Bruce's Spider and the Employee's Risk under a Pension System
通过半马尔可夫过程推广了布鲁斯蜘蛛爬墙模型，用于分析养老金制度下雇员因离职而无法获得既定养老金权益的概率，对精算师和养老金政策制定者有参考价值。
Robert the Bruce's spider' attempted to climb a wall of height s at unit speed. At each attempt, it reached either a height X < s and then slipped back to the base of the wall where it rested for a time Y before it started to climb again, or it reached the height s successfully. What is the distribution of the total time the spider takes to climb the wall successfully, if X and Y are non-negative random variables with distribution functions F(x) and G(x) respectively? This problem was introduced and solved by Barton and David [4] as a persistence problem in renewal theory, for negative exponential F(x) and a fixed resting time Y. Later, Dowton [7] reported a number of generalizations, including a reformulation as an alternating renewal process with absorption, using general distributions for climbing and resting times. He also suggested certain vehicle-rental and equipment-breakdown problems as possible application areas. In this note, the author presents a further generalization of the model through a semi-Markov process in the context of an employment pension system.2 The main aim is to determine the probability of no vested pension for a typical employee accruing from his or her career membership in employment pension plans under an arbitrary but fixed vesting requirement. The paper starts from the theoretical framework established in Sahin [8] where a two-state semi-Markov process was constructed to investigate the accumulation of pensionable service during an individual's working life. This theory has been recently extended to pension benefits and pension costs in Balcer and Sahin [3]. Results obtained in these studies are in terms of a number of basic functions representing the moments (in particular, mean and variance) of the underlying distributions.