Modelling approaches via (batch) markov modulated poisson processes

Resumen

This dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the (Batch) Markov Modulated Poisson process. Point processes arise in a wide range of situations in physics, biology, engineering, or economics. In general, the occurrence of events is defined depending on the context, but in many areas are defined by the occurrence of an event at a specific time. Sometimes, in order to simplify the models and obtain closed form expressions for the quantities of interest, the exponentiality and/or independence of the inter-event times is assumed. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice, and therefore, there is a need of more realistic models to fit the data. The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival process BMAP which has been widely used for the modeling of dependent and correlated simultaneous events (as arrivals, failures or risk events). Both BMMPP and BMAP allow for dependent and non-exponentially distributed inter-event times as well as for correlated batches, but at the same time they inherit the tractability of the Poisson processes. Therefore, they turn out suitable models to fit data with statistical features that differ form the classical Poisson assumptions. In spite of the large amount of works considering the BMAP subclasses of processes, still there are a number of open problems of interest that will be considered in this dissertation, which is organized as follows. In Chapter 1, a brief theoretical background that introduces the most important concepts and properties that are needed to carry out our analyses is presented. The markovian point processes and their main properties are introduced. In Chapter 2 the identifiability of the stationary BMMPP_m(K) is proven, where K is the maximum batch size and $m$ is the number of states of the underlying Markov chain. This is a powerful result for inferential issues. On the other hand, some findings related to the correlation and autocorrelation structures are provided. Chapter 3 focuses on exploring the possibilities of the BMMPP for the modeling of real phenomena involving point processes with group arrivals. The first result in this sense is the characterization of the BMMPP_2(K) by a set of moments related to the inter-event time and batch size distributions. This characterization leads to a sequential fitting approach via a moments matching method. The performance of the novel fitting approach is illustrated on both simulated and a real teletraffic data set, and compared to that of the EM algorithm. In addition, as an extension of the inference approach, the queue length distributions at departures in the queueing system BMMPP/M/1 is also estimated. Unlike Chapters 2 and 3, which are devoted to the Batch Markov Modulated Poisson Process, Chapter 4 presents an extension to the two-dimensional case of the Markov modulated Poisson process (MMPP), motivated by real failure data in a two-dimensional context. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-event times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence among the two sequences of inter-event times, while at the same time preserves the MMPP properties marginally. Such generalization is based on the Marshall-Olkin exponential distribution. Inference is undertaken for the new process through a method combining a matching moments approach and an ABC algorithm. The performance of the method is shown on simulated and real datasets representing failures of a public transport company. To conclude, Chapter 5 summarizes the most significant contributions of this dissertation, and also gives a short description of possible research lines.