47 research outputs found
Queues with delays in two-state strategies and Lévy input
We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires. © Applied Probability Trust 2008
Fully Unleashing the Power of Paying Multiplexing Only Once in Stochastic Network Calculus
The stochastic network calculus (SNC) holds promise as a versatile and
uniform framework to calculate probabilistic performance bounds in networks of
queues. A great challenge to accurate bounds and efficient calculations are
stochastic dependencies between flows due to resource sharing inside the
network. However, by carefully utilizing the basic SNC concepts in the network
analysis the necessity of taking these dependencies into account can be
minimized. To that end, we fully unleash the power of the pay multiplexing only
once principle (PMOO, known from the deterministic network calculus) in the SNC
analysis. We choose an analytic combinatorics presentation of the results in
order to ease complex calculations. In tree-reducible networks, a subclass of
general feedforward networks, we obtain a perfect analysis in terms of avoiding
the need to take internal flow dependencies into account. In a comprehensive
numerical evaluation, we demonstrate how this unleashed PMOO analysis can
reduce the known gap between simulations and SNC calculations significantly,
and how it favourably compares to state-of-the art SNC calculations in terms of
accuracy and computational effort. Motivated by these promising results, we
also consider general feedforward networks, when some flow dependencies have to
be taken into account. To that end, the unleashed PMOO analysis is extended to
the partially dependent case and a case study of a canonical example topology,
known as the diamond network, is provided, again displaying favourable results
over the state of the art
Queues with Lévy input and hysteretic control
We consider a (doubly) reflected Lévy process where the Lévy exponent is controlled by a hysteretic policy consisting of two stages. In each stage there is typically a different service speed, drift parameter, or arrival rate. We determine the steady-state performance, both for systems with finite and infinite capacity. Thereby, we unify and extend many existing results in the literature, focusing on the special cases of M/G/1 queues and Brownian motion. © The Author(s) 2009
On the modelling and performance measurement of service networks with heterogeneous customers
Service networks are common throughout the modern world, yet understanding how their individual services effect each other and contribute to overall system performance can be difficult. An important metric in these systems is the quality of service. This is an often overlooked measure when modelling and relates to how customers are affected by a service. Presented is a novel perspective for evaluating the performance of multi-class queueing networks through a combination of operational performance and service quality—denoted the “flow of outcomes”. Here, quality is quantified by customers moving between or remaining in classes as a result of receiving service or lacking service. Importantly, each class may have different flow parameters, hence the positive/negative impact of service quality on the system’s operational performance is captured. A fluid–diffusion approximation for networks of stochastic queues is used since it allows for several complex flow dynamics: the sequential use of multiple services; abandonment and possible rejoin; reuse of the same service; multiple customers classes; and, class and time dependent parameters. The scalability of the approach is a significant benefit since, the modelled systems may be relatively large, and the included flow dynamics may render the system analytically intractable or computationally burdensome. Under the right conditions, this method provides a framework for quickly modelling large time-dependent systems. This combination of computational speed and the “flow of outcomes” provides new avenues for the analysis of multi-class service networks where both service quality and operational efficiency interact
Concentration of measure and mixing for Markov chains
We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.Comment: 28 page
Monotonicity and error bounds for networks of Erlang loss queues
Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to (1) upper bounds for loss probabilities and \ud
(2) analytic error bounds for the accuracy of the approximation for various performance measures.\ud
The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications:\ud
(1)• pure loss networks as under (2)• GSM networks with fixed channel allocation as under.\ud
The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning
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