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Research shows that in a system model, when the production rate is adjusted based on the number of items in queue, the nature of the model changes from an open-loop queueing system to a closed-loop feedback control system. Service-rate adjustment can be implemented in a discrete event simulation model, but the effect of this adjustment has not been thoroughly analyzed in the literature. This research considers the design of feedback signals to generate realistic simulation models of production system behavior. A series of simulation experiments is conducted to provide practical guidance for simulation modelers on how adding a service-rate adjustment feedback loop to a queueing system affects system performance.

If either the arrival rate or the service rate in an M/M/S queue exhibit variability over time, then no steady state solution is available for examining the system behavior. The arrival and service rates can be represented through Fourier series approximations. This permits numerical approximation of the system characteristics over time. An example of an M/M/S representation of the operations of emergency treatment at Logan Regional hospital is presented. It requires numerical integration of the differential equation for L(t), the expected number of customers in the system at time t.

Thousands of call centers operate in the United States employing millions of people. Since personnel costs represent as much as 80% of the total operating expense of these centers, it is important for call center managers to determine an appropriate staffing level required to maintain the desired operating performance. Historically, queueing models serve an important role in this regard. The one most commonly used is the Erlang-C model. The Erlang-C model has several assumptions, however, which are required for the predicted performance measures to be valid. One assumption that has received significant attention from researchers is that callers have infinite patience and will not terminate a call until the service is complete regardless of the wait time. Since this assumption is not likely to occur in reality, researchers have suggested using Erlang-A instead. Erlang-A does consider caller patience and allows for calls to be abandoned prior to receiving service. However, the use of Erlang-A still requires an assumption that is very unlikely to occur in practice - the assumption that all agents provide service at the same rate. Discrete event simulation is used to examine the effects of agent heterogeneity on the operating performance of a call center compared to the theoretical performance measures obtained from Erlang-A. Based on the simulation results, it is concluded that variability in agent service rate does not materially affect call center performance except to the extent that the variability changes the average handle time of the call center weighted by the number of calls handled and not weighted by agent. This is true regardless of call center size, the degree of agent heterogeneity, and the distribution shape of agent variability. The implication for researchers is that it is unnecessary to search for an analytic solution to relax the Erlang-A assumption that agents provide service at the same rate. Several implications for managers are discussed including the reliability of using Erlang-A to determine staffing levels, the importance of considering the service rates of the agents rather than the average handle time, and the unintended consequence of call routing schemes which route calls to faster rather than slower agents.

In this thesis, we study the tradeoff of average delay with average service cost and average utility for both continuous time and discrete time single server queueing models without and with admission control. The continuous time and discrete time queueing models that we consider are motivated by cross-layer models for point-to-point links with random packet arrivals and fading at slow and fast time scales. Our studies are motivated by the need to optimally tradeoff the average delay of the packets (a network layer performance measure) with the average service cost of transmitting the packets, e.g. the average power required for transmission (a physical layer performance measure) under a lower bound constraint on the average throughput, in various point-to-point communication scenarios. The tradeoff problems are studied for a class of monotone and stationary scheduling policies and under the assumption that the service cost rate and utility rate are respectively convex and concave functions of the service rate and arrival rate. We also consider the problem of optimally trading off the average delay and average error rate of randomly arriving message symbols which are transmitted over a noisy point-to-point link, in which case the service cost function is non-convex. The solutions to the tradeoff problems that we address in the thesis are asymptotic in nature, and are similar in spirit to the Berry-Gallager asymptotic bounds. It is intuitive that to keep a queue stable under a lower bound constraint on the average utility a minimum number of customers have to be served per unit time. This in turn implies that queue stability requires a minimum average service cost expenditure. In the thesis we obtain an asymptotic characterization of the minimum average delay for monotone stationary policies subject to an upper bound constraint on the average service cost and a lower bound constraint on the average utility, in the asymptotic regime where the average service cost constraint is made arbitrarily close to the above minimum average service cost. In the thesis, we obtain asymptotic lower bounds on the minimum average delay for the cases for which lower bounds were previously not known. The asymptotic characterization of the minimum average delay for monotone stationary policies, for both continuous time and discrete time models, is obtained via geometric bounds on the stationary probability of the queue length, in the above asymptotic regime. The restriction to monotone stationary policies enables us to obtain an intuitive explanation for the behaviour of the asymptotic lower bounds using the above geometric bounds on the stationary probability distribution of the queue length. The geometric bounds on the stationary probability of the queue length also lead to a partial asymptotic characterization of the structure of any optimal monotone stationary policy, in the above asymptotic regime, which was not available in previous work. Furthermore, the geometric bounds on the stationary probability can be extended to analyse the tradeoff problem in other scenarios, such as for other continuous time queueing models, multiple user communication models, queueing models with service time control, and queueing models with general holding costs. Usually, queueing models with integer valued queue evolution, are approximated by queueing models with real valued queue evolution and strictly convex service cost functions for analytical tractability. Using the asymptotic bounds, we show that for some cases the average delay does not grow to infinity in the asymptotic regime, although the approximate model suggests that the average delay does grow to infinity. In other cases where the average delay does grow to infinity in the asymptotic regime, our results illustrate that the tradeoff behaviour of the approximate model is different from that of the original integer valued queueing model unless the service cost function is modelled as the piecewise linear lower convex envelope of the service cost function for the original model.

