# Mini-conference on Probability Theory & Its Applications

*Date:*

*Location: JH Michell Theatre @ Richard Berry Bldg, Parkville Campus of the University of Melbourne*

The next "micro-conference" on Probability Theory & Its Applications is proudly supported by MASCOS

This time there will be six (6) 40+5' talks start from 11am to 5:45pm.

### PROGRAM

Thursday, 15 November 2012, 11:00am +

Venue: JH Michell Theatre @ Richard Berry Bld, Parkville Campus, Melbourne Uni

11:00 - 11:40 | [11:45 end of question time] Talk 1: The Maximal Priority Process in the M/G/1 Accumulating Priority Queue. Speaker: Peter Taylor (Melbourne Uni) |

11:45 - 12:25 | [12:30 end of question time] Talk 2: On the limiting distribution of Pitman estimators. Speaker: Alexander Novikov (UTS) |

12:30 - 14:15pm | Lunch break |

14:15 - 14:55 | [15:00 end of question time] Talk 3: Sojourn measures for heavy-tailed random fields. Speaker: Andriy Olenko (La Trobe University) |

15:00 - 15:40 | [15:45 end of question time] Talk 4: Optimal control of convex switching systems. Speaker: Juri Hinz (UTS) |

15:45 - 16:15 | Tea/coffee break (tearoom, Richard Berry Bldg) |

16:15 - 16:55 | [17:00 end of question time] Talk 5: Reinforcement processes. Speaker: Andrea Collevecchio (Monash University) |

17:00 - 17:40 | [17:45 end of question time] Talk 6: Extinction probabilities of branching processes with countably infinitely many types. Speaker: Sophie Hautphenne (Melbourne Uni) |

18:30pm + | Dinner (venue TBA) |

### APPENDIX: Abstracts of the talks

**Talk 1: The Maximal Priority Process in the M/G/1 Accumulating Priority Queue.**

*Speaker*: Peter Taylor (The University of Melbourne)*Abstract*: It is often desirable to differentiate between waiting times experienced by different classes of customers in a queue. For example, in a hospital emergency department, patients are assigned to triage categories according to the acuity of their condition and the waiting time that they should receive depends on their triage category.

Waiting times cannot be controlled in a queue where customers of one class have absolute priority over customers of another class, and sometimes the service levels of high priority customers are better than they need to be at the expense of low priority customers. To address this problem, back in the 1960s, Kleinrock proposed that customers could have priority levels that increase linearly with their time in the queue and, when the server becomes free, it selects the customer with the highest accumulated priority for the next service.

For such a model, Kleinrock showed that the accumulation rates can be adjusted to achieve any ratios of expected waiting times that fall within a specified feasible set.

In this talk, we revisit Kleinrock's accumulating priority queue to derive its waiting time distributions, and study how the priority accumulation mechanism can be used to meet service levels defined in terms of tails of waiting time distributions.

[Joint work with David Stanford and Ilze Ziedins.]

**Talk 2: On the limiting distribution of Pitman estimators.**

*Speaker*: Alexander Novikov (UTS)*Abstract*: In some non-regular statistical estimation problems, the limiting likelihood processes are generated by fractional Brownian motion (fBm) with Hurst's parameter H, 0<H≤1. For these cases we present several analytical and numerical results on asymptotics of moments and distribution of Pitman estimators, also known as Bayesian estimators with a constant prior on real line.

We also present selected Monte Carlo simulation results for the limit variance and quantiles of Pitman estimators.

[Joint work with Nino Kordzakhia (Macquarie University)]

**Talk 3: Sojourn measures for heavy-tailed random fields.**

*Speaker*: Andriy Olenko (La Trobe University)

*Abstract*: In this talk limit theorems for the volumes of excursion sets of strongly dependent heavy-tailed random fields are presented.

Generalizations to sojourn measures above moving levels and for cross-correlated scenarios are discussed. Special attention is paid to Student and Fisher random fields. Some simulation results are shown.

**Talk 4: Optimal control of convex switching systems**

*Speaker*: Juri Hinz (UTS)*Abstract*: In this talk, we discuss the so-called convex switching systems. This is a class of specific Markov decision problems, which frequently appear in industrial applications with a challenging policy optimization, due to high dimensional state spaces.

In a convex switching system, the state dynamics is driven by a finite set of possible actions, and the state space consists of two

components: The first component takes a finite number of values and is under deterministic control, whereas the second component follows an uncontrolled linear state dynamics.

We show that under slight additional assumptions, the value functions of such systems exhibit convexity properties, which we utilize in the construction of efficient numerical policy optimization schemes.

Furthermore, we provide convergence results for such algorithms and illustrate our approach by a number of examples.

**Talk 5: Reinforcement processes.***Speaker*: Andrea Collevecchio (Monash University)

*Abstract*: Reinforced random walk (RRW) is a broad class of processes which jump between nearest neighbor vertices of graphs, and prefer visiting often visited ones over seldom visited ones. This "nostalgia"

makes RRW strongly dependent on the whole past, so it is not a Markov process. We present an overview of the theory of RRW defined on trees and $\Z^d$. We also present results involving a general class of preferential attachment models.

**Talk 6: Extinction probabilities of branching processes with countably infinitely many types***Speaker*: Sophie Hautphenne (The University of Melbourne)

*Abstract*: 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.

[Joint work with Guy Latouche (Université libre de Bruxelles), and Giang T. Nguyen (University of Adelaide)]