arXiv subjects

Sophie Hautphenne

Publications and source records attributed to Sophie Hautphenne.

25 records · Page 2Linked to original sources

arXiv2017

The time-dependent expected reward and deviation matrix of a finite QBD process

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples.

math.PR↗

arXiv2015

Sensitivity analysis of a branching process evolving on a network with application in epidemiology

We perform an analytical sensitivity analysis for a model of a continuous-time branching process evolving on a fixed network. This allows us to determine the relative importance of the model parameters to the growth of the population on the network. We then apply our results to the early stages of an influenza-like epidemic spreading among a set of cities connected by air routes in the United States. We also consider vaccination and analyze the sensitivity of the total size of the epidemic with respect to the fraction of vaccinated people. Our analysis shows that the epidemic growth is more sensitive with respect to transmission rates within cities than travel rates between cities. More generally, we highlight the fact that branching processes offer a powerful stochastic modeling tool with analytical formulas for sensitivity which are easy to use in practice.

physics.soc-ph↗

arXiv2015

How old is this bird? The age distribution under some phase sampling schemes

In this paper, we use a finite-state continuous-time Markov chain with one absorbing state to model an individual's lifetime. Under this model, the time of death follows a phase-type distribution, and the transient states of the Markov chain are known as phases. We then attempt to provide an answer to the simple question "What is the conditional age distribution of the individual, given its current phase"? We show that the answer depends on how we interpret the question, and in particular, on the phase observation scheme under consideration. We then apply our results to the computation of the age pyramid for the endangered Chatham Island black robin Petroica traversi during years of intensive conservation efforts in 1980-1989.

math.PR↗

arXiv2014

Lyapunov exponents for branching processes in a random environment: The effect of information

We consider multitype Markovian branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define dual processes and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound and we show that to add more information gives smaller lower bounds. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bound.

math.PR↗

arXiv2013

The Second Order Terms of the Variance Curves for Some Queueing Output Processes

We consider queueing output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve, indicating the variance of the number of served customers over a time interval. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to the results by finding explicit expressions for the second order approximation of the variance curve, namely the y-intercept of the linear asymptote. For M/M/1/K queues our results are based on the Drazin inverse of the generator. It turns out that by viewing output processes as MAPs (Markovian Arrival Processes) and considering the Drazin inverse, one can obtain explicit expressions for the y-intercept, together with some further insight regarding the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1 queues our results are based on a classic transform of D.J. Daley. In this case we represent the y-intercept of the variance curve in terms of the first three moments of the service time distribution. A further performance measure that we are able to calculate for both models, is the asymptotic covariance between the queue length and the number of arrivals or departures. In addition we shed light on a classic conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

math.PR↗

arXiv2012

Extinction probabilities of branching processes with countably infinitely many types

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

math.PR↗

arXiv2012

On the nature of Phase-Type Poisson distributions

Matrix-form Poisson probability distributions were recently introduced as one matrix generalization of Panjer distributions. We show in this paper that under the constraint that their representation is to be nonnegative, they have a physical interpretation as extensions of PH distributions, and we name this restricted family Phase-type Poisson. We use our physical interpretation to construct an EM algorithm-based estimation procedure.

math.PR↗