Poisson's equation for discrete-time quasi-birth-and-death processes

We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value

Aggregate matrix-analytic techniques and their applications

The complexity of computer systems affects the complexity of modeling techniques that can be used for their performance analysis. In this dissertation, we develop a set of techniques that are based on tractable analytic models and enable efficient performance analysis of computer systems. Our approach is three pronged: first, we propose new techniques to parameterize measurement data with Markovian-based stochastic processes that can be further used as input into queueing systems; second, we propose new methods to efficiently solve complex queueing models; and third, we use the proposed methods to evaluate the performance of clustered Web servers and propose new load balancing policies based on this analysis.;We devise two new techniques for fitting measurement data that exhibit high variability into Phase-type (PH) distributions. These techniques apply known fitting algorithms in a divide-and-conquer fashion. We evaluate the accuracy of our methods from both the statistics and the queueing systems perspective. In addition, we propose a new methodology for fitting measurement data that exhibit long-range dependence into Markovian Arrival Processes (MAPs).;We propose a new methodology, ETAQA, for the exact solution of M/G/1-type processes, (GI/M/1-type processes, and their intersection, i.e., quasi birth-death (QBD) processes. ETAQA computes an aggregate steady state probability distribution and a set of measures of interest. E TAQA is numerically stable and computationally superior to alternative solution methods. Apart from ETAQA, we propose a new methodology for the exact solution of a class of GI/G/1-type processes based on aggregation/decomposition.;Finally, we demonstrate the applicability of the proposed techniques by evaluating load balancing policies in clustered Web servers. We address the high variability in the service process of Web servers by dedicating the servers of a cluster to requests of similar sizes and propose new, content-aware load balancing policies. Detailed analysis shows that the proposed policies achieve high user-perceived performance and, by continuously adapting their scheduling parameters to the current workload characteristics, provide good performance under conditions of transient overload

Performability modelling of homogenous and heterogeneous multiserver systems with breakdowns and repairs

This thesis presents analytical modelling of homogeneous multi-server systems with reconfiguration and rebooting delays, heterogeneous multi-server systems with one main and several identical servers, and farm paradigm multi-server systems. This thesis also includes a number of other research works such as, fast performability evaluation models of open networks of nodes with repairs and finite queuing capacities, multi-server systems with deferred repairs, and two stage tandem networks with failures, repairs and multiple servers at the second stage. Applications of these for the popular Beowulf cluster systems and memory servers are also accomplished. Existing techniques used in performance evaluation of multi-server systems are investigated and analysed in detail. Pure performance modelling techniques, pure availability models, and performability models are also considered. First, the existing approaches for pure performance modelling are critically analysed with the discussions on merits and demerits. Then relevant terminology is defined and explained. Since the pure performance models tend to be too optimistic and pure availability models are too conservative, performability models are used for the evaluation of multi-server systems. Fault-tolerant multi-server systems can continue service in case of certain failures. If failure does not occur at a critical point (such as breakdown of the head processor of a farm paradigm system) the system continues serving in a degraded mode of operation. In such systems, reconfiguration and/or rebooting delays are expected while a processor is being mapped out from the system. These delay stages are also taken into account in addition to failures and repairs, in the exact performability models that are developed. Two dimensional Markov state space representations of the systems are used for performability modelling. Following the critical analysis of the existing solution techniques, the Spectral Expansion method is chosen for the solution of the models developed. In this work, open queuing networks are also considered. To evaluate their performability, existing modelling approaches are expanded and validated by simulations, for performability analysis of multistage open networks with finite queuing capacities. The performances of two extended modelling approaches are compared in terms of accuracy for open networks with various queuing capacities. Deferred repair strategies are becoming popular because of the cost reductions they can provide. Effects of using deferred repairs are analysed and performability models are provided for homogeneous multi-server systems and highly available farm paradigm multi-server systems. Since one of the random variables is used to represent the number of jobs in one of the queues, analytical models for performance evaluation of two stage tandem networks suffer because of numerical cumbersomeness. Existing approaches for modelling these systems are actually pure performance models since breakdowns and repairs cannot be considered. One way of modelling these systems can be to divide one of the random variables to present both the operative and non-operative states of the server in one dimension. However, this will give rise to state explosion problem severely limiting the maximum queue capacity that can be handled. In order to overcome this problem a new approach is presented for modelling two stage tandem networks in three dimensions. An approximate solution is presented to solve such a system. This approach manifests itself as a novel contribution for alleviating the state space explosion problem for large and/or complex systems. When two state tandem networks with feedback are modelled using this approach, the operative states can be handled independently and this makes it possible to consider multiple operative states at the second stage. The analytical models presented can be used with various parameters and they are extendible to consider systems with similar architectures. The developed three dimensional approach is capable to handle two stage tandem networks with various characteristics for performability measures. All the approaches presented give accurate results. Numerical solutions are presented for all models developed. In case the solution presented is not exact, simulations are performed to validate the accuracy of the results obtained

07101 Abstracts Collection -- Quantitative Aspects of Embedded Systems

From March 5 to March 9, 2007, the Dagstuhl Seminar 07101 ``Quantitative Aspects of Embedded Systems\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Recursive Probabilistic Models: efficient analysis and implementation

This thesis examines Recursive Markov Chains (RMCs), their natural extensions and connection to other models. RMCs can model in a natural way probabilistic procedural programs and other systems that involve recursion and probability. An RMC is a set of ordinary finite state Markov Chains that are allowed to call each other recursively and it describes a potentially infinite, but countable, state ordinary Markov Chain. RMCs generalize in a precise sense several well studied probabilistic models in other domains such as natural language processing (Stochastic Context-Free Grammars), population dynamics (Multi-Type Branching Processes) and in queueing theory (Quasi-Birth-Death processes (QBDs)). In addition, RMCs can be extended to a controlled version called Recursive Markov Decision Processes (RMDPs) and also a game version referred to as Recursive (Simple) Stochastic Games (RSSGs). For analyzing RMCs, RMDPs, RSSGs we devised highly optimized numerical algorithms and implemented them in a tool called PReMo (Probabilistic Recursive Models analyzer). PReMo allows computation of the termination probability and expected termination time of RMCs and QBDs, and a restricted subset of RMDPs and RSSGs. The input models are described by the user in specifically designed simple input languages. Furthermore, in order to analyze the worst and best expected running time of probabilistic recursive programs we study models of RMDPs and RSSGs with positive rewards assigned to each of their transitions and provide new complexity upper and lower bounds of their analysis. We also establish some new connections between our models and models studied in queueing theory. Specifically, we show that (discrete time) QBDs can be described as a special subclass of RMCs and Tree-like QBDs, which are a generalization of QBDs, are equivalent to RMCs in a precise sense. We also prove that for a given QBD we can compute (in the unit cost RAM model) an approximation of its termination probabilities within i bits of precision in time polynomial in the size of the QBD and linear in i. Specifically, we show that we can do this using a decomposed Newton’s method