This thesis is concerned with queueing models where demand is allowed to exceed the system capacity, and also with the capacity sizing and pricing problem for heterogeneous products and resources under demand uncertainty. Our aim is to improve productivity and profitability. In the first part of the thesis, we consider the dynamic assignment of servers to tasks in queueing networks where demand may exceed the capacity for service. The objective is to maximize the system throughput. We use fluid limit analysis to show that several quantities of interest, namely the maximum possible throughput, the maximum throughput for a given arrival rate, the minimum arrival rate that will yield a desired feasible throughput, and the optimal allocations of servers to classes for a given arrival rate and desired throughput, can be computed by solving linear programming problems. We develop generalized round robin policies for assigning servers to classes for a given arrival rate and desired throughput, and show that our policies achieve the desired throughput as long as this throughput is feasible for the arrival rate. We conclude with numerical examples that illustrate the points discussed and provide insights into the system behavior when the arrival rate deviates from the one the system is designed for. In the second part of the thesis, we consider the effects of inspection and repair stations on the production capacity and product quality in a serial line with possible inspection and repair following each operation. We consider multiple defect types and allow for possible inspection errors that are defect dependent. We construct a profit function that takes into account inspection, repair, and goodwill costs, as well as the capacity of each station. Then we compare the profitability of different inspection plans and discuss how to identify the optimal inspection plan. Finally, in the third part of the thesis, we consider the capacity and pricing decisions made by a monopolistic firm producing two heterogeneous products under demand uncertainty. The objective is to maximize profit. Our model incorporates dedicated and flexible resources, product substitutability, and processing rates that may depend on the product and on the resource type. We provide the optimum prices and production quantities as functions of resource capacities and demand intercepts. We also show that investment in flexible capacity is only desirable when it is optimal to invest in dedicated capacities for both products, and obtain upper bounds for the costs of the dedicated capacities that need to be satisfied for investment in the flexible resource. We conclude with numerical examples that illustrate the points discussed and provide insights into how the optimal capacities and expected production quantities, prices, and profit depend on various model parameters.

Peer to peer networks are becoming increasingly popular among Internet users as the downloading peers share the storage and upload bandwidth load of the system. This makes it possible for a large number of users to share a data file available at a server without the server upload bandwidth becoming a bottleneck. The P2P technology is being widely used not only for file sharing but also for video on demand, live streaming and IPTV. The delay deadlines are more stringent in live streaming and IPTV than those in file sharing as the traffic is real time. The performance perceived by a user depends upon whether the video stream is being downloaded at the streaming rate. Coolstreaming is the first large scale P2P IPTV system. We model the multi-channel Coolstreaming system via an open queueing network. The peer dynamics at a channel is modelled by a closed queueing network working at a faster rate. We compute the expected number of substreams in the overlay of New Coolstreaming which are not being received at the proper rate. The computation of the Markov chain with a very large state space is handled using the two time scale decomposition. Further we characterize the end to end delay encountered by a video stream originating from the server and received at a user of New Coolstreaming. Three factors contribute towards the delay. The first factor is the mean path length in terms of overlay hops of the partnership graph. The second factor is the mean number of routers between any two overlay peers in the network layer and the third factor is the queueing delay at a router in the Internet. The mean shortest path length in terms of overlay peers in the New Coolstreaming graph is shown to be O(logn)where nis the number of peers in the overlay. This is done by modelling the overlay by a random graph. The mean shortest path in terms of routers in the Internet’s router level topology is seen to be at most O(logNI)where NIis the number of routers in the Internet. We also discuss a method by which we can get the mean delay at a router in the Internet. Thus, the mean end to end delay in New Coolstreaming is shown to be upper bounded by O(lognlogNIE[W])where E[W]is the mean delay at a router in the Internet.

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Queueing theory is the mathematical study of queues or waiting lines, which are formed whenever demand for service exceeds the capacity to provide service. A queueing system is composed of customers, packets or calls that need some kind of service. These entities arrive at queueing system, join a queue if service is not immediately available and leave system after receiving service. There are also cases when customers, packets or calls leave system without joining queue or drop out without receiving service even after waiting for some time. Queueing network models with finite capacity have facilitated the analysis of discrete flow systems, such as computer systems, transportation networks, manufacturing systems and telecommunication networks, by providing powerful and realistic tools for performance evaluation and prediction. In wireless cellular systems mobility is the most important feature and continuous service is achieved by supporting handoff from one cell to another. Hand off is the process of changing channel associated with the current connection while a call is in progress. A handoff is required when a mobile terminal moves from one cell to another or the signal quality deteriorates in current cell. Since neighbouring cells use disjoint subset of frequency bands therefore negotiation must take place between mobile terminal, the current base station and next potential base station. A poorly designed handoff scheme significantly decreases quality of service (QOS). Different schemes have been devised and in these schemes handoff calls are prioritize. Also most of the performance evaluation techniques consider the case where the arrival process is poisson and service is exponential i.e. there is single arrival and single departure. Whereas in practice there is burstiness in cellular traffic i.e. there can be bulk arrivals and bulk departures. Other issue is that, assumptions made by stochastic process models are not satisfied. Most of the effort is concentrated on providing different interpretations of M/M queues rather than attempting to provide a new methodology. In this thesis performance evaluation of multi traffic cellular models i.e. non priority, priority and hand off calling scheme for bursty traffic are devised. Moreover extensions are carried out towards the analysis of a multi-traffic M/M queueing system and state probabilities are calculated analytically.

