On a class of renewal queueing and risk processes

Purpose – The purpose of this paper is to investigate how queueing theory has been applied to derive results for a Sparre Andersen risk process for which the claim inter‐arrival distribution is hyper Erlang. Design/methodology/approach – The paper exploits the duality results between the queueing theory and risk processes to derive explicit expressions for the ultimate ruin probability and moments of time to ruin in this renewal risk model. Findings – This paper derives explicit expressions for the Laplace transforms of the idle/waiting time distribution in GI /HE r ( k i , λ i )/1 and its dual HE r ( k i , λ i )/ G /1. As a consequence, an expression for the ultimate ruin probability is obtained in this model. The relationship between the time of ruin and busy period in M / G /1 queuing system is used to derive the expected time of ruin. Originality/value – The study of renewal risk process is mostly concentrated on Erlang distributed inter‐claim times. But the Erlang distributions are not dense in the space of all probability distributions and therefore, the paper cannot approximate an arbitrary distribution function by an Erlang one. To overcome this difficulty, the paper uses the hyper Erlang distributions, which can be used to approximate the distribution of any non‐negative random variable.