On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

Efficient duration modelling in the hierarchical hidden semi-Markov models and their applications

Modeling patterns in temporal data has arisen as an important problem in engineering and science. This has led to the popularity of several dynamic models, in particular the renowned hidden Markov model (HMM) [Rabiner, 1989]. Despite its widespread success in many cases, the standard HMM often fails to model more complex data whose elements are correlated hierarchically or over a long period. Such problems are, however, frequently encountered in practice. Existing efforts to overcome this weakness often address either one of these two aspects separately, mainly due to computational intractability. Motivated by this modeling challenge in many real world problems, in particular, for video surveillance and segmentation, this thesis aims to develop tractable probabilistic models that can jointly model duration and hierarchical information in a unified framework. We believe that jointly exploiting statistical strength from both properties will lead to more accurate and robust models for the needed task. To tackle the modeling aspect, we base our work on an intersection between dynamic graphical models and statistics of lifetime modeling. Realizing that the key bottleneck found in the existing works lies in the choice of the distribution for a state, we have successfully integrated the discrete Coxian distribution [Cox, 1955], a special class of phase-type distributions, into the HMM to form a novel and powerful stochastic model termed as the Coxian Hidden Semi-Markov Model (CxHSMM). We show that this model can still be expressed as a dynamic Bayesian network, and inference and learning can be derived analytically.Most importantly, it has four superior features over existing semi-Markov modelling: the parameter space is compact, computation is fast (almost the same as the HMM), close-formed estimation can be derived, and the Coxian is flexible enough to approximate a large class of distributions. Next, we exploit hierarchical decomposition in the data by borrowing analogy from the hierarchical hidden Markov model in [Fine et al., 1998, Bui et al., 2004] and introduce a new type of shallow structured graphical model that combines both duration and hierarchical modelling into a unified framework, termed the Coxian Switching Hidden Semi-Markov Models (CxSHSMM). The top layer is a Markov sequence of switching variables, while the bottom layer is a sequence of concatenated CxHSMMs whose parameters are determined by the switching variable at the top. Again, we provide a thorough analysis along with inference and learning machinery. We also show that semi-Markov models with arbitrary depth structure can easily be developed. In all cases we further address two practical issues: missing observations to unstable tracking and the use of partially labelled data to improve training accuracy. Motivated by real-world problems, our application contribution is a framework to recognize complex activities of daily livings (ADLs) and detect anomalies to provide better intelligent caring services for the elderly.Coarser activities with self duration distributions are represented using the CxHSMM. Complex activities are made of a sequence of coarser activities and represented at the top level in the CxSHSMM. Intensive experiments are conducted to evaluate our solutions against existing methods. In many cases, the superiority of the joint modeling and the Coxian parameterization over traditional methods is confirmed. The robustness of our proposed models is further demonstrated in a series of more challenging experiments, in which the tracking is often lost and activities considerably overlap. Our final contribution is an application of the switching Coxian model to segment education-oriented videos into coherent topical units. Our results again demonstrate such segmentation processes can benefit greatly from the joint modeling of duration and hierarchy

Traffic analysis and modeling in PMR systems

Cataloged from PDF version of article.Reliable knowledge of traffic in PMR (Private Mobile Radio) systems is essential for assessing the issues in migration from analog to digital and trunked PMR systems. In this work, we investigated two concepts. First, we modeled the service time distribution of conventional PMR networks by using teletraffic data of a conventional PMR network. It is found that the density of the service time is a shifted exponential which is delayed by 0.7 second. The mean service time is about 2.5 seconds. We showed that voice call arrivals to a transmission trunked PMR network are not Poisson distributed. Analytical and simulation methods based on M/G/C (a C server queue with Poisson input and general service) models may not model the system as well as G/G/C (a C server queue with general input and general service) models. Several trunked PMR systems have been designed over the last decade, most of which have symmetric downlink and uplink channel capacities. These systems may not be spectrally efficient in case of group or broadcast-based voice and data calls, a common feature of PMR systems. Second, we studied a new asymmetric PMR system comprising of a wideband OFDM (Orthogonal Frequency Division Multiplexing)-based downlink, such as Digital Audio Broadcasting (DAB) system. We found that for 2 % GoS and a mean service time of 2.86 second, PMR users that can be supported by the proposed system is 315000.Can, BaşakM.S

