We propose a situation in which a single employee is responsible for processing incoming claims to an insurance company that can be classified as being one of two possible types. More specifically, we consider a priority-based system having separate buffers to store high priority and low priority incoming claims. We construct a mathematical model and perform queueing analysis to evaluate the performance of this priority-based system, which incorporates the possibility of claims being redistributed, lost, or prematurely processed.

<p>Many types of services are provided using some equipment or machines, e.g. transportation systems using vehicles. Designs of these systems require capacity decisions, e.g., the number of vehicles. In my dissertation, I use many-server and conventional heavy-traffic limit theory to derive asymptotically optimal capacity decisions, giving the desired level of delay and availability performance with minimum investment. The results provide near-optimal solutions and insights to otherwise analytically intractable problems.</p> <p>The dissertation will comprise two essays. In the first essay, &ldquoAsymptotic Analysis of Delay-based Performance Metrics and Optimal Capacity Decisions for the Machine Repair Problem with Spares,&rdquo I study the M/M/R machine repair problem with spares. This system can be represented by a closed queuing network. Applications include fleet vehicles' repair and backup capacity, warranty services' staffing and spare items investment decisions. For these types of systems, customer satisfaction is essential; thus, the delays until replacements of broken units are even more important than delays until the repair initiations of the units. Moreover, the service contract may include conditions on not only the fill rate but also the probability of acceptable delay (delay being less than a specified threshold value).</p> <p>I address these concerns by expressing delays in terms of the broken-machines process; scaling this process by the number of required operating machines (or the number of customers in the system); and using many-server limit theorems (limits taken as the number of customers goes to infinity) to obtain the limiting expected delay and probability of acceptable delay for both delay until replacement and repair initiation. These results lead to an approximate optimization problem to decide on the repair and backup-capacity investment giving the minimum expected cost rate, subject to a constraint on the acceptable delay probability. Using the characteristics of the scaled broken-machines process, we obtain insights about the relationship between quality of service and capacity decisions. Inspired by the call-center literature, we categorize capacity level choice as efficiency-driven, quality-driven, or quality- and efficiency-driven. Hence, our study extends the conventional call center staffing problem to joint staffing and backup provisioning. Moreover, to our knowledge, the machine-repair problem literature has focused mainly on mean and fill rate measures of performance for steady-state cost analysis. This approach provides complex, nonlinear expressions not possible to solve analytically. The contribution of this essay to the machine-repair literature is the construction of delay-distribution approximations and a near-optimal analytical solution. Among the interesting results, we find that for capacity levels leading to very high utilization of both spares and repair capacity, the limiting distribution of delay until replacement depends on one type of resource only, the repair capacity investment.</p> <p>In the second essay, &ldquoDiffusion Approximations and Near-Optimal Design of a Make-to-Stock Queue with Perishable Goods and Impatient Customers,&rdquo I study a make-to-stock system with perishable inventory and impatient customers as a two-sided queue with abandonment from both sides. This model describes many consumer goods, where not only spoilage but also theft and damage can occur. We will refer to positive jobs as individual products on the shelf and negative jobs as backlogged customers. In this sense, an arriving negative job provides the service to a waiting positive job, and vice versa. Jobs that must wait in queue before potential matching are subject to abandonment. Under certain assumptions on the magnitude of the abandonment rates and the scaled difference between the two arrival rates (products and customers), we suggest approximations to the system dynamics such as average inventory, backorders, and fill rate via conventional heavy traffic limit theory.</p> <p>We find that the approximate limiting queue length distribution is a normalized weighted average of two truncated normal distributions and then extend our results to analyze make-to-stock queues with/without perishability and limiting inventory space by inducing thresholds on the production (positive) side of the queue. Finally, we develop conjectures for the queue-length distribution for a non-Markovian system with general arrival streams. We take production rate as the decision variable and suggest near-optimal solutions.</p> / Dissertation

