Richard J. Boucherie

A characterization of independence for competing Markov chains with applications to stochastic Petri nets

This paper shows that some of the recently obtained product form results for stochastic Petri nets can be obtained as a special case of a simple exclusion mechanism for the product process of a collection of Markov chains. >

RUNNING TIMES ON RAILWAY SECTIONS WITH HETEROGENEOUS TRAIN TRAFFIC

In contemporary railway traffic, different train services often run on the same track of a railway section. Since these train services generally travel at different speed, this may result in delays due to fast trains being caught behind slower ones. This note investigates these delays developing a stochastic model capturing both scheduled and unscheduled train movements. Running time distributions for each train service can be obtained solving a system of linear differential equations. This system can be solved explicitly when free running times for all train services are deterministic. The mathematical model falls in the class of infinite server resequencing queues, and the results generalise existing results for such queues, allowing dependence between consecutive interarrival times, as well as between interarrival times and service times. A case study of a railway section in the Dutch railway network illustrates the practical value of the model, both for long and short term railway planning.

On a Queueing NetWork Model for Cellular Mobile Telecommunications Networks

This paper presents a queueing network description of a cellular mobile communications network. These cellular networks are characterised by the call lengths, the fresh call arrival rate, and the handover rates, as well as the capacity restrictions on the number of calls because of limited bandwidth. In the resulting queueing network the routing probabilities depend on the age of the customers. This queueing network is amenable to analysis via known methods. In particular, insensitivity results are generalised to also include capacity restrictions under age-dependent routing and to include call holding time distributions of customers moving from one queue to another.

Estimation of performance measures for product form cellular mobile communications networks

This paper investigates cellular mobile communications networks. The purpose of the paper is twofold. First, it is shown that the restrictive assumption of reversible routing is not required for the network population distribution to be of product form. Different protocols with their specific ways of handling congestion, all of them yielding product form, are discussed. Second, the notoriously difficult task of obtaining performance measures derived from product form expressions is attacked by an efficient method based on importance sampling. This algorithm substantially speeds up the computational time required to estimate, for example, the probability that a call attempting a hand‐over is blocked. In addition, qualitative insight is gained into the network conditional on blocking in a specific cell: are neighbouring cells overloaded as well? The examples include networks with capacity constraints due to effective interference between cells, and a reasonably sized network containing 49 cells and 7 cell reuse groups.

On the arrival theorem for product form queueing networks with blocking

Abstract The arrival theorem is known to be valid for product form queueing networks that do not have capacity constraints at the queues. This paper investigates the validity of the arrival theorem for open and closed queueing networks with finite queues under product form conditions, and derives a necessary and sufficient condition for the validity of the arrival theorem under product form conditions. It will be shown that a product form alone is not a guarantee for the validity of the arrival theorem. Further conditions are required to retain its validity. The paper provides an extensive set of examples and counterexamples, and can therefore also be considered as a survey to provide insight for the validity of the arrival theorem.

Local balance in queueing networks with positive and negative customers

This paper investigates to what extent a recently developed new product form result for queueing networks with positive and negative customers fits into the class of product form queueing networks that satisfy a notion of partial or local balance. As such, this paper investigates whether this new product form is still a consequence of an appropriate notion of local balance. To this end, a new and non-standard type of local balance is introduced as an extension of standard local balance. This new type of local balance appears more restrictive and is no longer directly sufficient for global balance. Nevertheless, based on this new type of local balance, some extensions such as blocking phenomena for queueing networks with positive and negative customers can be concluded.

Norton's equivalent for queueing networks comprised of quasireversible components linked by state-dependent routing

Abstract This paper considers queueing networks comprised of quasireversible components linked by state-dependent routing. It is shown that the product form property is valid despite the state-dependent nature of the routing process. From this observation Nortons theorem for queueing networks is extended to include queueing networks with non-product form components and state-dependent routing, such as most notably including blocking of customers. The methods used to prove the results provide a combination of two approaches to obtain product form results: local balance and quasireversibility.

Markov decision processes in practice

It is over 30 years ago since D.J. White started his series of surveys on practical applications of Markov decision processes (MDP), over 20 years after the phenomenal book by Martin Puterman on the theory of MDP, and over 10 years since Eugene A. Feinberg and Adam Shwartz published their Handbook of Markov Decision Processes: Methods and Applications. In the past decades, the practical development of MDP seemed to have come to a halt with the general perception that MDP is computationally prohibitive. Accordingly, MDP is deemed unrealistic and is out of scope for many operations research practitioners. In addition, MDP is hampered by its notational complications and its conceptual complexity. As a result, MDP is often only briefly covered in introductory operations research textbooks and courses. Recently developed approximation techniques supported by vastly increased numerical power have tackled part of the computational problems; see, e.g., Chaps. 2 and 3 of this handbook and the references therein. This handbook shows that a revival of MDP for practical purposes is justified for several reasons: 1. First and above all, the present-day numerical capabilities have enabled MDP to be invoked for real-life applications. 2. MDP allows to develop and formally support approximate and simple practical decision rules. 3. Last but not least, MDP’s probabilistic modeling of practical problems is a skill if not art by itself.

The invariant measure of random walks in the quarter-plane: respresentation in geometric terms

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a finite linear combination of geometric terms and present conditions on the structure of these linear combinations such that the resulting measure may yield an invariant measure of a random walk. We demonstrate that each geometric term must individually satisfy the balance equations in the interior of the state space and further show that the geometric terms in an invariant measure must have a pairwise-coupled structure. Finally, we show that at least one of the coefficients in the linear combination must be negative.

A linear programming approach to error bounds for random walks in the quarter-plane

We consider the approximation of the performance of random walks in the quarter-plane. The approximation is in terms of a random walk with a product-form stationary distribution, which is obtained by perturbing the transition probabilities along the boundaries of the state space. A Markov reward approach is used to bound the approximation error. The main contribution of the work is the formulation of a linear program that provides the approximation error.