On the non-optimality of the FB discipline within the service time distribution class IMRL

It is known that, within the service time distribution class DHR, the FB discipline minimizes the mean delay in the M/G/1 queue among all work-conserving and non-anticipating service disciplines. It is also believed that a similar result is valid within a more general distribution class IMRL. However, we point out a flaw in the existing proof of this latter result that cannot be overcome. Moreover, by constructing a counter-example, we demonstrate that FB is not optimal within class IMRL. On the other hand, we prove that the mean delay for FB is smaller than that of PS within class IMRL giving a weaker version of an earlier hypothesis