Operations
research (OR) is an interdisciplinary
mathematical science that focuses on the
application of advanced analytical methods to
help make better decisions. Operations Research
methods are used in modelling a wide range of
industrial, environmental, and biological
systems of significance. Its latest technologies
use methods drawn from Probability, Optimization
and Simulation. Operations Research
is a highly
sought-after field of expertise by
industry, which contributes millions of dollars
in benefits and savings each year.
Operations Research 2 (KMA255) and Operations
Research 3 (KMA355) – these OR units develop
skills in the analysis, evaluation and
optimization of real-life systems using
deterministic or probabilistic methods, which
are essential for careers in Engineering,
Management, Finance, and Economics. The applied,
meaningful context of these units means that
they are also highly advantageous for careers in
Teaching.
KMA305 – this (Probabilistic) OR unit develops
skills that are essential in constructing models
for the analysis and performance assessment of
real-life systems with an element of
uncertainty, which, since unpredictability is
the common feature of many real-life systems of
great significance, are essential for careers in
the Physical and Biological Sciences,
Engineering, Computer Science, Finance, and
Economics.
Prospective students:
If you are interested in post-graduate study in
Operations Research, feel free to contact me to
discuss possibilities. Also, see the examples of
topics below.
Markov
Chains are the most important class
of stochastic models in the theory of
Probability, due to their powerful modeling
features and numerical tractability. Markovian-modulated
models are built on the concept of
Markov Chains. Their state space is
two-dimensional and consists of the phase
variable i(t) and the level
variable X(t). Phase is used to model the
state of some real-life environment, while
level is used to model its performance measure
at time t. We assume that the phase changes
according to some underlying Markov Chain.
Further, we assume that the rate at which the
level changes at time t depends on the current
phase i(t). For example, in modelling of a
telecommunications buffer, the phase may
represent the operating switch, while the
level may be the amount of data in the buffer.
The rate at which the buffer empties of fills
in, depends on which switch is on. See the
diagram of a simple two-phase model and a
graph illustrating the evolution of X(t) in
time for a model with 4 phases below.
Markovian-modulated models have attracted a
lot of interest due to their applicability to
a wide range of real-life systems of great
engineering or environmental significance,
well beyond applications in high-speed
telecommunications systems, from which
they were originally derived. Very
interesting results for this class of models
have been obtained in recent years. My recent
interest is in 2-dimensional models. The
applications of these include ad-hoc mobile
networks, the process of coral bleaching,
operation of hydro-power generation, amongst
other examples. I am interested in
constructing useful models, analyzing them
analytically, deriving mathematical
expressions for various important transient
and stationary performance measures and
building efficient algorithms for their
numerical evaluations.
My current
fun: Markovian-modulated Stochastic
2-Dimensional Models
Experience:
During my Masters and PhD research, I made
contributions to the theory of optimal design
and reliability of linear consecutive systems.
This class of systems is applicable throughout
industry, including, for example,
telecommunications, mining, space exploration
and medicine. Specifically, I have developed
several necessary conditions for the optimal
design of linear consecutive-k-out-of-n systems
and procedures to improve designs not satisfying
these conditions, as well as novel results for
variant optimal designs, a type of optimal
design that is particularly difficult to treat.
Both my Master and PhD theses were in the area
of linear consecutive systems.
In 2001, I gave two presentations on linear
consecutive-k-out-of-n systems at the
Optimization Day at the 16th National Conference
of the Australian Society for Operations
Research (ASOR). To see the slides from these
presentations, click here,
and then here.
In recent years, I have been involved in
research in matrix analytic models. This work
resulted in interesting contributions to the
development of fluid flow models, which is a
class of Markov-driven stochastic models. My
contributions include:
·development
of
expressions for various performance measures
and their physical interpretations,
·construction
of
several fast algorithms to evaluate these
performance measures,
·comparison
of
algorithms
with respect to their physical
interpretations, convergence rates, number of
iterations, complexity, and suitability for
analysis,
·treatment
of bounded and unbounded level-independent
models,
·analysis
of
fluid models with an element of
level-dependence,
·some
work on multi-dimensional fluid models.
I gave an invited talk on Fluid Models at the
Tutorial Workshop on Matrix-Analytic Methods for
Stochastic Modelling, organized by ARC Centre for
Excellence for Mathematics and Statistics of
Complex Systems in Melbourne in 2004. To see the
slides from this talk, click here.
