11,471 research outputs found
Markovian queues with correlated arrival processes
In an attempt to examine the effect of dependencies in the arrival process on the steady state
queue length process in single server queueing models with exponential service time distribution,
four different models for the arrival process, each with marginally distributed exponential interarrivals
to the queueing system, are considered. Two of these models are based upon the upper
and lower bounding joint distribution functions given by the Fréchet bounds for bivariate
distributions with specified marginals, the third is based on Downton’s bivariate exponential
distribution and fourthly the usual M/M/1 model. The aim of the paper is to compare conditions
for stability and explore the queueing behaviour of the different models
Birth/birth-death processes and their computable transition probabilities with biological applications
Birth-death processes track the size of a univariate population, but many
biological systems involve interaction between populations, necessitating
models for two or more populations simultaneously. A lack of efficient methods
for evaluating finite-time transition probabilities of bivariate processes,
however, has restricted statistical inference in these models. Researchers rely
on computationally expensive methods such as matrix exponentiation or Monte
Carlo approximation, restricting likelihood-based inference to small systems,
or indirect methods such as approximate Bayesian computation. In this paper, we
introduce the birth(death)/birth-death process, a tractable bivariate extension
of the birth-death process. We develop an efficient and robust algorithm to
calculate the transition probabilities of birth(death)/birth-death processes
using a continued fraction representation of their Laplace transforms. Next, we
identify several exemplary models arising in molecular epidemiology,
macro-parasite evolution, and infectious disease modeling that fall within this
class, and demonstrate advantages of our proposed method over existing
approaches to inference in these models. Notably, the ubiquitous stochastic
susceptible-infectious-removed (SIR) model falls within this class, and we
emphasize that computable transition probabilities newly enable direct
inference of parameters in the SIR model. We also propose a very fast method
for approximating the transition probabilities under the SIR model via a novel
branching process simplification, and compare it to the continued fraction
representation method with application to the 17th century plague in Eyam.
Although the two methods produce similar maximum a posteriori estimates, the
branching process approximation fails to capture the correlation structure in
the joint posterior distribution
Compound Markov counting processes and their applications to modeling infinitesimally over-dispersed systems
We propose an infinitesimal dispersion index for Markov counting processes.
We show that, under standard moment existence conditions, a process is
infinitesimally (over-) equi-dispersed if, and only if, it is simple
(compound), i.e. it increases in jumps of one (or more) unit(s), even though
infinitesimally equi-dispersed processes might be under-, equi- or
over-dispersed using previously studied indices. Compound processes arise, for
example, when introducing continuous-time white noise to the rates of simple
processes resulting in Levy-driven SDEs. We construct multivariate
infinitesimally over-dispersed compartment models and queuing networks,
suitable for applications where moment constraints inherent to simple processes
do not hold.Comment: 26 page
A Spatio-Temporal Point Process Model for Ambulance Demand
Ambulance demand estimation at fine time and location scales is critical for
fleet management and dynamic deployment. We are motivated by the problem of
estimating the spatial distribution of ambulance demand in Toronto, Canada, as
it changes over discrete 2-hour intervals. This large-scale dataset is sparse
at the desired temporal resolutions and exhibits location-specific serial
dependence, daily and weekly seasonality. We address these challenges by
introducing a novel characterization of time-varying Gaussian mixture models.
We fix the mixture component distributions across all time periods to overcome
data sparsity and accurately describe Toronto's spatial structure, while
representing the complex spatio-temporal dynamics through time-varying mixture
weights. We constrain the mixture weights to capture weekly seasonality, and
apply a conditionally autoregressive prior on the mixture weights of each
component to represent location-specific short-term serial dependence and daily
seasonality. While estimation may be performed using a fixed number of mixture
components, we also extend to estimate the number of components using
birth-and-death Markov chain Monte Carlo. The proposed model is shown to give
higher statistical predictive accuracy and to reduce the error in predicting
EMS operational performance by as much as two-thirds compared to a typical
industry practice
Relative fixed-width stopping rules for Markov chain Monte Carlo simulations
Markov chain Monte Carlo (MCMC) simulations are commonly employed for
estimating features of a target distribution, particularly for Bayesian
inference. A fundamental challenge is determining when these simulations should
stop. We consider a sequential stopping rule that terminates the simulation
when the width of a confidence interval is sufficiently small relative to the
size of the target parameter. Specifically, we propose relative magnitude and
relative standard deviation stopping rules in the context of MCMC. In each
setting, we develop sufficient conditions for asymptotic validity, that is
conditions to ensure the simulation will terminate with probability one and the
resulting confidence intervals will have the proper coverage probability. Our
results are applicable in a wide variety of MCMC estimation settings, such as
expectation, quantile, or simultaneous multivariate estimation. Finally, we
investigate the finite sample properties through a variety of examples and
provide some recommendations to practitioners.Comment: 24 page
- …