Diffusion Limits for Shortest Remaining Processing Time Queues with Multiple Customer Types

We consider a single-server queueing system with multiple customer types having bounded processing times in which users are scheduled according to the Shortest Remaining Processing Time (SRPT) discipline, with First In First Out (FIFO) as the tie-breaker. We assume that the processing times of jobs arriving in the system are bounded. We use probabilistic methods to find, under typical heavy traffic assumptions, a suitable approximation of the workload and queue length processes after a long time has passed and show that these processes are divided among the customer classes according to specific proportions, depending on their arrival rates and distributions of initial service times. Our results are confirmed by simulations.