A note on the event horizon for a processor sharing queue

In this note we identify a phenomenon for processor sharing queues that is unique to ones with time-varying rates. This property was discovered while correcting a proof in Hampshire, Harchol-Balter and Massey (Queueing Syst. 53(1-2), 19-30, 2006). If the arrival rate for a processor sharing queue has unbounded growth over time, then it is possible for the number of customers in a processor sharing queue to grow so quickly that a newly entering job never finishes. We define the minimum size for such a job to be the event horizon for a processor sharing queue. We discuss the use of such a concept and develop some of its properties. This short article serves both as errata for Hampshire, Harchol-Balter and Massey (Queueing Syst. 53(1-2), 19-30, 2006) and as documentation of a characteristic feature for some processor sharing queues with time varying rates.