- Location: The University of Melbourne
- Organiser: Kostya Borovkov

** Speakers and talks**

**Peter Taylor**(Melbourne Uni).*The Maximal Priority Process in the M/G/1 Accumulating Priority Queue.***Alexander Novikov**(UTS).*On the limiting distribution of Pitman estimators*.**Andriy Olenko**(La Trobe University).*Sojourn measures for heavy-tailed random fields*.**Juri Hinz**(UTS).*Optimal control of convex switching systems*.**Andrea Collevecchio**(Monash University).*Reinforcement processes*.**Sophie Hautphenne**(Melbourne Uni).*Extinction probabilities of branching processes with countably infinitely many types.*

**Peter Taylor.** * The Maximal Priority Process in the M/G/1 Accumulating Priority Queue.*

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.

**Alexander Novikov** (UTS). *On the limiting distribution of Pitman estimators.*

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)]

**Andriy Olenko** (La Trobe University). *Sojourn measures for heavy-tailed random fields.*

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.

**Juri Hinz (UTS)**. *Optimal control of convex switching systems*

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.

**Andrea Collevecchio** (Monash University). * Reinforcement processes.*

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.

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