Analytical modelling and characterization of Weighted Fair Queueing (WFQ) have recently received considerable attention by several researches since WFQ offers the minimum delay and optimal fairness guarantee. However, all previous work on WFQ has focused on developing approximations of the scheduler with an infinite buffer because of supposed scalability problems in the WFQ computation. The main aims of this thesis are to study WFQ system, by providing an analytical WFQ model which is a theoretical construct based on a form of processor sharing for finite capacity. Furthermore, the solutions for classes with Poisson arrivals and exponential service are derived and verified against global balance solution. This thesis shows that the analytical models proposed can give very good results under particular conditions which are very close to WFQ algorithms, where accuracy of the models is verified by simulations of WFQ model. Simulations were performed with QNAP-2 simulator. In addition, the thesis presents several performance studies signifying the power of the proposed analytical model in providing an accurate delay bounds to a large number of classes. These results are not able to cover all unsolved issues in the WFQ system. They represent a starting point for the research activities that the Author will conduct in the future. The author believes that the most promising research activities exist in the scheduler method to provide statistical guarantees to multi-class services. The author is convinced that alternative software, for example, on the three class model buffer case, is able to satisfy the large number of buffer because of the software limitation in this thesis. While they can be a good topic for long-term research, the short-medium term will show an increasing interest in the modification of the WFQ models to provide differentiated services.

Today, the radio spectrum is rarely fully utilized. This problem is valid in more domains, e.g., time, frequency and geographical location. To provide an efficient utilization of the radio spectrum, the Cognitive Radio Networks (CRNs) have been advanced. The key idea is to open up the licensed spectrum to unlicensed users, thus allowing them to use the so-called spectrum opportunities as long as they do not harmfully interfere with licensed users. An important focus is laid on the limitation of previously reported research efforts, which is due to the limited consideration of the problem of competition among unlicensed users for spectrum access in heterogeneous CRNs. A software framework is introduced, which is called PRioritized Opportunistic spectrum Access System (PROAS). In PROAS, the heterogeneity aspects of CRNs are specifically expressed in terms of cross-layer design and various wireless technologies. By considering factors like ease of implementation and efficiency of control, PROAS provides priority scheduling based solutions to alleviate the competition problem of unlicensed users in heterogenous CRNs. The advanced solutions include theoretical models, numerical analysis and experimental simulations for performance evaluation. By using PROAS, three particular CRN models are studied, which are based on ad-hoc, mesh-network and cellular-network technologies. The reported results show that PROAS has the ability to bridge the gap between research results and the practical implementation of CRNs.

This dissertation focuses on reexamining traditional management problems that emerge in service systems where customers or jobs queue for service. In particular, we investigate how a manger should make maintenance and routing decisions in settings where there is a departure from traditional modeling assumptions. In many cases, the performance evaluation of a management problems has, at its heart, a complex, infinite Markov chain which must be solved before any optimization can begin. Unfortunately, most Markov chains are not analytically tractable. In the first essay, we address the solution of infinite state Markov chains. We focus on class M Markov chains, a broad class of chains which is representative of a wide array of problems arising in the management of computer, service, and manufacturing systems where queueing parameters change over time according to a restricted stochastic pattern. We develop a new method, called Clearing Analysis on Phases, for the limiting probability distribution of such chains in exact closed form. In the second essay, we apply the CAP method to answer the question of how a manager should maintain a system in a setting where an online customer-facing service is vulnerable to persistent malware infections. These infections can cause performance degradation and facilitate data theft, both of which have monetary repercussions. Infections can go undetected and can only be removed by a timeconsuming cleanup procedure, which takes the service offline and causes all existing jobs to be discarded without service. In particular, we provide recommendations for when (and in response to what events) a manager should initiate cleanup procedures by solving an infinite state maintenance problem. We quantify the efficiency of various cleanup (maintenance) policies by proposing a revenue model which incorporates both delay-based pricing and data theft costs. In the third essay, we examine queueing systems in call centers and answer the question of a how a manager should route customers to strategic staff who choose their own service rates in response to workload incentives. We address this problem using game theoretic techniques. In particular, we introduce a utility model where the servers choose their service rate in order to maximize a tradeoff between an “effort cost” and a “value of idleness.” We find that relaxing the classical assumption that all servers work at a fixed rate renders traditional routing policies inadequate. Our approach allows us to recommend novel routing policies that are both fair for the staff and efficient for the customers. In the fourth essay we look at web server farms and answer the question of how jobs should be immediately routed to computer servers in a setting where some jobs are more valuable or more important than others. Such settings arise when some jobs are generated by users who are paying for a premium service. We address how a manager should incorporate information about a job’s value when making routing decisions in order to minimize expected value-weighted response times. The heterogeneity in job values greatly the dimensionality of this problem. Via a combination of exact analysis, asymptotic analysis, and simulation, we are able to deduce many unexpected results regarding routing.

