Nested Fork-Join Queuing Networks and Their Application to Mobility Airfield Operations Analysis

A single-chain nested fork-join queuing network (FJQN) model of mobility airfield ground processing is proposed. In order to analyze the queuing network model, advances on two fronts are made. First, a general technique for decomposing nested FJQNs with probabilistic forks is proposed, which consists of incorporating feedback loops into the embedded Markov chain of the synchronization station, then using Marie\u27s Method to decompose the network. Numerical studies show this strategy to be effective, with less than two percent relative error in the approximate performance measures in most realistic cases. The second contribution is the identification of a quick, efficient method for solving for the stationary probabilities of the λn/Ck/r/N queue. Unpreconditioned Conjugate Gradient Squared is shown to be the method of choice in the context of decomposition using Marie\u27s Method, thus broadening the class of networks where the method is of practical use. The mobility airfield model is analyzed using the strategies described above, and accurate approximations of airfield performance measures are obtained in a fraction of the time needed for a simulation study. The proposed airfield modeling approach is especially effective for quick-look studies and sensitivity analysis

Bose--Einstein Condensation in the Large Deviations Regime with Applications to Information System Models

We study the large deviations behavior of systems that admit a certain form of a product distribution, which is frequently encountered both in Physics and in various information system models. First, to fix ideas, we demonstrate a simple calculation of the large deviations rate function for a single constraint (event). Under certain conditions, the behavior of this function is shown to exhibit an analogue of Bose--Einstein condensation (BEC). More interestingly, we also study the large deviations rate function associated with two constraints (and the extension to any number of constraints is conceptually straightforward). The phase diagram of this rate function is shown to exhibit as many as seven phases, and it suggests a two--dimensional generalization of the notion of BEC (or more generally, a multi--dimensional BEC). While the results are illustrated for a simple model, the underlying principles are actually rather general. We also discuss several applications and implications pertaining to information system models

Transform-domain analysis of packet delay in network nodes with QoS-aware scheduling

In order to differentiate the perceived QoS between traffic classes in heterogeneous packet networks, equipment discriminates incoming packets based on their class, particularly in the way queued packets are scheduled for further transmission. We review a common stochastic modelling framework in which scheduling mechanisms can be evaluated, especially with regard to the resulting per-class delay distribution. For this, a discrete-time single-server queue is considered with two classes of packet arrivals, either delay-sensitive (1) or delay-tolerant (2). The steady-state analysis relies on the use of well-chosen supplementary variables and is mainly done in the transform domain. Secondly, we propose and analyse a new type of scheduling mechanism that allows precise control over the amount of delay differentiation between the classes. The idea is to introduce N reserved places in the queue, intended for future arrivals of class 1

Loss systems in a random environment

We consider a single server system with infinite waiting room in a random environment. The service system and the environment interact in both directions. Whenever the environment enters a prespecified subset of its state space the service process is completely blocked: Service is interrupted and newly arriving customers are lost. We prove an if-and-only-if-condition for a product form steady state distribution of the joint queueing-environment process. A consequence is a strong insensitivity property for such systems. We discuss several applications, e.g. from inventory theory and reliability theory, and show that our result extends and generalizes several theorems found in the literature, e.g. of queueing-inventory processes. We investigate further classical loss systems, where due to finite waiting room loss of customers occurs. In connection with loss of customers due to blocking by the environment and service interruptions new phenomena arise. We further investigate the embedded Markov chains at departure epochs and show that the behaviour of the embedded Markov chain is often considerably different from that of the continuous time Markov process. This is different from the behaviour of the standard M/G/1, where the steady state of the embedded Markov chain and the continuous time process coincide. For exponential queueing systems we show that there is a product form equilibrium of the embedded Markov chain under rather general conditions. For systems with non-exponential service times more restrictive constraints are needed, which we prove by a counter example where the environment represents an inventory attached to an M/D/1 queue. Such integrated queueing-inventory systems are dealt with in the literature previously, and are revisited here in detail

The evaluation of computer performance by means of state-dependent queueing network models

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Queueing-Theoretic End-to-End Latency Modeling of Future Wireless Networks

The fifth generation (5G) of mobile communication networks is envisioned to enable a variety of novel applications. These applications demand requirements from the network, which are diverse and challenging. Consequently, the mobile network has to be not only capable to meet the demands of one of these applications, but also be flexible enough that it can be tailored to different needs of various services. Among these new applications, there are use cases that require low latency as well as an ultra-high reliability, e.g., to ensure unobstructed production in factory automation or road safety for (autonomous) transportation. In these domains, the requirements are crucial, since violating them may lead to financial or even human damage. Hence, an ultra-low probability of failure is necessary. Based on this, two major questions arise that are the motivation for this thesis. First, how can ultra-low failure probabilities be evaluated, since experiments or simulations would require a tremendous number of runs and, thus, turn out to be infeasible. Second, given a network that can be configured differently for different applications through the concept of network slicing, which performance can be expected by different parameters and what is their optimal choice, particularly in the presence of other applications. In this thesis, both questions shall be answered by appropriate mathematical modeling of the radio interface and the radio access network. Thereby the aim is to find the distribution of the (end-to-end) latency, allowing to extract stochastic measures such as the mean, the variance, but also ultra-high percentiles at the distribution tail. The percentile analysis eventually leads to the desired evaluation of worst-case scenarios at ultra-low probabilities. Therefore, the mathematical tool of queuing theory is utilized to study video streaming performance and one or multiple (low-latency) applications. One of the key contributions is the development of a numeric algorithm to obtain the latency of general queuing systems for homogeneous as well as for prioritized heterogeneous traffic. This provides the foundation for analyzing and improving end-to-end latency for applications with known traffic distributions in arbitrary network topologies and consisting of one or multiple network slices.Es wird erwartet, dass die fünfte Mobilfunkgeneration (5G) eine Reihe neuartiger Anwendungen ermöglichen wird. Allerdings stellen diese Anwendungen sowohl sehr unterschiedliche als auch überaus herausfordernde Anforderungen an das Netzwerk. Folglich muss das mobile Netz nicht nur die Voraussetzungen einer einzelnen Anwendungen erfüllen, sondern auch flexibel genug sein, um an die Vorgaben unterschiedlicher Dienste angepasst werden zu können. Ein Teil der neuen Anwendungen erfordert hochzuverlässige Kommunikation mit niedriger Latenz, um beispielsweise unterbrechungsfreie Produktion in der Fabrikautomatisierung oder Sicherheit im (autonomen) Straßenverkehr zu gewährleisten. In diesen Bereichen ist die Erfüllung der gestellten Anforderungen besonders kritisch, da eine Verletzung finanzielle oder sogar personelle Schäden nach sich ziehen könnte. Eine extrem niedrige Ausfallwahrscheinlichkeit ist daher von größter Wichtigkeit. Daraus ergeben sich zwei wesentliche Fragestellungen, welche diese Arbeit motivieren. Erstens, wie können extrem niedrige Ausfallwahrscheinlichkeiten evaluiert werden. Ihr Nachweis durch Experimente oder Simulationen würde eine extrem große Anzahl an Durchläufen benötigen und sich daher als nicht realisierbar herausstellen. Zweitens, welche Performanz ist für ein gegebenes Netzwerk durch unterschiedliche Konfigurationen zu erwarten und wie kann die optimale Konfiguration gewählt werden. Diese Frage ist insbesondere dann interessant, wenn mehrere Anwendungen gleichzeitig bedient werden und durch sogenanntes Slicing für jeden Dienst unterschiedliche Konfigurationen möglich sind. In dieser Arbeit werden beide Fragen durch geeignete mathematische Modellierung der Funkschnittstelle sowie des Funkzugangsnetzes (Radio Access Network) adressiert. Mithilfe der Warteschlangentheorie soll die stochastische Verteilung der (Ende-zu-Ende-) Latenz bestimmt werden. Dies liefert unterschiedliche stochastische Metriken, wie den Erwartungswert, die Varianz und insbesondere extrem hohe Perzentile am oberen Rand der Verteilung. Letztere geben schließlich Aufschluss über die gesuchten schlimmsten Fälle, die mit sehr geringer Wahrscheinlichkeit eintreten können. In der Arbeit werden Videostreaming und ein oder mehrere niedriglatente Anwendungen untersucht. Zu den wichtigsten Beiträgen zählt dabei die Entwicklung einer numerischen Methode, um die Latenz in allgemeinen Warteschlangensystemen für homogenen sowie für priorisierten heterogenen Datenverkehr zu bestimmen. Dies legt die Grundlage für die Analyse und Verbesserung von Ende-zu-Ende-Latenz für Anwendungen mit bekannten Verkehrsverteilungen in beliebigen Netzwerktopologien mit ein oder mehreren Slices