A successive censoring algorithm for a system of connected QBD-processes

We consider a Markov Chain in which the state space is partitioned into sets where both transitions within sets and between sets have a special structure. Transitions within each set constitute a finite Quasi-Birth-and-Death-process, and transitions between sets are restricted to four types of transitions. We present a successive censoring algorithm, based on Matrix Analytic Methods, to obtain the stationary distribution of this system of connected QBD-processes

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations. We show that the quantitative termination decision problem for QBDs (namely, ``is $G_{u,v} \geq 1/2$?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard

Joint queue length distribution of multi-class, single-server queues with preemptive priorities

In this paper we analyze an MN/MN/1 queueing system with N customer classes and preemptive priorities between classes, by using matrix-analytic techniques. This leads to an exact method for the computation of the steady state joint queue length distribution. We also indicate how the method can be extended to models with multiple servers and other priority rules

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations. We show that the quantitative termination decision problem for QBDs (namely, ``is $G_{u,v} \geq 1/2$?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata and Pushdown Systems

to appear in QEST 2008We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD (even a null-recurrent one), we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results for pPDSs) that for the more general TL-QBDs this bound fails badly. Specifically, in the worst case Newton's method ``converges linearly'' to the termination probabilities for TL-QBDs, but requires exponentially many iterations in the encoding size of the TL-QBD to approximate these probabilities within any non-trivial constant error $c < 1$. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for monotone systems of polynomial equations

Performance evaluation of warehouses with automated storage and retrieval technologies.

In this dissertation, we study the performance evaluation of two automated warehouse material handling (MH) technologies - automated storage/retrieval system (AS/RS) and autonomous vehicle storage/retrieval system (AVS/RS). AS/RS is a traditional automated warehouse MH technology and has been used for more than five decades. AVS/RS is a relatively new automated warehouse MH technology and an alternative to AS/RS. There are two possible configurations of AVS/RS: AVS/RS with tier-captive vehicles and AVS/RS with tier-to-tier vehicles. We model the AS/RS and both configurations of the AVS/RS as queueing networks. We analyze and develop approximate algorithms for these network models and use them to estimate performance of the two automated warehouse MH technologies. Chapter 2 contains two parts. The first part is a brief review of existing papers about AS/RS and AVS/RS. The second part is a methodological review of queueing network theory, which serves as a building block for our study. In Chapter 3, we model AS/RSs and AVS/RSs with tier-captive vehicles as open queueing networks (OQNs). We show how to analyze OQNs and estimate related performance measures. We then apply an existing OQN analyzer to compare the two MH technologies and answer various design questions. In Chapter 4 and Chapter 5, we present some efficient algorithms to solve SOQN. We show how to model AVS/RSs with tier-to-tier vehicles as SOQNs and evaluate performance of these designs in Chapter 6. AVS/RS is a relatively new automated warehouse design technology. Hence, there are few efficient analytical tools to evaluate performance measures of this technology. We developed some efficient algorithms based on SOQN to quickly and effectively evaluate performance of AVS/RS. Additionally, we present a tool that helps a warehouse designer during the concepting stage to determine the type of MH technology to use, analyze numerous alternate warehouse configurations and select one of these for final implementation

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

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

Analysis, Design, and Construction of Nucleic Acid Devices

Nucleic acids present great promise as building blocks for nanoscale devices. To achieve this potential, methods for the analysis and design of DNA and RNA need to be improved. In this thesis, traditional algorithms for analyzing nucleic acids at equilibrium are extended to handle a class of pseudoknots, with examples provided relevant to biologists and bioengineers. With these analytical tools in hand, nucleic acid sequences are designed to maximize the equilibrium probability of a desired fold. Upon analysis, it is concluded that both affinity and specificity are important when choosing a sequence; this conclusion holds for a wide range of target structures and is robust to random perturbations to the energy model. Applying the intuition gained from these studies, a process called hybridization chain reaction (HCR) is invented, and sequences are chosen that experimentally verify this phenomenon. In HCR, a small number of DNA or RNA molecules trigger a system wide configurational change, allowing the amplification and detection of specific, nucleic acid sequences. As an extension, HCR is combined with a pre-existing aptamer domain to successfully construct an ATP sensor, and the groundwork is laid for the future development of sensors for other small molecules. In addition, recent studies on multi-stranded algorithms and improvements to HCR are included in the appendices. Not only will these advancements increase our understanding of biological RNAs, but they will also provide valuable tools for the future development of nucleic acid nanotechnologies