43 research outputs found
A large-deviations analysis of the GI/GI/1 SRPT queue
We consider a GI/GI/1 queue with the shortest remaining processing time
discipline (SRPT) and light-tailed service times. Our interest is focused on
the tail behavior of the sojourn-time distribution. We obtain a general
expression for its large-deviations decay rate. The value of this decay rate
critically depends on whether there is mass in the endpoint of the service-time
distribution or not. An auxiliary priority queue, for which we obtain some new
results, plays an important role in our analysis. We apply our SRPT-results to
compare SRPT with FIFO from a large-deviations point of view.Comment: 22 page
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
Sojourn time asymptotics in processor sharing queues
This paper addresses the sojourn time asymptotics for a GI/GI/âą queue operating under the
Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the
logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize
distribution has a light tail. Whereas upper bounds on the decay rate can be derived under
fairly general conditions, the establishment of the corresponding lower bounds requires that the
service process satisfies a samplepath large-deviation principle. We show that the class of
allowed service processes includes the case where the service rate is modulated by a Markov
process. Finally, we extend our results to a similar system operation under the Discriminatory
Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations
techniques
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
PSBS: Practical Size-Based Scheduling
Size-based schedulers have very desirable performance properties: optimal or
near-optimal response time can be coupled with strong fairness guarantees.
Despite this, such systems are very rarely implemented in practical settings,
because they require knowing a priori the amount of work needed to complete
jobs: this assumption is very difficult to satisfy in concrete systems. It is
definitely more likely to inform the system with an estimate of the job sizes,
but existing studies point to somewhat pessimistic results if existing
scheduler policies are used based on imprecise job size estimations. We take
the goal of designing scheduling policies that are explicitly designed to deal
with inexact job sizes: first, we show that existing size-based schedulers can
have bad performance with inexact job size information when job sizes are
heavily skewed; we show that this issue, and the pessimistic results shown in
the literature, are due to problematic behavior when large jobs are
underestimated. Once the problem is identified, it is possible to amend
existing size-based schedulers to solve the issue. We generalize FSP -- a fair
and efficient size-based scheduling policy -- in order to solve the problem
highlighted above; in addition, our solution deals with different job weights
(that can be assigned to a job independently from its size). We provide an
efficient implementation of the resulting protocol, which we call Practical
Size-Based Scheduler (PSBS). Through simulations evaluated on synthetic and
real workloads, we show that PSBS has near-optimal performance in a large
variety of cases with inaccurate size information, that it performs fairly and
it handles correctly job weights. We believe that this work shows that PSBS is
indeed pratical, and we maintain that it could inspire the design of schedulers
in a wide array of real-world use cases.Comment: arXiv admin note: substantial text overlap with arXiv:1403.599
Is Tail-Optimal Scheduling Possible?
This paper focuses on the competitive analysis of scheduling disciplines in a large deviations setting. Although there are policies that are known to optimize the sojourn time tail under a large class of heavy-tailed job sizes (e.g., processor sharing and shortest remaining processing time) and there are policies known to optimize the sojourn time tail in the case of light-tailed job sizes (e.g., first come first served), no policies are known that can optimize the sojourn time tail across both light- and heavy-tailed job size distributions. We prove that no such work-conserving, nonanticipatory, nonlearning policy exists, and thus that a policy must learn (or know) the job size distribution in order to optimize the sojourn time tail
Scheduling for the tail: Robustness versus optimality
When scheduling to minimize the sojourn time tail, the goals of optimality and robustness are seemingly at odds. Over the last decade, results have emerged which show that scheduling disciplines that are near-optimal under light (exponential) tailed workload distributions do not perform well under heavy (power) tailed workload distributions, and vice-versa. Very recently, it has been shown that this conflict between optimality and robustness is fundamental, i.e., no policy that does not learn information about the workload can be optimal across both light-tailed and heavy-tailed workloads. In this paper we show that one can exploit very limited workload information (the system load) in order to design a scheduler that provides robust performance across heavy-tailed and light-tailed workloads