A single server queue with renewal arrivals and i.i.d. service times will be considered. Two service disciplines will be discussed.
Under the Shortest Remaining Processing Time (SRPT), the server gives preemptive priority to the job which can be completed first. In contrast, the server working under the Longest Remaining Time First (LRTF) policy assigns preemptive priority to the job with the longest remaining processing time. Both SRPT and LRTF are examples of so-called size-based service disciplines, analyzed in the operations research and computer science literature. In a queuing system of this kind, information about the remaining service time of each task needs to be stored. Accordingly, it is convenient to model the evolution of such a queue using measure-valued state descriptors.
In this talk, I will focus on fluid (i.e., functional law of large numbers) limits for the evolution of the systems described above. After reviewing the results of Down, Gromoll and Puha (2009) on fluid models and fluid limits for a single SRPT queue, I will discuss their counterparts for a multiple-input single server SRPT queue with K customer classes (joint work with Ewa Sokołowska). I will also present analogous results for a single queue under the LRTF service policy.