Bounds on the mean and squared coefficient of variation of phase-type distributions

We consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.

Extensions of Markov Modulated Poisson Processes and Their Applications to Deep Earthquakes

<p>The focus of this thesis is on the Markov modulated Poisson process (MMPP) and its extensions, aiming to propose appropriate statistical models for the occurrence patterns of main New Zealand deep earthquakes. Such an attempt might be beyond the scope of the MMPP and its extensions, however we hope its main patterns can be characterized by current models proposed in three parts of the thesis. The first part of the thesis is concerned with introductions and preliminaries of discrete time hidden Markov models (HMMs) and MMPP. The exibility in model formulation and openness in model framework of HMMs are reviewed in this part, suggesting also possible extensions of MMPP. The second part of the thesis is mainly about several extensions of MMPP. One extension of MMPP is by associating each occurrence of MMPP with a mark. Such an extension is potentially useful for spatial-temporal modelling or other point processes with marks. A special case of this type of extension is by allowing the multiple observations of MMPP synchronized together under the same Markov chain. This extension opens the possibility of modelling multiple point process observations with weak dependence. The third extension is motivated by the attempt to describe small scale temporal clustering existing in the deep earthquakes via treating the recognized aftershocks as marks which itself forms a finite point process. The rest of the second part focuses on some information theoretical aspects of MMPPs such as the entropy rate of the underlying Markov chain and observed point process respectively and their mutual information rate. A conjecture on the possible links between mutual information rate of MMPP and the Fisher information of the estimated parameters is suggested. The second part on extensions of MMPP is featured by the derivation of the likelihood and complete likelihood, parameter estimation via EM algorithm, state smoothing estimation and model evaluation through systematic applications of rescaling theory of multivariate point processes and marked point processes. The third part of the thesis includes the applications of these methods to the deep earthquakes in New Zealand. We first evaluate the data coverage, catalogue completeness and explore its descriptive characteristics and empirical properties such as epicentral distributions, depth distributions and magnitude distributions. Clustering behavior is studied via the second order moment analysis of point processes in the chapter 8. We also apply, the stress release models and the ETAS models which are usually used for shallow earthquakes, to the New Zealand deep earthquakes and provide tentative explanations of why they are not satisfactory for the deep earth-quakes. The chapter 9 is on the applications of MMPP and its extensions to the New Zealand deep earthquakes. Conclusions and future studies are presented in chapter 10.</p>

Extensions of Markov Modulated Poisson Processes and Their Applications to Deep Earthquakes

<p>The focus of this thesis is on the Markov modulated Poisson process (MMPP) and its extensions, aiming to propose appropriate statistical models for the occurrence patterns of main New Zealand deep earthquakes. Such an attempt might be beyond the scope of the MMPP and its extensions, however we hope its main patterns can be characterized by current models proposed in three parts of the thesis. The first part of the thesis is concerned with introductions and preliminaries of discrete time hidden Markov models (HMMs) and MMPP. The exibility in model formulation and openness in model framework of HMMs are reviewed in this part, suggesting also possible extensions of MMPP. The second part of the thesis is mainly about several extensions of MMPP. One extension of MMPP is by associating each occurrence of MMPP with a mark. Such an extension is potentially useful for spatial-temporal modelling or other point processes with marks. A special case of this type of extension is by allowing the multiple observations of MMPP synchronized together under the same Markov chain. This extension opens the possibility of modelling multiple point process observations with weak dependence. The third extension is motivated by the attempt to describe small scale temporal clustering existing in the deep earthquakes via treating the recognized aftershocks as marks which itself forms a finite point process. The rest of the second part focuses on some information theoretical aspects of MMPPs such as the entropy rate of the underlying Markov chain and observed point process respectively and their mutual information rate. A conjecture on the possible links between mutual information rate of MMPP and the Fisher information of the estimated parameters is suggested. The second part on extensions of MMPP is featured by the derivation of the likelihood and complete likelihood, parameter estimation via EM algorithm, state smoothing estimation and model evaluation through systematic applications of rescaling theory of multivariate point processes and marked point processes. The third part of the thesis includes the applications of these methods to the deep earthquakes in New Zealand. We first evaluate the data coverage, catalogue completeness and explore its descriptive characteristics and empirical properties such as epicentral distributions, depth distributions and magnitude distributions. Clustering behavior is studied via the second order moment analysis of point processes in the chapter 8. We also apply, the stress release models and the ETAS models which are usually used for shallow earthquakes, to the New Zealand deep earthquakes and provide tentative explanations of why they are not satisfactory for the deep earth-quakes. The chapter 9 is on the applications of MMPP and its extensions to the New Zealand deep earthquakes. Conclusions and future studies are presented in chapter 10.</p>

Optimal Open-Loop Routing and Threshold-Based Allocation in TWO Parallel QUEUEING Systems with Heterogeneous Servers

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics.