Bo Friis Nielsen

A Markovian approach for modeling packet traffic with long-range dependence

We present a simple Markovian framework for modeling packet traffic with variability over several time scales. We present a fitting procedure for matching second-order properties of counts to that of a second-order self-similar process. Our models essentially consist of superpositions of two-state Markov modulated Poisson processes (MMPPs). We illustrate that a superposition of four two-state MMPPs suffices to model second-order self-similar behavior over several time scales. Our modeling approach allows us to fit to additional descriptors while maintaining the second-order behavior of the counting process. We use this to match interarrival time correlations.

Vertical migration and dispersion of sprat (Sprattus sprattus) and herring (Clupea harengus) schools at dusk in the Baltic Sea

In populations of herring (Clupea harengus) or sprat (Sprattus sprattus), one typically observes a pattern of schools forming at dawn and dispersing at dusk, usually combined with vertical migration. This behaviour influences interactions with other species; hence a better understanding of the processes could contribute to deeper insight into ecosystem dynamics. This paper reports field measurements of the dispersal at dusk and examines two hypotheses through statistical modelling: that the vertical migration and the dissolution of schools is determined by decrease in light intensity, and that the dissolution of schools can be modelled by diffusion, i.e. active repulsion is not required. The field measurements were obtained during 3 days in March at one location in the Baltic Sea and included continuous hydroacoustical monitoring, trawl samples, and hydrographical CTD data. Echogram patterns were analysed using the school detection module in Echoview ® and local light intensities were calculated using a model for surface illuminance. The data and the analysis support that schools migrate upwards during dusk, possibly trying to remain aggregated by keeping the local light intensities above a critical threshold, that schools initiate their dissolution when ambient light intensity drops below this critical threshold, and that fish subsequently swim in an uncorrelated random walk pattern.

An application of superpositions of two state Markovian source to the modelling of self-similar behaviour

We present a modelling framework and a fitting method for modelling second order self-similar behaviour with the Markovian arrival process (MAP). The fitting method is based on fitting to the autocorrelation function of counts a second order self-similar process. It is shown that with this fitting algorithm it is possible closely to match the autocorrelation function of counts for a second order self-similar process over 3-5 time-scales with 8-16 state MAPs with a very simple structure, i.e. a superposition of 3 and 4 interrupted Poisson processes (IPP) respectively and a Poisson process. The fitting method seems to work well over the entire range of the Hurst (1951) parameter.

Quasi-Birth-and-Death Processes with Rational Arrival Process Components

This paper introduces the concept of a Quasi-Birth-and-Death process (QBD) with Rational Arrival Process (RAP) components. We use the physical interpretation of the prediction process of the RAP, developed by Asmussen and Bladt, and develop an analysis that parallels the analysis of a traditional QBD. Further, we present an algorithm for the numerical evaluation of the matrix G. As an example, we consider two queues where the arrival process and the sequence of service times are taken from two dependent RAPs, that are not Markovian Arrival Processes.

Phase-type models of channel-holding times in cellular communication systems

In this paper, we derive the distribution of the channel-holding time when both cell-residence and call-holding times are phase-type distributed. Furthermore, the distribution of the number of handovers, the conditional channel-holding time distributions, and the channel-holding time when cell residence times are correlated are derived. All distributions are of phase type, making them very general and flexible. The channel-holding times are of importance in performance evaluation and simulation of cellular mobile communication systems.

Queues with waiting time dependent service

Motivated by service levels in terms of the waiting-time distribution seen, for instance, in call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steady-state waiting-time distributions for both models. The results are illustrated with numerical examples.

On the time reversal of Markovian Arrival Processes

Abstract We study the point process obtained by reversing time in a stationary Markovian Arrival Process (MAP). That process is also a MAP. We show that the most frequently used classical statistical descriptors of point processes are insensitive to the orientation of the time-axis. Therefore they fail to distinguish between a MAP and its reverse. That is the case for the second order descriptors of the counting and interval processes. Actually, for a MAP and its reverse the marginal distributions of the counting and interval processes agree. Using simple examples, we demonstrate that the behavior of two queues, one with a MAP and the other with its reverse as input streams, can be very different. This, in spite of the agreement of most of the standard descriptors. These findings illustrate the limitations of most of the standard descriptors of point processes in predicting queueing behavior. Quantification of non-reversibility could lead to new informative descriptors.

First in Line Waiting Times as a Tool for Analysing Queueing Systems

We introduce a new approach to modelling queueing systems where the priority or the routing of customers depends on the time the first customer has waited in the queue. This past waiting time of the first customer in line, WFIL, is used as the primary variable for our approach. A Markov chain is used for modelling the system where the states represent both the number of free servers and a discrete approximation to WFIL. This approach allows us to obtain waiting time distributions for complex systems, such as the N-design routing scheme widely used, e.g., in call centers and systems with dynamic priorities.

On the Construction of Bivariate Exponential Distributions with an Arbitrary Correlation Coefficient

In this article we use the concept of multivariate phase-type distributions to define a class of bivariate exponential distributions. This class has the following three appealing properties. Firstly, we may construct a pair of exponentially distributed random variables with any feasible correlation coefficient (also negative). Secondly, the class satisfies that any linear combination (projection) of the marginal random variables is a phase-type distribution. The latter property is partially important for the development of hypothesis testing in linear models. Finally, it is easy to simulate the exponential random vectors.

A computational framework for a quasi birth and death process with a continuous phase variable

A computational framework for implementing Tweedies operator geometric approach is developed, and its feasibility illustrated through some examples. The work is motivated by the following facts which are of interest to performance analysis of B-ISDN: (a) phase type models with a continuum of phases provide a framework to incorporate heavy tailed stationary queue length distributions and long range correlations; (b) the set of popular models based on chaotic maps developed by Erramilli, Pruthi, and Singh, fall under that category as special cases.