Information-theoretic arguments focus on modeling the reliability of information transmission, assuming availability of infinite data at sources, thus ignoring randomness in message generation times at the respective sources. However, in information transport networks, not only is reliable transmission important, but also stability, i.e., finiteness of mean delay in- curred by messages from the time of generation to the time of successful reception. Usually, delay analysis is done separately using queueing-theoretic arguments, whereas reliable information transmission is studied using information theory. In this thesis, we investigate these two important aspects of data communication jointly by suitably combining models from these two fields. In particular, we model scheduled communication of messages , that arrive in a random process, (i) over multiaccess channels, with either independent decoding or joint decoding, and (ii) over degraded broadcast channels. The scheduling policies proposed permit up to a certain maximum number of messages for simultaneous transmission. In the first part of the thesis, we develop a multi-class discrete-time processor-sharing queueing model, and then investigate the stability of this queue. In particular, we model the queue by a discrete-time Markov chain defined on a countable state space, and then establish (i) a sufficient condition for c-regularity of the chain, and hence positive recurrence and finiteness of stationary mean of the function c of the state, and (ii) a sufficient condition for transience of the chain. These stability results form the basis for the conclusions drawn in the thesis. The second part of the thesis is on multiaccess communication with random message arrivals. In the context of independent decoding, we assume that messages can be classified into a fixed number of classes, each of which specifies a combination of received signal power, message length, and target probability of decoding error. Each message is encoded independently and decoded independently. In the context of joint decoding, we assume that messages can be classified into a fixed number of classes, each of which specifies a message length, and for each of which there is a message queue. From each queue, some number of messages are encoded jointly, and received at a signal power corresponding to the queue. The messages are decoded jointly across all queues with a target probability of joint decoding error. For both independent decoding and joint decoding, we derive respective discrete- time multiclass processor-sharing queueing models assuming the corresponding information-theoretic models for the underlying communication process. Then, for both the decoding schemes, we (i) derive respective outer bounds to the stability region of message arrival rate vectors achievable by the class of stationary scheduling policies, (ii) show for any mes- sage arrival rate vector that satisfies the outer bound, that there exists a stationary “state-independent” policy that results in a stable system for the corresponding message arrival process, and (iii) show that the stability region of information arrival rate vectors, in the limit of large message lengths, equals an appropriate information-theoretic capacity region for independent decoding, and equals the information-theoretic capacity region for joint de-coding. For independent decoding, we identify a class of stationary scheduling policies, for which we show that the stability region in the limit of large maximum number of simultane-ous transmissions is independent of the received signal powers, and each of which achieves a spectral efficiency of 1 nat/s/Hz in the limit of large message lengths. In the third and last part of the thesis, we show that the queueing model developed for multiaccess channels with joint decoding can be used to model communication over degraded broadcast channels, with superposition encoding and successive decoding across all queues. We then show respective results (i), (ii), and (iii), stated above.

Materialfluss-Systeme für den Stückgut-Transport auf der Basis von Stetigförderern sind meist modular aufgebaut. Das Verhalten gleichartiger Materialfluss-Elemente unterscheidet sich durch technische Parameter (z.B. geometrische Größen) und durch unterschiedliche logistische Belastungen der Elemente im System. Durch die in der Arbeit getroffenen Modellannahmen werden für die Elemente nur lokale Steuerungsregeln zugelassen und für das System Blockierfreiheit vorausgesetzt. Das Verhalten eines Materialfluss-Elements hängt dann nicht mehr von Zuständen anderer Elemente des Systems ab sondern nur noch von den stochastischen Prozessen des Eintreffens von Transporteinheiten. Die Auslastung eines Elements, die Quantile der Warteschlangenlängen an seinen Eingängen und die Variationskoeffizienten seiner Abgangsströme sind statistische Kenngrößen. Sie hängen im Wesentlichen nur von der Klasse des Elements, seinen technischen Parametern, den Parametern der Eingangsströme und der lokalen Transportmatrix ab. Diese funktionellen Abhängigkeiten sind im Allgemeinen nicht analytisch handhabbar. Da diese Funktionen stetig differenzierbar und beschränkt sind und von relativ viele Eingansgrößen anhängen, sind neuronale Netze gut geeignet für numerische Näherungen. Mit Hilfe von einfachen neuronalen Netzen können die statistischen Kenngrößen numerisch approximiert werden. Aus einzelnen Teilmodellen kann ein hybrides Modell des gesamten Systems zusammengesetzt werden. Anhand von einigen Beispielen wird die Güte der Modellierung bewertet. / Material flow systems are normally built with a modular structure. The behavoir of similar elements only differs by technical parameters (e.g. geometriy), and by different logistic loads of the elements in the system. In this paper, a new model is being developed for a non-blocking system with non-global control rules. The behavior of a flow of a material flow element is assumed not to depend on the conditions of other elements of the system, but only on stochastic processes of the arrival of transportation units. The rate of utilization of an element, the quantiles of the queue lengths at its inputs, and the dispersion of its output stream are statistic characteristics. They depend only on the type of the element, its technical parameters, the parameters of the input streams, and the local transportation matrix. These functional dependencies are not analytically manageable. But due to their properties, neural nets are well suited for numeric approximations of these statistic functions. The single models can be used to compose a hybrid model of the whole system. A few examples show the quality of the new modeling technique.

