Iain M MacPhee
Department of Mathematical Sciences
Iain MacPhee passed away on 13th January 2012. This page will be left as it was.
A brief mathematical biography of Iain can be found here.
A one-day meeting was held in memory of Iain on 3 April 2014: see
here.
 |
Job: |
Lecturer |
| Address: |
Dept. of Mathematical Sciences,
University of Durham,
South Road,
Durham DH1 3LE,
England |
| FAX: |
(0191 or +44 191) 334 3051 |
| E-mail: |
i.m.macphee@durham.ac.uk |
| Phone: |
(0191 or +44 191) 334 3106 |
My research interests and links to publications
I have interests in a range of probability and operations research
areas. My recent work is on problems such as
Some years ago now I worked on
Since his arrival in Durham I have worked with
Mikhail Menshikov on
various problems concerning the long-term behaviour
(recurrence, transience) of discrete time random walks. Our
approach is via a collection of results on local expected drifts of
the process (Foster's criterion is a simple result of this type)
rather than using stationarity conditions or transform theory. This
allows us to relax assumptions about homogeneity of jump distributions
though we do need conditions on the moments of the jumps.
Very complex models can be studied in this way. We have applied these
techniques to interacting particle systems with hybrid interaction,
long term behaviour of random walk in the plane with mild regularity
conditions, various phenomena displayed by random walks on
two-dimensional complexes (collections of quarter planes connected at
their boundaries) inspired by models from computer science.
In recent years Misha and I, together with
Andrew Wade
who has recently moved to Strathclyde University in Glasgow, have put
a great deal of effort into various models using local expected
drifts and Lyapunov functions to study stability and long-term behaviour.
We found some stability results for a recent interacting particle
system with mixed voter-exclusion moves. A draft of "Passage-time
moments and hybrid zones for the exclusion-voter model" can be found
on arxiv at can be found
here . This paper has been accepted by Bernoulli to appear in
2010. We still have some ideas to explore in the particle system model.
We have also established necessary and sufficient conditions for
a.s. finite exit times of weakly homogeneous random
walks from cones. We have a large number of results which we have
recently organised into two papers. A preliminary draft, "Exit times
from cones for non-homogeneous random walk with asymptotically zero
drift" can be found
here . A rather
different version of some of the results here, which we've called
Angular asymptotics ...
has been accepted by
Markov Chains and Related Fields to appear in 2010.
Models inspired by polling systems
Polling systems have many queues with a single server and a rule to
determine the order the queues are serviced in and when the server
should change queue. A random walk on a complex made by connecting a
set of orthants at their boundaries provides a natural model of any
queueing systems where occasional control decisions e.g. movement of
the server in a polling system or more generally re-arrangement of
servers in a queueing system, change the process dynamics in a
wholesale way.
In 2006/07 Misha and I worked with
Dimitri Petritis and
Serguei Yu Popov
on a polling system model (exponential service times and Poisson
arrivals) with parameter regeneration. By this we mean
that the arrival and service rates can change each time the server
switches to another queue. Our results have now appeared in the Annals
of Applied Probability: the case where there are only two stations in 2007 and the
general case in
2008. This model displays some interesting behaviour. Even though all
service times have exponential moments there can be a thick region of
the parameter space where the process is null recurrent. Similarly
there can be regions where the process is positive recurrent but
return times to the empty state have only finitely many moments. This
heavy-tail type behaviour arises because the system sometimes has
an overall arrival rate larger than the service rate so that extremely
large queues develop.
In 2003/04 we wrote, together with Serguei Popov and
Stas Volkov, a
paper on periodic behaviour for a type of stochastic billiards model
that arises from projecting the trajectories of a transient three
queue exhaustive polling system onto the unit simplex. It appeared in
the Annals of Applied Probability in 2006 and can be downloaded from
Project Euclid
here. We expected that under a greedy switching rule (go to the
queue with the most work) the server would eventually visit the
queues in some periodic order but found that chaotic switching could
occur on a parameter set of measure zero.
My first
paper with Misha appeared in the Annals of Applied Probability in
2003. It contains a detailed analysis of the possible behaviours of a
critical random walk on a complex of quarter planes joined at their
boundaries - with just two quarter planes this would model a polling
system with two queues.
