On optimal right-of-way policies at a single-server station when insertion of idle times is permitted

A general stream of n types of customers arrives at a Single Server station where service is non-preemptive, the server may undergo Poisson breakdowns and insertion of idle times is allowed. If ξ(k) and c(k) are, respectively, the expected service time and sojourn cost per unit time of a type k customer (1≤k≤n), call k "V.I.P." type if ξ(k)/c(k) = min_1≤i≤n[ξ(i)/sbc(i)]. We show that any right-of-way service policy can be improved by a policy that grants V.I.P. customers priority over all others, and never inserts idle time when a V.I.P. customer is present. We further show that if the arrival stream is Poisson, the so-called "cμ" priority rule (applied with no delays) is optimal in the class of all service policies, and not just among those of a priority nature.