Perfect and Nearly Perfect Sampling of Work-conserving Queues

We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers. Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called nearly perfect sampling. Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method. Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems

Long range dependence in network traffic and the closed loop behaviour of buffers under adaptive window control

We consider an Internet link carrying http-like traffic, i.e., transfers of finite volume files arriving at random time instants. These file transfers are controlled by an adaptive window protocol (AWP); an example of such a protocol is TCP. We provide analysis for the auto-covariance function of the AWP-controlled traffic into the link's buffer; this traffic, in general, cannot be represented by an on-off process. The analysis establishes that, for TCP-controlled transfer of Pareto-distributed file sizes with infinite second moment, the traffic into the link buffer is long range-dependent (LRD). We also develop an analysis for obtaining the stationary distribution of the link buffer occupancy under an AWP-controlled transfer of files sampled from some distribution. For any AWP, the analysis provides us with the Laplace-Stieltjes transform (LST) of the distribution of the link buffer occupancy process in terms of the functions defining the AWP and the file size distribution. The analysis also provides a necessary and a sufficient condition for the finiteness of the mean link buffer content; these conditions again have explicit dependence on the AWP used and the file size distribution. This establishes the sensitivity of the buffer occupancy process to the file size distribution. Combining the results from the above analyses, we provide various examples in which the closed loop control of an AWP results in finite mean link buffer occupancy even though the file sizes are Pareto-distributed (with infinite second moment), and the traffic into the link buffer is long range-dependent (with Hurst parameters which would suggest an infinite mean queue occupancy under open loop analysis). We also study the effect of window reductions due to active queue management and find that window reductions lead to further lightening of the tail of buffer occupancy distribution. The significance of this work is three-fold: (i) by looking at the window evolution as a function of the amount of data served and not as a function of time, this work provides a new framework for analysing various processes related to the link buffer under AWP-controlled transfer of files with a general file size distribution; (ii) it indicates that the buffer behaviour in the Internet may not be as poor as predicted from an open loop analysis of a queue fed with LRD traffic; and (iii) it shows that the buffer behaviour (and hence the throughput performance for finite buffers) is sensitive to the distribution of file sizes

Queueing Systems with Heavy Tails

VI+227hlm.;24c

Statistical Applications in Healthcare Systems

This thesis consists of three contributing manuscripts related to waiting times with possible applications in health care. The first manuscript is inspired by a practical problem related to decision making in an emergency department (ED). As short-run predictions of ED censuses are particularly important for efficient allocation and management of ED resources we model ED changes and present estimations for short term (hourly) ED censuses at each time point. We present a Markov-chain based algorithm to make census predictions in near future. Considering the variation in arrival pattern and service requirements, we apply and compare three models which best describe our data. We provide hourly predictions up to 24 hours in a day which will provide suggestions to ED managers on how to prevent over-crowding in their system. We illustrate our approach using 22 months data obtained from the ED of a hospital in south western Ontario. The next two manuscripts extend the theory underlying the Accumulating Priority queues (APQs). We focus on the queues with two classes of customers and Poisson arrivals. The first work in this topic derives the stationary waiting time distributions for the class of lowest priority customers in an Affine Accumulating Priority queues (Affine APQs). APQs were first studied by Kleinrock (1964) and later revisited by Stanford et al (2014) where they obtained explicit solution for the Laplace Stieltjes Transform (LST) of the stationary waiting times for all classes of customers. All subsequent publications on APQs, have assumed that all arriving customers accumulate priority credits over time starting from the same initial value (assumed, without loss of generality, to be 0). Whereas, our model studies Affine APQs which assume different initial priorities (without loss of generality in a two-class setting we assume the lowest class starts with 0 credit and the higher class customers with positive credit a. In this work we determine the waiting time distributions for the lower class of customers with Poisson arrivals and general service and present some numerical results for special cases of M/M/1, M/M/c and M/D/1. Inspired by health care applications, we have also considered a particular optimization problem related to the Affine APQ model, in order to select the optimum accumulation rate which allows for the lowest class customers to meet their associated KPIs. We next focus on the Analysis of the Maximum priority processes in the context of Affine APQ. Maximum Priority Processes were first introduced in the context of APQs in Stanford et al (2014). We derive the LST of the stationary steady state distributions of the Maximum Priority Processes as recursive functions and derive the explicit solutions for the LSTs in classical APQ (i.e. a = 0). We employ this argument to present a new approach to determine the LST of waiting time distribution for an APQ with two-classes of customers under the M=M=1 discipline. Since the Analysis of the Maximum Priority Processes in this work is done for the general class of Affine APQs, it has provided the grounds for future researches to obtain the LST of the waiting time distributions in the Affine APQs

APPLICATION OF MODELING AND SIMULATION IN A MANUFACTURING SYSTEM

The aim ofthis project is to develop a simulation model of an air conditioners manufacturing system with a discrete event simulation tool. The model would be utilized as a decision support system for the investigation of improving the process by implementing several options like cost cutting and simplifying operation. This report discusses steps in the development of a simulation model for a manufacturing system using the DES tool, ARENA. A modeling procedure for the development of manufacturing simulation model is presented. The current manufacturing system model is developed to ascertain its limitations and problems to achievethe production target. The steps include data gathering, model building, verification and validation. Several experiments were conducted to recognize parameters useful in the interpretation of the simulation data like the warm up period, run length and number of repetition. The results show that the manufacturing system was improved by 40% by speeding up parts delivery to the system, whilst the waiting time andqueue at each station can be improved by proper line balancing. The findings demonstrates the ability if the approach to provide potential solution to the decision maker

Scheduling for today’s computer systems: bridging theory and practice

Scheduling is a fundamental technique for improving performance in computer systems. From web servers to routers to operating systems, how the bottleneck device is scheduled has an enormous impact on the performance of the system as a whole. Given the immense literature studying scheduling, it is easy to think that we already understand enough about scheduling. But, modern computer system designs have highlighted a number of disconnects between traditional analytic results and the needs of system designers. In particular, the idealized policies, metrics, and models used by analytic researchers do not match the policies, metrics, and scenarios that appear in real systems. The goal of this thesis is to take a step towards modernizing the theory of scheduling in order to provide results that apply to today’s computer systems, and thus ease the burden on system designers. To accomplish this goal, we provide new results that help to bridge each of the disconnects mentioned above. We will move beyond the study of idealized policies by introducing a new analytic framework where the focus is on scheduling heuristics and techniques rather than individual policies. By moving beyond the study of individual policies, our results apply to the complex hybrid policies that are often used in practice. For example, our results enable designers to understand how the policies that favor small job sizes are affected by the fact that real systems only have estimates of job sizes. In addition, we move beyond the study of mean response time and provide results characterizing the distribution of response time and the fairness of scheduling policies. These results allow us to understand how scheduling affects QoS guarantees and whether favoring small job sizes results in large job sizes being treated unfairly. Finally, we move beyond the simplified models traditionally used in scheduling research and provide results characterizing the effectiveness of scheduling in multiserver systems and when users are interactive. These results allow us to answer questions about the how to design multiserver systems and how to choose a workload generator when evaluating new scheduling designs