<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>

This research considers the transmission of real-time data within a wireless local area network (WLAN). Exact and approximate analytic network evaluation techniques are examined. The suitability of using a given technique in a particular situation is discussed. Simulation models are developed to study the performance of our protocol RT-MAC (real-time medium access control). RT-MAC is a novel, simple, and elegant MAC protocol for use in transmitting real-time data in point to point ad hoc WLAN. Our enhancement of IEEE 802.11, RT-MAC, achieves dramatic reductions in mean delay, missed deadlines, and packet collisions by selectively discarding packets and sharing station state information. For example, in a 50 station network with a normalized offered load of 0.7, mean delay is reduced from more than 14 seconds to less than 45 ms, late packets are reduced from 76% to less than 1%, and packet collisions are reduced from 36% to less than 1%. Stations using RT-MAC are interoperable with stations using IEEE 802.11. In networks with both RT-MAC and IEEE 802.11 stations, significant performance improvements were seen even when more than half of the stations in the network were not RT-MAC stations. The effect of the wireless channel and its impact on the ability of a WLAN to meet packet deadlines is evaluated. It is found that, in some cases, other factors such as the number of stations in the network and the offered load are more significant than the condition of the wireless channel. Regression models are developed from simulation data to predict network behavior in terms of throughput, mean delay, missed deadline ratio, and collision ratio. Telemetry, avionics, and packetized voice traffic models are considered. The applicability of this research is not limited to real-time wireless networks. Indeed, the collision reduction algorithm of RT-MAC is independent of the data being transported. Furthermore, RT-MAC would perform equally well in wired networks. Incorporating the results of this research into existing protocols will result in immediate and dramatic improvements in network performance. / Ph. D.

