We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

The ‘Price of Anarchy’ states that the performance of multi-agent service systems degrades with the agents’ selfishness (anarchy). We investigate a service model in which both customers and the firm are strategic. We find that, for a Stackelberg game in which the server invests in capacity before customers decide whether or not to join, there can be a ‘Benefit of Anarchy’, that is, customers acting selfishly can have a greater overall utility than customers who are coordinated to maximize their overall utility. We also show that customer anarchy can be socially beneficial, resulting in a ‘Social Benefit of Anarchy’. We show that such phenomena are rather general and can arise in multiple settings (e.g. in both profit-maximizing and welfare-maximizing firms, in both capacity-setting and price-setting firms, and in both observable and unobservable queues). However, the underlying mechanism leading to the Benefit of Anarchy can differ significantly from one setting to another.

We would like to correct the statement of Lemma 4.1 in [BDK+18].

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.