown distribution, and the \(s\) as well as the \(\tau\) are i.i.d. The authors begin by pointing out that this equation is applicable to queueing network problems in areas such as operations research and computer systems. They then apply the theory of random marked flows to state conditions under which there is weak convergence to the minimal solution of the above equation. They further characterize this solution in terms of the unique probability one solution to another equation. It follows also that if the fundamental stability condition (earlier required for existence) is violated, then no solution to the generalized Lindley equation exists. Finally, the nature of the solution is further characterized and described.
scientific article; zbMATH DE number 691018