Stability Analysis of a MAP/M/s Cluster Model by Matrix-Analytic Method

Morozov, Evsey; Rumyantsev, Alexander

doi:10.1007/978-3-319-46433-6_5

Cited by 14 publications

(10 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A few papers address an FCFS multiserver-job model with more than two servers in an analytic (non-numerical) manner [1,20,22]. Each of these papers assumes that all jobs have service times drawn from a single distribution, regardless of the number of servers required by the job.…”

Section: Prior Multiserver-job Modelsmentioning

confidence: 99%

Stability for Two-class Multiserver-job Systems

Grosof¹,

Scheller‐Wolf²

2020

Preprint

View full text Add to dashboard Cite

Much is known in the dropping setting, where jobs are immediately discarded if they require more servers than are currently available. However, very little is known in the more practical setting where jobs queue instead.In this paper, we derive a closed-form analytical expression for the stability region of a two-class (nondropping) multiserver-job system where each class of jobs requires a distinct number of servers and requires a distinct exponential distribution of service time, and jobs are served in first-come-first-served (FCFS) order. This is the first result of any kind for an FCFS multiserver-job system where the classes have distinct service distributions. Our work is based on a technique that leverages the idea of a "saturated" system, in which an unlimited number of jobs are always available.Our analytical formula provides insight into the behavior of FCFS multiserver-job systems, highlighting the huge wastage (idle servers while jobs are in the queue) that can occur, as well as the nonmonotonic effects of the service rates on wastage. 1 The data was published in a scaled form [27]. We rescale the data so the smallest job in the trace uses one normalized CPU.

show abstract

Section: Prior Multiserver-job Modelsmentioning

confidence: 99%

Stability for Two-class Multiserver-job Systems

Grosof¹,

Scheller‐Wolf²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…In 2017, Rumyantsev and Morozov [38] derived the stability region for the multi-server job model where, for all class 𝑖, all jobs have the same exponential service duration 𝑆 𝑖 ∼ Exp(𝜇). This work was generalized in [34] and [2] to allow for more general arrival processes, still under the assumption that all jobs have the same exponential service duration. Very recently, Grosof et al [13] derived a simple closed-form expression for the stability region for the multi-server job model where jobs have different exponential service time durations.…”

Section: Related Workmentioning

confidence: 99%

“…Attempts to derive the steady-state distribution have assumed highly simplified systems with only 𝑛 = 2 servers [8,12], where solutions are already highly complex, involving roots to a quartic equation. Even characterizing the stability region of the system is an open problem except for the special cases where all jobs have the same service rates [2,34,38], or where there are only two job classes with different service rates [13].…”

Section: Introductionmentioning

confidence: 99%

Zero Queueing for Multi-Server Jobs

Wang¹,

Xie²

2020

Preprint

View full text Add to dashboard Cite

Cloud computing today is dominated by multi-server jobs. These are jobs that request multiple servers simultaneously and hold onto all of these servers for the duration of the job. Multi-server jobs add a lot of complexity to the traditional one-job-per-server model: an arrival might not "fit" into the available servers and might have to queue, blocking later arrivals and leaving servers idle. From a queueing perspective, almost nothing is understood about multi-server job queueing systems; even understanding the exact stability region is a very hard problem.In this paper, we investigate a multi-server job queueing model under scaling regimes where the number of servers in the system grows. Specifically, we consider a system with multiple classes of jobs, where jobs from different classes can request different numbers of servers and have different service time distributions, and jobs are served in first-come-first-served order. The multi-server job model opens up new scaling regimes where both the number of servers that a job needs and the system load scale with the total number of servers. Within these scaling regimes, we derive the first results on stability, queueing probability, and the transient analysis of the number of jobs in the system for each class. In particular we derive sufficient conditions for zero queueing. Our analysis introduces a novel way of extracting information from the Lyapunov drift, which can be applicable to a broader scope of problems in queueing systems.1 Importantly, we note that we are assuming that jobs are served in FCFS order, rather than trying to "pack" jobs into servers.FCFS is the default scheduling policy used throughout the cloud-computing industry when running multi-server jobs [27]. Even when there are multiple priority classes of jobs, as in [42], within each class the jobs are served in FCFS order.

show abstract

“…Attempts to derive the steady-state distribution have assumed highly simplified systems. Even characterizing the stability region of the system is an open problem except for the special cases where all jobs have the same service rates [1,8,9], or where there are only two job classes with different service rates [4].…”

Section: Introductionmentioning

confidence: 99%

Zero Queueing for Multi-Server Jobs

Wang

Xie

2021

Abstract Proceedings of the 2021 ACM SIGMETRICS / International Conference on Measurement and Modeling of Computer Systems

View full text Add to dashboard Cite

Cloud computing today is dominated by multi-server jobs. These are jobs that request multiple servers simultaneously and hold onto all of these servers for the duration of the job. Multi-server jobs add a lot of complexity to the traditional one-server-per-job model: an arrival might not "fit" into the available servers and might have to queue, blocking later arrivals and leaving servers idle. From a queueing perspective, almost nothing is understood about multiserver job queueing systems; even understanding the exact stability region is a very hard problem.In this paper, we investigate a multi-server job queueing model under scaling regimes where the number of servers in the system grows. Specifically, we consider a system with multiple classes of jobs, where jobs from different classes can request different numbers of servers and have different service time distributions, and jobs are served in first-come-first-served order. The multi-server job model opens up new scaling regimes where both the number of servers that a job needs and the system load scale with the total number of servers. Within these scaling regimes, we derive the first results on stability, queueing probability, and the transient analysis of the number of jobs in the system for each class. In particular we derive sufficient conditions for zero queueing. Our analysis introduces a novel way of extracting information from the Lyapunov drift, which can be applicable to a broader scope of problems in queueing systems.

show abstract

Stability Analysis of a MAP/M/s Cluster Model by Matrix-Analytic Method

Cited by 14 publications

References 15 publications

Stability for Two-class Multiserver-job Systems

Stability for Two-class Multiserver-job Systems

Zero Queueing for Multi-Server Jobs

Zero Queueing for Multi-Server Jobs

Contact Info

Product

Resources

About