Resumen de (Batch) Markovian arrival processes: the identifiability issue and other applied aspects
Joanna Virginia Rodríguez Cesar
This dissertation is mainly motivated by the problem of statistical modeling via a specific point process, namely, the Batch Markovian arrival processes. Point processes arise in a wide range of situations of our daily activities, such as people arriving to a bank, claims of an insurance company or failures in a system. They are defined by the occurrence of an event at a specific time, where the event occurrences may be understood from different perspectives, either by the arrival of a person or group of people in a waiting line, the different claims to the insurance companies or failures occurring in a system. Point processes are defined in terms of one or several stochastic processes which implies more versatility than mere single random variables, for modeling purposes. A traditional assumption when dealing with the analysis of point processes is that the occurrence of events are independent and identically distributed, which considerably simplifies the theoretical calculations and computational complexity, and again because of simplicity, the Poisson process has been widely considered in stochastic modelling. However, the independence and exponentiability assumptions become unrealistic and restrictive in practice. For example, in teletraffic or insurance contexts it is usual to encounter dependence amongst observations, high variability, arrivals occurring in batches, and therefore, there is a need of more realistic models to fit the data. In particular, in this dissertation we investigate new theoretical and applied properties concerning the (batch) Markovian arrival processes, or (B)MAP, which is well known to be a versatile class of point process that allows for dependent and non-exponentially distributed inter-event times as well as correlated batches. They inherit the tractability of the Poisson processes, and turn out suitable models to fit data with statistical features that differ from the classical Poisson assumptions. In addition, in spite of the large amount of works considering the BMAP, still there are a number of open problems which are of interest and which shall be considered in this dissertation. This dissertation is organized as follows. In Chapter 1, we present a brief theoretical background that introduces the most important concepts and properties that are needed to carry out our analyses. We give a theoretical background of point processes and describe them from a probabilistic point of view. We introduce the Markovian point processes and its main properties, and also provide some point process estimation backdrop with a review of recent works. An important problem to consider when the statistical inference for any model is to be developed is the uniqueness of its representation, the identifiability problem. In Chapter 2 we analyze the identifiability of the non-stationary two-state MAP. We prove that, when the sample information is given by the inter-event times, then, the usual parametrization of the process is redundant, that is, the process is nonidentifiable. We present a methodology to build an equivalent non-stationary two-state MAPs from any fixed one. Also, we provide a canonical and unique parametrization of the process so that the redundant versions of the same process can be reduced to its canonical version. In Chapter 3 we study an estimation approach for the parameters of the non-stationary version of the MAP under a specific observed information. The framework to be considered is the modelling of the failures of N electrical components that are identically distributed, but for which it is not reasonable to assume that the operational times related to each component are independent and identically distributed. We propose a moments matching estimation approach to fit the data to the non-stationary two-state MAP. A simulated and a real data set provided by the Spanish electrical group Iberdrola are used to illustrate the approach. Unlike Chapters 2 and 3, which are devoted to the Markovian arrival process, Chapters 4 and 5 focus on its arrivals-in-batches counterpart, the BMAP. The capability of modeling non-exponentially distributed and dependent inter-event times as well as correlated batches makes the BMAP suitable in different real-life settings as teletraffic, queueing theory or actuarial contexts, to name a few. In Chapter 4 we analyze the identifiability issue of the BMAP. Specifically, we explore the identifiability of the stationary two-state BMAP noted as BMAP2(k), where k is the maximum batch arrival size, under the assumptions that both the inter-event times and batches sizes are observed. It is proven that for k ? 2 the process cannot be identified. The proof is based on the construction of an equivalent BMAP2(k) to a given one, and on the decomposition of a BMAP2(k) into k BMAP2(2)s. In Chapter 5 we study the auto-correlation functions of the inter-event times and batch sizes of the BMAP. This chapter examines the characterization of both auto-correlation functions for the stationary BMAP2(k), for k ? 2, where four behavior patterns are identified for both functions for the BMAP2(2). It is proven that both auto-correlation functions decrease geometrically as the time lag increases. Also, the characterization of the autocorrelation functions has been extended for the general BMAPm(k) case, m ? 3. To conclude, Chapter 6 summarizes the most significant contributions of this dissertation, and also give a short description of possible research lines.
