Diffusion limits for shortest remaining processing time queues

We present a heavy traffic analysis for a single server queue with renewal arrivals and generally distributed i.i.d. service times, in which the server employs the Shortest Remaining Processing Time (SRPT) policy. Under typical heavy traffic assumptions, we prove a diffusion limit theorem for a measure-valued state descriptor, from which we conclude a similar theorem for the queue length process. These results allow us to make some observations on the queue length optimality of SRPT. In particular, they provide the sharpest illustration of the well-known tension between queue length optimality and quality of service for this policy.Comment: 19 pages; revised, fixed typos. To appear in Stochastic System

Invariance of fluid limits for the Shortest Remaining Processing Time and Shortest Job First policies

We consider a single-server queue with renewal arrivals and i.i.d. service times, in which the server employs either the preemptive Shortest Remaining Processing Time (SRPT) policy, or its non-preemptive variant, Shortest Job First (SJF). We show that for given stochastic primitives (initial condition, arrival and service processes), the model has the same fluid limit under either policy. In particular, we conclude that the well-known queue length optimality of preemptive SRPT is also achieved, asymptotically on fluid scale, by the simpler-to-implement SJF policy. We also conclude that on fluid scale, SJF and SRPT achieve the same performance with respect to response times of the longest-waiting jobs in the system.Comment: 24 page

Load Balancing in the Non-Degenerate Slowdown Regime

We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a classical load balancing policy for queueing systems with multiple parallel servers. Parallel server queueing systems are regularly analysed and dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling regime. However, when jobs must be dispatched to a server upon arrival, we advocate the Non-Degenerate Slowdown regime (NDS) to compare different load-balancing rules. In this paper we identify novel diffusion approximation and timescale separation that provides insights into the performance of JSQ. We calculate the price of irrevocably dispatching jobs to servers and prove this to within 15% (in the NDS regime) of the rules that may manoeuvre jobs between servers. We also compare ours results for the JSQ policy with the NDS approximations of many modern load balancing policies such as Idle-Queue-First and Power-of-$d$-choices policies which act as low information proxies for the JSQ policy. Our analysis leads us to construct new rules that have identical performance to JSQ but require less communication overhead than power-of-2-choices.Comment: Revised journal submission versio

SRPT Scheduling Discipline in Many-Server Queues with Impatient Customers

The shortest-remaining-processing-time (SRPT) scheduling policy has been extensively studied, for more than 50 years, in single-server queues with infinitely patient jobs. Yet, much less is known about its performance in multiserver queues. In this paper, we present the first theoretical analysis of SRPT in multiserver queues with abandonment. In particular, we consider the M/GI/s+GI queue and demonstrate that, in the many-sever overloaded regime, performance in the SRPT queue is equivalent, asymptotically in steady state, to a preemptive two-class priority queue where customers with short service times (below a threshold) are served without wait, and customers with long service times (above a threshold) eventually abandon without service. We prove that the SRPT discipline maximizes, asymptotically, the system throughput, among all scheduling disciplines. We also compare the performance of the SRPT policy to blind policies and study the effects of the patience-time and service-time distributions

Fluid Limits for Shortest Job First with Aging

We investigate fluid scaling of single server queueing systems under the shortest job first with aging (SJFA) scheduling policy. We use the measure-valued Skorokhod map to characterize the fluid limit for SJFA queues with a general aging rule and establish convergence results to the fluid limit. We treat in detail examples of linear and exponential aging