A rapidez na realização do atendimento às vítimas é uma das maiores necessidades de serviços de atendimento médico de urgência, e o tempo decorrido entre o instante da ocorrência da solicitação pelo serviço e o início do atendimento, denominado tempo de resposta, é um dos principais fatores que influenciam o desempenho do sistema. Este tempo depende de uma reunião de fatores como condições de tráfego, dia e período do dia, número de veículos disponíveis e suas localizações, capacitação profissional da equipe, etc. Apresenta-se neste trabalho uma análise do desempenho do serviço oferecido na cidade de Campinas-SP, tratando o problema por meio do modelo hipercubo de filas, que considera as variações aleatórias dos processos de chegadas e atendimento dos chamados. Sua aplicação produz uma ampla variedade de indicadores de desempenho para o sistema, que são comparados com os valores reais observados, para validar a hipótese de aplicação do modelo. Os resultados de sua aplicação para configurações operacionais alternativas, tais como descentralização e aumento do número de ambulâncias, mostraram uma elevação significativa do nível de serviço oferecido ao usuário. Conclui-se que o modelo constitui uma importante ferramenta de análise para este tipo de sistema, auxiliando na tomada de decisões estratégicas e operacionais do sistema. / One of the major concerns of medical emergency systems is to provide the fastest possible medical attention for the victims. The time elapsed between the emergency call and the assistance, called the response time, is one of the main factors that influence the system\'s performance. This time lapse depends on traffic conditions, the day of the week and time of day, the number of available vehicles and their location, the rescue team\'s professional qualifications, etc. This work consists of an analysis of the performance of the emergency service available in Campinas, SP, and deals with the problem using the hypercube queuing model, which considers stochastic variations of the arrival and assistance processes. The application of this model produces a wide variety of system performance indicators, which are compared with the real observed values to validate the model\'s hypothetical application. Application of the model in alternative operational scenarios, such as decentralization and a greater number of ambulances, showed a significant increase in the quality of the service offered to the user. It was concluded that the model constitutes an important analytical tool for this type of system, serving as an aid for strategic and operational decision-making.

<div>Non-stationary point processes are often used to model systems whose rates vary over time. Estimating underlying rate functions is important for input to a discrete-event simulation along with various statistical analyses. We study nonparametric estimators to the marked point process, the infinite-server queueing model, and the transitory queueing model. We conduct statistical inference for these estimators by establishing a number of asymptotic results.</div><div><br></div><div>For the marked point process, we consider estimating the offered load to the system over time. With direct observations of the offered load sampled at fixed intervals, we establish asymptotic consistency, rates of convergence, and asymptotic covariance through a Functional Strong Law of Large Numbers, a Functional Central Limit Theorem, and a Law of Iterated Logarithm. We also show that there exists an asymptotically optimal interval width as the sample size approaches infinity.</div><div><br></div><div>The infinite-server queueing model is central in many stochastic models. Specifically, the mean number of busy servers can be used as an estimator for the total load faced to a multi-server system with time-varying arrivals and in many other applications. Through an omniscient estimator based on observing both the arrival times and service requirements for n samples of an infinite-server queue, we show asymptotic consistency and rate of convergence. Then, we establish the asymptotics for a nonparametric estimator based on observations of the busy servers at fixed intervals.</div><div><br></div><div>The transitory queueing model is crucial when studying a transitory system, which arises when the time horizon or population is finite. We assume we observe arrival counts at fixed intervals. We first consider a natural estimator which applies an underlying nonhomogeneous Poisson process. Although the estimator is asymptotically unbiased, we see that a correction term is required to retrieve an accurate asymptotic covariance. Next, we consider a nonparametric estimator that exploits the maximum likelihood estimator of a multinomial distribution to see that this estimator converges appropriately to a Brownian Bridge.</div>