41 research outputs found
Performance of the Gittins Policy in the G/G/1 and G/G/k, With and Without Setup Times
How should we schedule jobs to minimize mean queue length? In the preemptive
M/G/1 queue, we know the optimal policy is the Gittins policy, which uses any
available information about jobs' remaining service times to dynamically
prioritize jobs. For models more complex than the M/G/1, optimal scheduling is
generally intractable. This leads us to ask: beyond the M/G/1, does Gittins
still perform well?
Recent results indicate that Gittins performs well in the M/G/k, meaning that
its additive suboptimality gap is bounded by an expression which is negligible
in heavy traffic. But allowing multiple servers is just one way to extend the
M/G/1, and most other extensions remain open. Does Gittins still perform well
with non-Poisson arrival processes? Or if servers require setup times when
transitioning from idle to busy?
In this paper, we give the first analysis of the Gittins policy that can
handle any combination of (a) multiple servers, (b) non-Poisson arrivals, and
(c) setup times. Our results thus cover the G/G/1 and G/G/k, with and without
setup times, bounding Gittins's suboptimality gap in each case. Each of (a),
(b), and (c) adds a term to our bound, but all the terms are negligible in
heavy traffic, thus implying Gittins's heavy-traffic optimality in all the
systems we consider. Another consequence of our results is that Gittins is
optimal in the M/G/1 with setup times at all loads.Comment: 41 page
Optimal Scheduling in the Multiserver-job Model under Heavy Traffic
Multiserver-job systems, where jobs require concurrent service at many
servers, occur widely in practice. Essentially all of the theoretical work on
multiserver-job systems focuses on maximizing utilization, with almost nothing
known about mean response time. In simpler settings, such as various known-size
single-server-job settings, minimizing mean response time is merely a matter of
prioritizing small jobs. However, for the multiserver-job system, prioritizing
small jobs is not enough, because we must also ensure servers are not
unnecessarily left idle. Thus, minimizing mean response time requires
prioritizing small jobs while simultaneously maximizing throughput. Our
question is how to achieve these joint objectives.
We devise the ServerFilling-SRPT scheduling policy, which is the first policy
to minimize mean response time in the multiserver-job model in the heavy
traffic limit. In addition to proving this heavy-traffic result, we present
empirical evidence that ServerFilling-SRPT outperforms all existing scheduling
policies for all loads, with improvements by orders of magnitude at higher
loads.
Because ServerFilling-SRPT requires knowing job sizes, we also define the
ServerFilling-Gittins policy, which is optimal when sizes are unknown or
partially known.Comment: 32 pages, to appear in ACM SIGMETRICS 202
Uniform Bounds for Scheduling with Job Size Estimates
We consider the problem of scheduling to minimize mean response time in M/G/1 queues where only estimated job sizes (processing times) are known to the scheduler, where a job of true size s has estimated size in the interval [? s, ? s] for some ? ? ? > 0. We evaluate each scheduling policy by its approximation ratio, which we define to be the ratio between its mean response time and that of Shortest Remaining Processing Time (SRPT), the optimal policy when true sizes are known. Our question: is there a scheduling policy that (a) has approximation ratio near 1 when ? and ? are near 1, (b) has approximation ratio bounded by some function of ? and ? even when they are far from 1, and (c) can be implemented without knowledge of ? and ??
We first show that naively running SRPT using estimated sizes in place of true sizes is not such a policy: its approximation ratio can be arbitrarily large for any fixed ? < 1. We then provide a simple variant of SRPT for estimated sizes that satisfies criteria (a), (b), and (c). In particular, we prove its approximation ratio approaches 1 uniformly as ? and ? approach 1. This is the first result showing this type of convergence for M/G/1 scheduling.
We also study the Preemptive Shortest Job First (PSJF) policy, a cousin of SRPT. We show that, unlike SRPT, naively running PSJF using estimated sizes in place of true sizes satisfies criteria (b) and (c), as well as a weaker version of (a)
Performance Analysis of Modified SRPT in Multiple-Processor Multitask Scheduling
In this paper we study the multiple-processor multitask scheduling problem in
both deterministic and stochastic models. We consider and analyze Modified
Shortest Remaining Processing Time (M-SRPT) scheduling algorithm, a simple
modification of SRPT, which always schedules jobs according to SRPT whenever
possible, while processes tasks in an arbitrary order. The M-SRPT algorithm is
proved to achieve a competitive ratio of for
minimizing response time, where denotes the ratio between maximum job
workload and minimum job workload, represents the ratio between maximum
non-preemptive task workload and minimum job workload. In addition, the
competitive ratio achieved is shown to be optimal (up to a constant factor),
when there are constant number of machines. We further consider the problem
under Poisson arrival and general workload distribution (\ie, system),
and show that M-SRPT achieves asymptotic optimal mean response time when the
traffic intensity approaches , if job size distribution has finite
support. Beyond finite job workload, the asymptotic optimality of M-SRPT also
holds for infinite job size distributions with certain probabilistic
assumptions, for example, system with finite task workload
SEH: Size Estimate Hedging for Single-Server Queues
For a single server system, Shortest Remaining Processing Time (SRPT) is an
optimal size-based policy. In this paper, we discuss scheduling a single-server
system when exact information about the jobs' processing times is not
available. When the SRPT policy uses estimated processing times, the
underestimation of large jobs can significantly degrade performance. We propose
a simple heuristic, Size Estimate Hedging (SEH), that only uses jobs' estimated
processing times for scheduling decisions. A job's priority is increased
dynamically according to an SRPT rule until it is determined that it is
underestimated, at which time the priority is frozen. Numerical results suggest
that SEH has desirable performance when estimation errors are not unreasonably
large
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