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.

In this paper we consider a system with two carousels operated by one picker. The items to be picked are randomly located on the carousels and the pick times follow a phase-type distribution. The picker alternates between the two carousels, picking one item at a time. Important performance characteristics are the waiting time of the picker and the throughput of the two carousels. The waiting time of the picker satisfies an equation very similar to Lindley's equation for the waiting time in the P H/U/1 queue. Although the latter equation has no simple solution, we show that the one for the waiting time of the picker can be solved explicitly. Furthermore, it is well known that the mean waiting time in the P H/U/1 queue depends on to the complete interarrival time distribution, but numerical results show that, for the carousel system, the mean waiting time and throughput are rather insensitive to the pick-time distribution.

We consider a system consisting of a server alternating between two service points+ At both service points, there is an infinite queue of customers that have to undergo a preparation phase before being served+ We are interested in the waiting time of the server+ The waiting time of the server satisfies an equation very similar to Lindley's equation for the waiting time in the GI0G01 queue+ We will analyze this Lindley-type equation under the assumptions that the preparation phase follows a phase-type distribution, whereas the service times have a general distribution+ If we relax the condition that the server alternates between the service points, then the model turns out to be the machine repair problem+ Although the latter is a wellknown problem, the distribution of the waiting time of the server has not been studied yet+ We derive this distribution under the same setting and we compare the two models numerically+ As expected, the waiting time of the server is, on average, smaller in the machine repair problem than in the alternating service system, but they are not stochastically ordered+

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Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a prespecified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.