The master’s thesis solves models of queueing systems, which use the property of Markov chains. The queueing system is a system, where the objects enter into this system in random moments and require the service. This thesis solves specifically such models of queueing systems, in which the intervals between the objects incomings and service time have exponential distribution. In the theoretical part of the master’s thesis I deal with the topics stochastic process, queueing theory, classification of models and description of the models having Markovian property. In the practical part I describe realization and function of the program, which solves simulation of chosen model M/M/m. At the end I compare results which were calculated in analytic way and by simulation of the model M/M/m.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-10-26T17:09:15Z No. of bitstreams: 1 TeseSCF.pdf: 10747360 bytes, checksum: 8a9b640ece219d76af8f66468c70af0a (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:25:43Z (GMT) No. of bitstreams: 1 TeseSCF.pdf: 10747360 bytes, checksum: 8a9b640ece219d76af8f66468c70af0a (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-11-08T18:25:51Z (GMT) No. of bitstreams: 1 TeseSCF.pdf: 10747360 bytes, checksum: 8a9b640ece219d76af8f66468c70af0a (MD5) / Made available in DSpace on 2016-11-08T18:25:59Z (GMT). No. of bitstreams: 1 TeseSCF.pdf: 10747360 bytes, checksum: 8a9b640ece219d76af8f66468c70af0a (MD5) Previous issue date: 2016-06-22 / Não recebi financiamento / This work deals with the analysis of queueing systems for Call Centers regarding the possibility of the customer abandon the system before being served, due to his/her impatience. The Call Centers are service organizations that predominantly serve customers by phone calls and are considered a particular kind of Contact Centers that serve their costumers through phone, fax, e-mail, chat, mobile devices and other communication channels. From a business point of view, the main concern on the management of the Call Center is the offer quality service with minimum cost. The perception of the quality of services offered is expressed, by customer, for example, through the abandonment of queue, which is considered one of the most important operational measures to evaluate the performance of a Call Center. This work aims to present and apply analytical queueing models with abandonment, represented by generic probability distributions (particularly mixed distributions), as an effective analysis approach to represent the problem of congestion in Call Center systems and support decisions of dimensioning and operations on these systems. Two studies were conducted with Call Center extracted data of a company located in the countryside of São Paulo State, Brazil and another located overseas, in Israel. The parameters (e.g, arrival rate, service rate, abandonment rate) and some measures of performance (e.g, average waiting time, waiting probability, abandonment probability and traffic intensity) were established based on these data. These sampling measures were equated with the same measures achieved by the analytical queueing models M/M/c+G, M/Gc/1+G and M/G/c+G considered in this research, using the parameters obtained empirically and mixed, Exponential, Fatigue Life, Normal and Lognormal distributions to represent the abandonment (patience) of users. It was observed that in some cases, depending on the considered performance measure, the queueing models with mixed distributions for the abandonment have better results (minor deviations, compared to the real data) than their corresponding with nonmixed distributions. It was observed, also, that independently of the analyzed performance measures, the Call Centers may be represented by a analytical queueing model with a mixed distribution for abandonment times, which behaves better than the others. There were no evidences that the mixed distributions to represent the abandonment times were the best in all analyzed performance measures, but they were always competitive. It was used, also, an experimental model of discrete simulation that properly represents the Call Center in order to check the results of the analytical models and explore alternative scenarios. The same scenarios were considered with the theoretical model and the performance measures achieved by the simulation and by the theoretical models were compared, showing the potential of the use of the approaches based on analytical models with abandonment for the Call Centers analysis. / Este trabalho trata da análise de sistemas de filas para Call Center considerando a possibilidade do cliente abandonar o sistema antes de ser servido, devido a sua impaciência. Os Call Centers são organizações de serviço que predominantemente servem os clientes via chamada telefônica e são considerados um tipo particular dos Centros de Contato, que atendem seus clientes por meio de telefone, fax, e-mail, chat, dispositivos móveis e outros canais de comunicação. Do ponto de vista dos negócios, a principal preocupação na gestão de um Call Center é oferecer serviço de qualidade com mínimo custo. A percepção da qualidade dos serviços oferecidos é manifestada, pelo cliente, por exemplo, por meio do abandono da fila de espera, que é considerado uma das medidas operacionais mais importantes para avaliar o desempenho de um Call Center. O objetivo deste trabalho é apresentar e aplicar modelos analíticos de filas com abandono, representado por distribuições genéricas (particularmente distribuições mistas), como uma abordagem de análise efetiva para representar o problema de congestão em sistemas de Call Centers e apoiar decisões de dimensionamento e operação nesses sistemas. Dois estudos foram conduzidos com dados extraídos do Call Center de uma empresa localizada no interior do Estado de São Paulo e de outro localizado no exterior, em Israel. Os parâmetros (por exemplo, taxa de chegada, taxa de serviço e taxa de abandono) e algumas medidas de desempenho (por exemplo, tempo médio de espera, probabilidade de esperar, probabilidade de abandonar e intensidade de tráfego) são determinadas com esses dados. Essas medidas amostrais são comparadas com as mesmas medidas obtidas por meio dos modelos analíticos de fila M/M/c+G, M/Gc/1+G e M/G/c+G considerados nesse estudo, que utilizam os parâmetros obtidos empiricamente e as distribuições mistas, Exponencial, Fatigue Life, Normal e Lognormal para representar o abandono (paciência) dos usuários. Observou-se como resultado, que em alguns casos, dependendo da medida de desempenho considerada, os modelos de fila com distribuição mista para o abandono têm apresentado resultados melhores (desvios menores em relação aos dados reais) que os seus correspondentes com distribuições não mistas. Observou-se, também, que independentemente das medidas de desempenho analisadas, os Call Centers podem ser representados por um modelo analítico de fila com uma distribuição mista para os tempos de abandono, que se comporta melhor do que os outros. Não houve evidências que as distribuições mistas para representar os tempos de abandono foram as melhores em todas as medidas de desempenho analisadas, mas foram sempre competitivas. Utilizou-se, também, um modelo experimental de simulação discreta, que representa apropriadamente o Call Center, para verificar os resultados dos modelos analíticos e explorar cenários alternativos. Os mesmos cenários foram considerados com o modelo teórico e as medidas de desempenho obtidas pela simulação e pelo modelo teórico foram comparadas, mostrando o potencial do uso de abordagens baseadas em modelos analíticos com abandono para análise de Call Centers.

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