Dažnai realių stochastinių sistemų dinamikos negalime aprašyti Markovo procesu, nes operacijų trukmės paprastai nėra pasiskirstę pagal eksponentinį dėsnį. Darbe buvo išnagrinėtas tokių atsitiktinių dydžių aproksimavimas dviejų eksponentinių atsitiktinių dydžių mišiniu. Paprasčiausioms sistemoms kartais galima gauti analizines formules sistemos būsenų stacionarioms tikimybėms suskaičiuoti, tačiau daugeliui sistemų to padaryti negalima. Būtent tokių sistemų tyrimui, panaudojus aproksimavimo algoritmus, buvo sukurta programinė įranga, kuri leidžia modeliuoti daugelį stochastinių sistemų. Magistro darbo užduotis: Sukurti stochastinių sistemų modelių aproksimavimo Markovo modeliais algoritmus ir programinę įrangą. Buvo iškelti tokie tikslai: Ištirti pasiskirstymo funkcijų aproksimavimo eksponentinių skirstinių mišiniu galimybes; Sukurti universalią programinę priemonę, kuri pagal pateiktą sistemos aprašymą, skaičiuotų jos stacionariąsias tikimybes bei funkcionavimo charakteristikas; Sukurtos programinės priemonės pagalba, sudaryti ir ištirti aptarnavimo sistemų ir vertybinių popierių įkainojimo modelius. Sukurta programinė įranga pasižymi universalumu ir paprastumu vartotojui. Sistemos funkcionavimą galima aprašyti turint minimalias C++ Builder programavimo kalbos žinias. Magistro darbe sukurta programinė įranga buvo pritaikyta aptarnavimo sistemoms modeliuoti, akcijų kainų dinamikai aprašyti bei opcionams įkainoti. / Application of numerical methods with approximation allows to extend a class of systems represented by Markovian processes under investigation compared with analytical methods. So a goal was stated to create algorithms for modeling stochastic systems approximating them by Markovian models. To reach this goal the following tasks were solved: Analyze possibilities to approximate stochastic systems’ models by Markovian models; Create a multipurpose software that would calculate stationary probabilities for given system described in an event-based language; Apply created software for models of service systems and stock valuation. Created software is universal and easy-to-use for anyone that has at least basic knowledge in C++ language. This software was applied for modeling of service systems, for description of share price variability as Markovian process and for option pricing.

We consider a single-part, two-echelon supply chain problem for spare parts. The network consists of a single manufacturing plant, a set of service centers (SCs) and a set of customers. Both echelons keep spare parts using the base-stock replenishment policy. The plant behaves as an M/M/1 queueing system and has limited production and storage capacity. Demand faced by each SC follows an independent Poisson process. The problem is to determine optimal location-allocation and optimal base-stock levels at both echelons while satisfying the target service levels and customer preferences of SCs. We develop a mixed integer non-linear programming model and use cutting-plane method to optimize the inventory-location decisions. We present an exact solution procedure for the inventory stocking problem and demonstrate the limitations of using traditional inventory models like METRIC-like and Approximate in case of high utilization rates. We show the effectiveness of our proposed cutting-plane algorithm and provide important managerial insights for spare parts management.