Approximation of Message Inter-Arrival and Inter-Departure Time Distributions in IMS/NGN Architecture Using Phase-Type Distributions, Journal of Telecommunications and Information Technology, 2013, nr 3

Currently it is assumed that requirements of the information society for delivering multimedia services will be satisfied by the Next Generation Network (NGN) architecture, which includes elements of the IP Multimedia Subsystem (IMS) solution. In order to guarantee Quality of Service (QoS), NGN has to be appropriately designed and dimensioned. Therefore, proper traffic models should be proposed and applied. This requires determination of queuing models adequate to message inter-arrival and inter-departure time distributions in the network. In the paper the above mentioned distributions in different points of a single domain of NGN are investigated, using a simulation model developed according to the latest standards and research. Relations between network parameters and obtained message inter-arrival as well as inter-departure time distributions are indicated. Moreover, possibility of approximating the above mentioned distributions using phase-type distributions is investigated, which can be helpful in identifying proper queuing models and constructing an analytical model suitable for NGN

Long Term Evolution-Advanced and Future Machine-to-Machine Communication

Long Term Evolution (LTE) has adopted Orthogonal Frequency Division Multiple Access (OFDMA) and Single Carrier Frequency Division Multiple Access (SC-FDMA) as the downlink and uplink transmission schemes respectively. Quality of Service (QoS) provisioning is one of the primary objectives of wireless network operators. In LTE-Advanced (LTE-A), several additional new features such as Carrier Aggregation (CA) and Relay Nodes (RNs) have been introduced by the 3rd Generation Partnership Project (3GPP). These features have been designed to deal with the ever increasing demands for higher data rates and spectral efficiency. The RN is a low power and low cost device designed for extending the coverage and enhancing spectral efficiency, especially at the cell edge. Wireless networks are facing a new challenge emerging on the horizon, the expected surge of the Machine-to-Machine (M2M) traffic in cellular and mobile networks. The costs and sizes of the M2M devices with integrated sensors, network interfaces and enhanced power capabilities have decreased significantly in recent years. Therefore, it is anticipated that M2M devices might outnumber conventional mobile devices in the near future. 3GPP standards like LTE-A have primarily been developed for broadband data services with mobility support. However, M2M applications are mostly based on narrowband traffic. These standards may not achieve overall spectrum and cost efficiency if they are utilized for serving the M2M applications. The main goal of this thesis is to take the advantage of the low cost, low power and small size of RNs for integrating M2M traffic into LTE-A networks. A new RN design is presented for aggregating and multiplexing M2M traffic at the RN before transmission over the air interface (Un interface) to the base station called eNodeB. The data packets of the M2M devices are sent to the RN over the Uu interface. Packets from different devices are aggregated at the Packet Data Convergence Protocol (PDCP) layer of the Donor eNodeB (DeNB) into a single large IP packet instead of several small IP packets. Therefore, the amount of overhead data can be significantly reduced. The proposed concept has been developed in the LTE-A network simulator to illustrate the benefits and advantages of the M2M traffic aggregation and multiplexing at the RN. The potential gains of RNs such as coverage enhancement, multiplexing gain, end-to-end delay performance etc. are illustrated with help of simulation results. The results indicate that the proposed concept improves the performance of the LTE-A network with M2M traffic. The adverse impact of M2M traffic on regular LTE-A traffic such as voice and file transfer is minimized. Furthermore, the cell edge throughput and QoS performance are enhanced. Moreover, the results are validated with the help of an analytical model