Ilze Ziedins

Virtual Partitioning by Dynamic Priorities: Fair and Efficient Resource-Sharing by Several Services

We propose a scheme for sharing an unbuffered resource, such as bandwidth or capacity, by various services. The scheme assigns a nominal capacity to each service class and implements a form of virtual partitioning by means of state-dependent priorities. That is, instead of each class of traffic having a fixed priority, as in traditional trunk reservation schemes, the priorities depend on the state of the system. An approximate method of analysis based on fixed point equations is given. Numerical results are obtained from the approximation, exact computations and simulations. The results show that the scheme is robust, fair and efficient.

Hierarchical virtual partitioning: algorithms for virtual private networking

This paper discusses the hierarchical virtual partitioning (HVP) concept, which facilitates the sharing of a resource, such as a link, by multiple customers, each having various service classes. Calls of each service class have distinctive bandwidth requirements, arrival rates, and mean holding times, and quality of service (QoS) is in terms of call blocking probabilities. The algorithms devised provide for the desired mix of fairness, robustness, and multiplexing efficiency at each of two levels — namely, at the customer level and at the level of the various services of each customer. The HVP scheme relies on nominal allocations of capacities to each customer and service, as well as priorities in the call admission process implemented by dynamic trunk reservations. The paper gives an approximate method of analysis based on fixed-point equations. In addition, a reward/penalty paradigm is devised to reflect the issues in virtual private networking, and the performance of our algorithms is compared to the optimal solution. The results show that the HVP scheme is fair, efficient, and robust.Hierarchical virtual partitioning facilitates sharing of a resource, such as a link, by multiple customers, each with multiple service classes. Calls of each service class have distinctive bandwidth requirements, arrival rates and mean holding times. The quality of service is in terms of call blocking probabilities. The algorithms allow the desired mix of fairness, robustness and multiplexing efficiency at each of two levels, namely, customers and the various services of each customer. The scheme relies on nominal allocations of capacities to each customer and service, and priorities in the call admission process, which are implemented by dynamic trunk reservations. An approximate method of analysis based on fixed point equations is given. A reward/penalty paradigm is devised to reflect the issues in virtual private networking and the performance of our algorithm compared to the optimal solution. The results show that the scheme is fair, efficient and robust.

Conceptual Challenges in Coordinating Theoretical and Data-Centered Estimates of Probability.

A core component of informal statistical inference is the recognition that judgments based on sample data are inherently uncertain. This implies that instruction aimed at developing informal inference needs to foster basic probabilistic reasoning. In this article, we analyze and critique the now-common practice of introducing students to both “theoretical” and “experimental” probability, typically with the hope that students will come to see the latter as converging on the former as the number of observations grows. On the surface of it, this approach would seem to fit well with objectives in teaching informal inference. However, our in-depth analysis of one eighth-graders reasoning about experimental and theoretical probabilities points to various pitfalls in this approach. We offer tentative recommendations about how some of these issues might be addressed.

The Downs-Thomson Paradox: Existence, Uniqueness and Stability of User Equilibria

Consider a network where two routes are available for users wishing to travel from a source to a destination. On one route (which could be viewed as private transport) service slows as traffic increases. On the other (which could be viewed as public transport) the service frequency increases with demand. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. Using the model proposed by Calvert [10], with a ⋅|M|1 queue corresponding to the private transport route, and a bulk-service infinite server queue modelling the public transport route, we give a complete analysis of this system in the setting of probabilistic routing. We obtain the user equilibria (which are not always unique), and determine their stability.

A Modelling Perspective on Probability

In this chapter, we argue for three interconnected ways of thinking about probability—“true” probability, model probability, and empirical probability—and for attention to notions of “good”, “poor” and “no” model. We illustrate these ways of thinking from the simple situation of throwing a die to the more complex situation of modelling bed numbers in an intensive care unit, which applied probabilists might consider. We then propose a reference framework for the purpose of thinking about the teaching and learning of probability from a modelling perspective and demonstrate with examples the thinking underpinning the framework. Against this framework we analyse a theory-driven and a data-driven learning approach to probability modelling used by two research groups in the probability education field. The implications of our analysis of these research groups’ approach to learning probability and of our framework and ways of thinking about probability for teaching are discussed.

Asymptotically optimal control of parallel tandem queues with loss

We consider admission and routing controls for a system of N parallel tandem queues with finite buffers as N becomes large, with the aim of minimizing costs due to loss. We obtain the fluid limit as N→∞, and solve a related optimization problem. Asymptotically, for N large, the optimal cost and associated control take one of two forms, depending on the ratio between the cost of blocking an arrival at entry and discarding after service at the first queue.

Admission controls for Erlang's loss system with service times distributed as a finite sum of exponential random variables

It is known that a threshold policy (or trunk reservation policy) is optimal for Erlangs loss system under certain assumptions. In this paper we examine the robustness of this policy under departures from the standard assumption of exponential service times (call holding times) and give examples where the optimal policy has a generalized trunk reservation form.

Monotonicity properties of user equilibrium policies for parallel batch systems

We study a simple network with two parallel batch-service queues, where service at a queue commences when the batch is full and each queue is served by infinitely many servers. A stream of general arrivals observe the current state of the system on arrival and choose which queue to join to minimize their own expected transit time. We show that for each set of parameter values there exists a unique user equilibrium policy and that it possesses various monotonicity properties. User equilibrium policies for probabilistic routing are also discussed and compared with the state-dependent setting.

A Paradox in a Queueing Network with State-Dependent Routing and Loss

Consider a network of parallel finite tandem queues with two stages, where each arrival attempts to minimize its own cost due to loss. It is known that the user optimal and asymptotic system optimal policies may differ—we give examples showing that they may differ for finite systems and that as the service rate is increased at the second stage the user optimal policy may change in such a way that the total expected cost due to loss increases.

Two-Link Approximation Schemes for Loss Networks with Linear Structure and Trunk Reservation

Loss networks have long been used to model various types of telecommunication network, including circuit-switched networks. Such networks often use admission controls, such as trunk reservation, to optimize revenue or stabilize the behaviour of the network. Unfortunately, an exact analysis of such networks is not usually possible, and reduced-load approximations such as the Erlang Fixed Point (EFP) approximation have been widely used. The performance of these approximations is typically very good for networks without controls, under several regimes. There is evidence, however, that in networks with controls, these approximations will in general perform less well. We propose an extension to the EFP approximation that gives marked improvement for a simple ring-shaped network with trunk reservation. It is based on the idea of considering pairs of links together, thus making greater allowance for dependencies between neighbouring links than does the EFP approximation, which only considers links in isolation.