The thesis investigates how Agent-Based Modeling and Simulation (ABMS) and Queueing Theory (QT) techniques help manage mass gathering (MG) crowds. The techniques are applied to Hajj MG, which is one of the most complex annual MG, with a focus on its challenging Tawaf ritual. The objective is to develop a Tawaf Decision Support System (DSS) to better understand Tawaf crowd dynamics and discover decisions that lead to superior performance. TawafSIM is an ABMS model in the DSS, which simulates macro-level Tawaf crowd dynamics through micro-level pilgrim modeling to explore the impact of crowd characteristics, facility layout, and management preferences on emergent crowd behaviours with respect to throughput, satisfaction, health, and safety. Whereas, TawafQT is a QT model in the DSS to explore the impact of pilgrim arrival rate and Tawaf throughput on expected arrival, departure, and waiting times along with average queue length in the Tawaf waiting area. The thesis provides several contributions, including the following. First, it is the only Tawaf research to use a hybrid ABMS and QT approach. Second, TawafSIM is a comprehensive Tawaf simulator. It incorporates features for pilgrim characteristics, facility design, and management preferences. It calculates eight metrics for Tawaf performance, which includes one for throughput, three for satisfaction, one for health, and three for safety. It is the only Tawaf simulator to estimate satisfaction and spread of infectious disease. It conducts 42 simulation experiments in 12 categories. It generates observations for emergent, tipping point, expected, and counter intuitive behaviours. It recommends a default scenario as the best decision along with a small subset of alternative scenarios, which provide above average Tawaf performance. It generates a Tawaf Crowd Management Guide to better understand Tawaf crowd dynamics and how to pursue above average Tawaf performance under different conditions. Third, TawafQT is the only study of the Tawaf waiting area. It uses an accurate queueing model with finite source, single service, and PH type distribution, which is not only applicable to the Tawaf and other Hajj related queueing systems but also to any queueing system, which has finite population and single service characteristics.

The thesis investigates how Agent-Based Modeling and Simulation (ABMS) and Queueing Theory (QT) techniques help manage mass gathering (MG) crowds. The techniques are applied to Hajj MG, which is one of the most complex annual MG, with a focus on its challenging Tawaf ritual. The objective is to develop a Tawaf Decision Support System (DSS) to better understand Tawaf crowd dynamics and discover decisions that lead to superior performance. TawafSIM is an ABMS model in the DSS, which simulates macro-level Tawaf crowd dynamics through micro-level pilgrim modeling to explore the impact of crowd characteristics, facility layout, and management preferences on emergent crowd behaviours with respect to throughput, satisfaction, health, and safety. Whereas, TawafQT is a QT model in the DSS to explore the impact of pilgrim arrival rate and Tawaf throughput on expected arrival, departure, and waiting times along with average queue length in the Tawaf waiting area. The thesis provides several contributions, including the following. First, it is the only Tawaf research to use a hybrid ABMS and QT approach. Second, TawafSIM is a comprehensive Tawaf simulator. It incorporates features for pilgrim characteristics, facility design, and management preferences. It calculates eight metrics for Tawaf performance, which includes one for throughput, three for satisfaction, one for health, and three for safety. It is the only Tawaf simulator to estimate satisfaction and spread of infectious disease. It conducts 42 simulation experiments in 12 categories. It generates observations for emergent, tipping point, expected, and counter intuitive behaviours. It recommends a default scenario as the best decision along with a small subset of alternative scenarios, which provide above average Tawaf performance. It generates a Tawaf Crowd Management Guide to better understand Tawaf crowd dynamics and how to pursue above average Tawaf performance under different conditions. Third, TawafQT is the only study of the Tawaf waiting area. It uses an accurate queueing model with finite source, single service, and PH type distribution, which is not only applicable to the Tawaf and other Hajj related queueing systems but also to any queueing system, which has finite population and single service characteristics.

A scheduling algorithm has the primary role in implementing the quality of service guaranteed to each flow by managing buffer space and selecting which packet to send next with a fair share of network. In this thesis, some scheduling algorithms for packet switching networks are studied. For evaluating their delay, jitter and throughput performances, a discrete event simulator has been developed. It has been seen that fair scheduling provides, fair allocation of bandwidth, lower delay for sources using less than their full share of bandwidth and protection from ill-behaved resources.

在這一篇論文裡，我們討論兩個階段的封閉式等候線網路，其中服務時間的機率分配都是Phase type分配。我們猜測服務時間的機率分配和離開時間間隔的機率分配滿足一組聯立方程組。然後，我們推導出非邊界狀態的穩定機率可以被表示成 product-form的線性組合，而每個product-form可以用聯立方程組的根來構成。利用非邊界狀態的穩定機率， 我們可以求出邊界狀態的機率。最後我們建立一個求穩定機率的演算過程。利用這個演算方法，可以簡化求穩定機率的複雜度。 / In this thesis, we are concerned with the property of a two-stage closed system in which the service times are identically of phase type. We first conjecture that the 　Laplace-Stieltjes Transforms (LST) of service time distributions may satisfy a system of equations. Then we present that the stationary probabilities on the unboundary states can be written as a linear combination of product-forms. Each component of these products can be expressed in terms of roots of the system of equations. Finally, we establish an algorithm to obtain all the stationary probabilities. The algorithm is expected to work well for relatively large customers in the system.