Grants:
ARC Discovery Project DP110101663
2011-2013
Advanced matrix analytic methods with applications
with Prof Peter G Taylor, Prof Nigel G Bean, Dr
Sophie M Hautphenne, Dr Mark W Fackrell, Prof Guy
G Latouche
$600,000
Australian Research Council (ARC) Discovery
Project DP0770388
2007-2009
The Use of stochastic fluid models for the
evaluation of applications-driven sample path
integrals
with Prof Nigel Bean
$198,000
Conferences:
2012 - A&NZ
PW - Australia and New Zealand Probability
Workshop, January 23–27, Auckland,
NZ. my
talk
2011 - MAM
7 - Seventh International on Matrix Analytic
Methods in Stochastic Models, June 13–16, New
York, USA. my
talkPostcards
from NY
2008-2010 - Dr Ren Yong (currently a Professor in
the Department of Mathematics at the Anhui Normal
University)
Publications:
Journal articles:
N.G.
Bean
and M.M. O'Reilly. A stochastic fluid model
driven by an uncountable-state process,
which is a stochastic fluid model itself:
Stochastic Fluid-Fluid model.. Submitted.
N.G.
Bean,
M.M. O'Reilly and Yong Ren. Second-order
Markov Reward Models driven by QBD
Processes. Submitted.
N.G.
Bean
and M.M. O'Reilly. Stochastic
2-Dimensional Fluid Model. Submitted.
M.M.
O'Reilly
and Z. Palmowski. Numerical analysis of
linear stochastic double fluid models. In
preparation.
M.M.
O'Reilly
and Z. Palmowski. Loss rate for stochastic
double fluid models. In
preparation.
M.M.
O'Reilly.
2-Stage stochastic fluid model for the
operation of a chocolate factory line. In
preparation.
N.G.
Bean,
M.M. O'Reilly and J. Sargison. Stochastic
fluid flow model of the operations and
maintenance of power generation systems. IEEE
Transactions on Power Systems, 25
(3) pp. 1361-1374, 2010.
A.E.R.
Helfgott,
N.G. Bean, S. Connolly, A. Baird and M.M.
O’Reilly. Modelling the resilience of coral
reefs to global climate change: A stochastic
fluid model of the adaptive bleaching
hypothesis on the great barrier reef. In
preparation.
N.G.
Bean,
M.M. O'Reilly and P.G. Taylor. Hitting
probabilities and hitting times for
stochastic fluid flows: the Bounded Model. Probability in
the Engineering and Informational Sciences,
23 (1) : 121-147, 2009.
N.G. Bean and, M.M. O'Reilly. N.G. Bean and
M.M. O'Reilly. Performance measures of a
multi-layer Markovian fluid model. Annals of
Operations Research, 160 (1): 99-120,
2008.
N.G. Bean, M.M. O'Reilly and P.G. Taylor.
Algorithms for the Laplace-Stieltjes
transforms of the first return probabilities
for stochastic fluid flows. Methodology and
Computing in Applied Probablility, 10
(3): 381-408, 2008.
N.G.
Bean,
M.M. O'Reilly and P.G. Taylor. Hitting
probabilities and hitting times for
stochastic fluid flows. Stochastic
Processes and Their Applications
115(9): 1530-1556, 2005.
N.G.
Bean,
M.M. O'Reilly and P.G. Taylor. Algorithms
for the first return probabilities for
stochastic fluid flows. Stochastic
Models 21(1): 149-184, 2005.
Book
chapters:
M.
O’Reilly. Optimal design of linear
consecutive k-out-of-n systems. Springer
Optimization and Its Applications, Springer,
C Pearce and E Hunt (ed), New York, pp.
307-326, 2009.
M.
O’Reilly. The (k+1)-th component of linear
consecutive-k-out-of-n systems. Springer
Optimization and Its Applications, Springer,
C Pearce and E Hunt (ed), New York, pp.
327-342, 2009.
M.
O’Reilly. Variant optimal designs of linear
consecutive-k-out-of-n systems. In: Industrial
Mathematics and Statistics, J.C. Misra
, editor, Narosa Publishing House, New
Delhi, 486-502, 2003.
Theses:
PhD
thesis:
"Necessary conditions for the optimal design
of linear consecutive systems", University
of Adelaide, October 2001.
Masters
thesis: "A hypothesis by Derman,
Lieberman and Ross", Wroclaw University,
1987.
Other
contributions:
J.G. Sumner and P.D. Jarvis. Using the tangle:
a consistent construction of phylogenetic
distance matrices for quartels.
General
method for computing times in an M/G/1 queue.
Teletraffic Research Centre, Department of
Applied Mathematics, University of Adelaide,
October 1996.
