Simple and efficient importance sampling scheme for a tandem queue with server slow-down

This paper considers importance sampling as a tool for rare-event simulation. The system at hand is a so-called tandem queue with slow-down, which essentially means that the server of the first queue (or: upstreanm queue) switches to a lower speed when the second queue (downstream queue) exceeds some threshold. The goal is to assess to what extent such a policy succeeds in protecting the first queue, and therefore we focus on estimating the probability of overflow in the downstream queue.\ud It is known that in this setting importance sampling with traditional state-independent distributions performs poorly. More sophisticated state-dependent schemes can be shown to be asymptotically efficient, but their implementation may be problematic, as for each state the new measure has to be computed. This paper presents an algorithm that is considerably simpler than the fully state-dependent scheme; it requires low computational effort, but still has high efficiency

Backpressure-based control protocols: design and computational aspects

Congestion control in packet-based networks is often realized by feedback protocols. In this paper we assess their performance under a back-pressure mechanism that has been proposed and standardized for Ethernet metropolitan networks. In such a mechanism the service rate of an upstream queue is reduced when the downstream queue is congested, in order to protect the downstream queue. We study a Markovian model that captures the essentials of the protocol, but at the same time allows for numerical analysis. We first derive explicit results for the stability condition of the model (which turns out to be nontrivial). Then we present logarithmic estimates of the probability of buffer overflow in the second queue, which are subsequentially used when devising an efficient simulation procedure based on importance sampling. We conclude the paper by presenting a number of numerical results, and some general design guidelines

State-dependent importance sampling for a Jackson tandem network

This paper considers importance sampling as a tool for rare-event simulation. The focus is on estimating the probability of overflow in the downstream queue of a Jacksonian two-node tandem queue – it is known that in this setting ‘traditional’ state-independent importance-sampling distributions perform poorly. We therefore concentrate on developing a state-dependent change of measure, that we prove to be asymptotically efficient.\ud More specific contributions are the following. (i) We concentrate on the probability of the second queue exceeding a certain predefined threshold before the system empties. Importantly, we identify an asymptotically efficient importance-sampling distribution for any initial state of the system. (ii) The choice of the importance-sampling distribution is backed up by appealing heuristics that are rooted in large-deviations theory. (iii) Our method for proving asymptotic efficiency is substantially more straightforward than some that have been used earlier. The paper is concluded by simulation experiments that show a considerable speed up

Numerical analysis of stochastic biochemical reaction networks

Numerical solution of the chemical master equation for stochastic reaction networks typically suffers from the state space explosion problem due to the curse of dimensionality and from stiffness due to multiple time scales. The dimension of the state space equals the number of molecular species involved in the reaction network and the size of the system of differential equations equals the number of states in the corresponding continuous-time Markov chain, which is usually enormously huge and often even infinite. Thus, efficient numerical solution approaches must be able to handle huge, possibly infinite and stiff systems of differential equations efficiently. In this thesis, we present efficient techniques for the numerical analysis of the biochemical reaction networks. We present an approximate numerical integration approach that combines a dynamical state space truncation procedure with efficient numerical integration schemes for systems of ordinary differential equations including adaptive step size selection based on local error estimates. We combine our dynamical state space truncation with the method of conditional moments, and present the implementation details and numerical results. We also incorporate ideas from importance sampling simulations into a non-simulative numerical method that approximates transient rare event probabilities based on a dynamical truncation of the state space. Finally, we present a maximum likelihood method for the estimation of the model parameters given noisy time series measurements of molecular counts. All approaches presented in this thesis are implemented as part of the tool STAR, which allows to model and simulate the biochemical reaction networks. The efficiency and accuracy is demonstrated by numerical examples.Numerische Lösungen der chemischen Master-Gleichung für stochastische Reaktionsnetzwerke leiden typischerweise an dem Zustandsraumexplosionsproblem aufgrund der hohen Dimensionalität und der Steifigkeit durch mehrfache Zeitskalen. Die Dimension des Zustandsraumes entspricht der Anzahl der molekularen Spezies von dem Reaktionsnetzwerk und die Größe des Systems von Differentialgleichungen entspricht der Anzahl der Zustände in der entsprechenden kontinuierlichen Markov-Kette, die in der Regel enorm gross und oft sogar unendlich gross ist. Daher müssen numerische Methoden in der Lage sein, riesige, eventuell unendlich grosse und steife Systeme von Differentialgleichungen effizient lösen zu können. In dieser Arbeit beschreiben wir effiziente Methoden für die numerische Analyse biochemischer Reaktionsnetzwerke. Wir betrachten einen inexakten numerischen Integrationsansatz, bei dem eine dynamische Zustandsraumbeschneidung und ein Verfahren mit einem effizienten numerischen Integrationsschema für Systeme von gewöhnlichen Differentialgleichungen benutzt werden. Wir kombinieren unsere dynamische Zustandsraumbeschneidungsmethode mit der Methode der bedingten Momente und beschreiben die Implementierungdetails und numerischen Ergebnisse. Wir benutzen auch Ideen des importance sampling für eine nicht-simulative numerische Methode, die basierend auf der Zustandsraumbeschneidung die Wahrscheinlichkeiten von seltenen Ereignissen berechnen kann. Schließlich beschreiben wir eine Maximum-Likelihood-Methode für die Schätzung der Modellparameter bei verrauschten Zeitreihenmessungen von molekularen Anzahlen. Alle in dieser Arbeit beschriebenen Ansätze sind in dem Software-Tool STAR implementiert, das erlaubt, biochemische Reaktionsnetzwerke zu modellieren und zu simulieren. Die Effizienz und die Genauigkeit werden durch numerische Beispiele gezeigt