10 research outputs found
Computable bounds in fork-join queueing systems
In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze.
This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non blocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing 'makes sense' from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions
СЕТИ МАССОВОГО ОБСЛУЖИВАНИЯ ПРОИЗВОЛЬНОЙ ТОПОЛОГИИ С ДЕЛЕНИЕМ И СЛИЯНИЕМ ТРЕБОВАНИЙ: СЛУЧАЙ БЕСКОНЕЧНОПРИБОРНЫХ СИСТЕМ ОБСЛУЖИВАНИЯ
В работе рассматривается класс открытых сетей массового обслужи-
вания произвольной топологии, который является развитием классиче-
ских fork-join сетей массового обслуживания, используемых в качестве
математических моделей стохастических систем с параллельным и рас-
пределенным принципом функционирования (GRID-системы, RAID-
массивы, MapReduce и т. д.)
Системы обслуживания в рассматриваемой сети поделены на три типа
в зависимости от их назначения: бесконечноприборные базовые систе-
мы, дивайдеры, интеграторы. Наличие бесконечного числа обслужива-
ющих приборов в базовых системах позволяет существенно упростить
анализ и рассмотреть сети обслуживания с произвольной топологией.
Требование, поступающее в дивайдер, делится на некоторое число частей — фрагментов. Полученные фрагменты обслуживаются независимо друг от друга в базовых системах сети, переходят по сети. Каждый
из фрагментов может снова поделиться на фрагменты при поступлении в дивайдер. Объединение фрагментов происходит в интеграторах. Так, перед уходом из сети все фрагменты объединяются в одном из интеграторов в исходное требование. Главным результатом работы стал метод получения распределения
длительности пребывания требований в изучаемой сети массового обслуживани
Approximations and Bounds for (n, k) Fork-Join Queues: A Linear Transformation Approach
Compared to basic fork-join queues, a job in (n, k) fork-join queues only
needs its k out of all n sub-tasks to be finished. Since (n, k) fork-join
queues are prevalent in popular distributed systems, erasure coding based cloud
storages, and modern network protocols like multipath routing, estimating the
sojourn time of such queues is thus critical for the performance measurement
and resource plan of computer clusters. However, the estimating keeps to be a
well-known open challenge for years, and only rough bounds for a limited range
of load factors have been given. In this paper, we developed a closed-form
linear transformation technique for jointly-identical random variables: An
order statistic can be represented by a linear combination of maxima. This
brand-new technique is then used to transform the sojourn time of non-purging
(n, k) fork-join queues into a linear combination of the sojourn times of basic
(k, k), (k+1, k+1), ..., (n, n) fork-join queues. Consequently, existing
approximations for basic fork-join queues can be bridged to the approximations
for non-purging (n, k) fork-join queues. The uncovered approximations are then
used to improve the upper bounds for purging (n, k) fork-join queues.
Simulation experiments show that this linear transformation approach is
practiced well for moderate n and relatively large k.Comment: 10 page
Collaborative Uploading in Heterogeneous Networks: Optimal and Adaptive Strategies
Collaborative uploading describes a type of crowdsourcing scenario in
networked environments where a device utilizes multiple paths over neighboring
devices to upload content to a centralized processing entity such as a cloud
service. Intermediate devices may aggregate and preprocess this data stream.
Such scenarios arise in the composition and aggregation of information, e.g.,
from smartphones or sensors. We use a queuing theoretic description of the
collaborative uploading scenario, capturing the ability to split data into
chunks that are then transmitted over multiple paths, and finally merged at the
destination. We analyze replication and allocation strategies that control the
mapping of data to paths and provide closed-form expressions that pinpoint the
optimal strategy given a description of the paths' service distributions.
Finally, we provide an online path-aware adaptation of the allocation strategy
that uses statistical inference to sequentially minimize the expected waiting
time for the uploaded data. Numerical results show the effectiveness of the
adaptive approach compared to the proportional allocation and a variant of the
join-the-shortest-queue allocation, especially for bursty path conditions.Comment: 15 pages, 11 figures, extended version of a conference paper accepted
for publication in the Proceedings of the IEEE International Conference on
Computer Communications (INFOCOM), 201