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A non-increasing Lindley-type equation

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.