A non-increasing Lindley-type equation

Abstract

Abstract. In this paper we study the Lindley-type equation W = max{0, B − A − W }. Its main characteristic is that it is a non-increasing monotone function in its main argument W . Our main goal is to derive a closed-form expression of the steady-state distribution of W . In general this is not possible, so we shall state a sufficient condition that allows us to do so. We also examine stability issues, derive the tail behaviour of W , and briefly discuss how one can iteratively solve this equation by using a contraction mapping.

An Operator Property of the Distribution of a Nonhomogeneous Poisson Process with Applications

“…Then, we have w(x) = (1 + sin x) e −sx and w(x + y) = e −sx e −sy + e −sx sin x e −sy cos y + e −sx cos x e −sy sin y , see also some details in Asmussen (2003, p. 89) and Vlasiou (2007).…”

“…In the case where the function w(x) is a tail distribution, the decomposition given in Eq. 3 have been used independently by Willmot (2007), Vlasiou (2007) and Lefévre (2007). In fact, in the renewal model of risk theory, Willmot (2007), see also Landriault and Willmot (2008), used a similar factorization as Eq.…”

“…3 for the tail of claim size distribution and studied some analytical expressions for the Gerber-Shiu discounted penalty function. Furthermore, Vlasiou (2007) studying a Lindleytype equation in a queueing system of one server and two service point used a similar decomposition. Finally, Lefévre (2007), used this type of factorization studying polynomial structures and recursions in the discrete compound Poisson process with curved boundaries.…”

Server Waiting Times in Infinite Supply Polling Systems With Preparation Times

“…A similar result under more general assumptions on the distribution of A and B seems hard to obtain, unless N = 2 (cf. [23]). …”

“…is not smaller than each of the arguments in the outer maximum operator in the right-hand side of (23). For the first argument, we have by using (22) and (21), respectively, that…”