Most recently I have been looking at the supermarket model together
with Misha and Marina
Vachkovskaia. In this model
arrivals are routed to queues through a neighbourhood structure. The
long term dynamics of a random walk based on this model shows some
interesting clustering behaviour which we have tried to explain
here .
From 2004 to 2006 I worked with a PhD student Lisa Müller on
identifying criteria for stability and instability of two queue
systems which have a range of different service regimes - these each
have Poisson arrival streams and exponential service times but the
parameters vary between regimes. Our aim was to specify all sets of
system parameters such that there exist control policies (state
dependent rules for deciding which service regime to apply) under
which the system is stable or alternatively when the system is
unstable under any control.
The notion of service regimes is a useful one as it allows very
general purpose models but lets us describe the decision problem in a
simple manner. It also fits together well with the available Lyapunov
function or semi-martingale methods, see e.g. Fayolle, Malyshev and
Menshikov's
book , which deal very well with systems where the
transition law is homogeneous over large regions.
In the course of the work it became clear to me that under some mild
requirements on the available controls the very natural conditions we
had found for stability of the two queue system could be extended to
larger systems. The conditions are described in terms of the convex
hull of the set of one step drift vectors (one for each service
regime). The resulting paper appeared in Queueing Systems, V52 #3 in
2006. Here is a link to
an
earlier draft if you are interested in reading about our model
and results.
We have also looked at extensions of our result for N queue sytems to
Lu-Kumar type systems where some queues only receive jobs forwarded
from other servers within the system - such systems do not satisfy the
boundary reflection condition used in the QSys paper but there is a
simple condition that permits the results to be extended. We have
written a further paper which appeared in Methodology and Computing in
Applied Probability V9 #3 in 2007 (here is a link to
a
draft).
It contains an application of our ideas to a model
studied by Nino-Mora and Glazebrook (J. Appl. Prob 2000).
We also have some results for the model with phase type service
distributions and some ideas for treating switching times of various
types. Lisa in now working the cycling organisation Sustrans so
progress has slowed but I plan to return to this work when I have time.
My colleague
Frank Coolen is very interested in foundational issues in
statistics and decision theory and their application within operations
research problems. He has many publications in the field of
nonparametric predictive inference (NPI). In discussion with
Pauline
Coolen-Schrijner it became clear that there was a a
potentially elegant solution to an interesting application of NPI to a
system reliability model. Pauline and I made a good start on this
problem and a joint paper with Pauline and Frank on this work has appeared
in the Journal of Risk and Reliability, V222, pp 463-476.
I am now supervising another graduate student, Ahmad Aboalkhair, who
started in May 2008. We are working together with Frank Coolen on
extending the system reliability model discussed above. Progress so
far has been good and we have had a paper proving an extension
of the results with Coolen-Schrijner and Coolen accepted by JRR.
I worked for quite a while on admissions and routing problems in
queueing networks, particularly loss networks where arrivals that
overflow buffers are lost. I have a paper with
Ilze Ziedins , `Admission controls for loss networks with
diverse routing' in
Stochastic
Networks , a 1996 O.U.P. book edited by Kelly, Zachary and
Ziedins.
From 1993 to 1995 I had a PhD student called Ben Jordan who became
very interested in search problems, particularly the problem of
searching a finite number of sites for a target that moves from site
to site. Each search of site i costs C_i and fails to find the target
a) if it is not there; b) with chance a_i even if it is there.
The problem with only 2 sites and Markovian dynamics had a conjectured
solution due to Sheldon Ross sometime in the 1970s. Before each search
calculate p, the probability that the target is at site 1. Ross
suggests we search site 1 precisely when p > = P* (a threshold that
depends on the costs, overlook probabilities and transition matrix)
and otherwise search site 2.
Ben and I wrote a paper establishing the conjecture for almost all
transition matrices which was published in PEIS V9, pp159-182. Here is
a
.pdf file of a draft of the paper for those who do not have easy
access to the 1995 issues of PEIS.
Ben did a lot of numerical work and symbolic manipulation on the two
site problem but also on cases with three and four sites. Alas he
went to work as a currency trader before completing this work.
Last updated: 19/3/09