78 research outputs found
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Periodic Little's law
In this dissertation, we develop the theory of the periodic Little's law (PLL) as well as discussing one of its applications. As extensions of the famous Little's law, the PLL applies to the queueing systems where the underlying processes are strictly or asymptotically periodic. We give a sample-path version, a steady-state stochastic version and a central-limit-theorem version of the PLL in the first part. We also discuss closely related issues such as sufficient conditions for the central-limit-theorem version of the PLL and the weak convergence in countably infinite dimensional vector space which is unconventional in queueing theory.
The PLL provides a way to estimate the occupancy level indirectly. We show how to construct a real-time predictor for the occupancy level inspired by the PLL as an example of its applications, which has better forecasting performance than the direct estimators
Polling systems with regularly varying service and/or switchover times
We consider a polling system consisting of K queues and a single server S who visits the queues in a cyclic order. The polling discipline in each queue is the gated or exhaustive service discipline. We investigate the tail behaviour of the waiting time distributions at the various queues in the case that at least one of the service time or switchover time distributions has a regularly varying tail
Scalable Performance Analysis of Massively Parallel Stochastic Systems
The accurate performance analysis of large-scale computer and communication systems is directly
inhibited by an exponential growth in the state-space of the underlying Markovian performance
model. This is particularly true when considering massively-parallel architectures
such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative
performance measures such as passage-time distributions from performance models of
these systems is critical for providers of these services. Indeed, without such an ability, they
remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have
any confidence of honouring. Additionally, this must be possible in a short enough period of
time to allow many different parameter combinations in a complex system to be tested. If we
can achieve this rapid performance analysis goal, it will enable service providers and engineers
to determine the cost-optimal behaviour which satisfies the SLAs.
In this thesis, we develop a scalable performance analysis framework for the grouped PEPA
stochastic process algebra. Our approach is based on the approximation of key model quantities
such as means and variances by tractable systems of ordinary differential equations (ODEs).
Crucially, the size of these systems of ODEs is independent of the number of interacting entities
within the model, making these analysis techniques extremely scalable. The reliability of our
approach is directly supported by convergence results and, in some cases, explicit error bounds.
We focus on extracting passage-time measures from performance models since these are very
commonly the language in which a service level agreement is phrased. We design scalable analysis
techniques which can handle passages defined both in terms of entire component populations
as well as individual or tagged members of a large population.
A precise and straightforward specification of a passage-time service level agreement is as important
to the performance engineering process as its evaluation. This is especially true of
large and complex models of industrial-scale systems. To address this, we introduce the unified
stochastic probe framework. Unified stochastic probes are used to generate a model augmentation
which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis,
we deploy these probes to define many detailed and derived performance measures that can
be automatically and directly analysed using rapid ODE techniques. In this way, we tackle
applicable problems at many levels of the performance engineering process: from specification
and model representation to efficient and scalable analysis
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