Transient analysis of a queue with system disasters and customer impatience

A single server queue with Poisson arrivals and exponential service times is studied. The system suffers disastrous breakdowns at an exponential rate, resulting in the loss of all running and waiting customers. When the system is down, it undergoes a repair mechanism where the repair time follows an exponential distribution. During the repair time any new arrival is allowed to join the system, but the customers become impatient when the server is not available for a long time. In essence, each customer, upon arrival, activates an individual timer, which again follows an exponential distribution with parameter ξ. If the system is not repaired before the customer’s timer expires, the customer abandons the queue and never returns. The time-dependent system size probabilities are presented using generating functions and continued fractions.     

Dimensional analysis,modelling and symmetry†

In this paper the author discusses the controversial question of the future of ‘ new ‘ mathematics.     

Stakeholder preference elicitation and modelling in multi-criteria decision analysis – A case study on urban water supply

Integration of multiple objectives to evaluate the alternative operating rules for urban water supply reservoir systems can be effectively accomplished by multi-criteria decision aid techniques, where preference elicitation and modelling plays an important role. This paper describes a preference elicitation and modelling procedure involving the multi-criteria outranking method PROMETHEE in evaluating these alternative operating rules. The Melbourne water supply system was considered as the case study. Eight performance measures (PMs) were identified under four main objectives to evaluate the system performance under alternative operating rules. Three major hypothetical stakeholder groups namely, resource managers, water users, and environmental interest groups were considered in decision-making. An interviewer-assisted questionnaire survey was used to derive the preference functions and weights of the PMs. The evaluation of alternative operating rules is not covered in this paper, rather an approach to elicit and model stakeholder preferences in decision-making is described.     

Logical soft systems modelling for information source analysis—The case of Hongkong Telecom

The paper describes the use of logico-linguistic modelling, a logically enhanced version of Soft System Methodology, to identify the information needed to support decisions made in a marketing department of a Hong Kong telecommunications company. This case indicates that logical complexity need not be a barrier to stakeholder comprehension. However, cross-cultural difficulties were encountered when using the basic SSM method.     

Income redistribution policies and decision support systems — Problems and potentials

Recent trends in Western industrial countries towards a re-examination of the means and goals of the welfare state have increased the demand for studies of the redistributive effects of government policies. The purpose of this paper is to discuss some problems and potentials of decision support systems in this field, with emphasis on the problems of value judgements and evaluation criteria and with reference to the author's experiences from the development of an interactive computer model of the Swedish tax and transfer system. The basic structure of the model and some examples of its application are briefly described.     

Risk taking by decision makers—using card-based decision gaming to develop models of behaviour

This paper describes an experimental method for determining the value of different types of information to military decision makers. The experimental method used a simple scenario and a set of serials constructed from cards, each presenting a single piece of information and presented sequentially. Each of a number of pairs of players were taken through the scenario and asked to judge how they would respond to the situation. This paper extends on the method presented in previous papers to consider the case of a decision in which the response can both increase and decrease as more information is presented. This allows a more general military problem to be considered, that of risk-taking behaviour in response to the possibility of chemical, biological, radiological or nuclear events.     

The Leibniz catenary and approximation of e — an analysis of his unpublished calculations

Leibniz published his Euclidean construction of a catenary in Acta Eruditorum of June 1691, but he was silent about the methods used to discover it. He explained how he used his differential calculus only in a private letter to Rudolph Christian von Bodenhausen and specified a number that was key to his construction, 2.7182818, with no clue about how he calculated it. Apparently, the calculations were never divulged to anyone but were discovered later among his personal papers. They may be the earliest record of an accurate approximation of the number we label e and a demonstration of its role as the base of the natural logarithm and exponential function.This, at that time, was a remarkably precise estimate for e, accomplished more than 22 years before Roger Cotes published e to 12 significant digits, and some 57 years before Euler's treatment of the logarithm in his Introductio in Analysin Infinitorum. The Leibniz construction reveals a hyperbolic cosine built on an exponential curve based on his estimated value, which implies that he understood the number as the base of his logarithmic curve. The sheets of arithmetic used by Leibniz preserved at the Gottfried Wilhelm Leibniz Bibliothek (GWLB) in Hannover, confirm this.Those sheets show how Leibniz calculated e and applied it to his catenary construction. The data actually yield e to 12 significant figures: 2.71828182845, missed by Leibniz because of a misplaced decimal point. We summarize the construction and examine the worksheets. The unpublished methods seem entirely modern to us and could serve as enrichening examples in modern calculus texts.     

Mathematical analysis of a tumor invasion model—global existence and stability

This work studies an outstanding reaction–diffusion system modeling tumor invasion, with interactions among tumor tissue, acid concentration and normal tissue. This model has very different features from the models extensively studied in the mathematics literature. The most challenge issue for mathematical analysis of the present model is the existence of classical solution, since the diffusion of tumor tissue is influenced by the density of normal cells and diffusion degeneracy arises when normal cells are at the carrying capacity. A rigorous proof of global existence and uniqueness of classical solutions is presented. Moreover, we study global dynamics of the solution, and show asymptotic stability of the four possible constant equilibria under various scenarios.     

Powergraph — a network-based approach to modelling and managing corporate strategic conflict and co-operation

Strategic relationships between companies are characterised by a combination of conflict and co-operation. Often the behaviour of partner/rival companies will appear arbitrary, almost irrational because of the necessary limitations on communication between the parties. Understanding, predicting and managing situations such as major project teamings or acquisition struggles is essential for the satisfactory governance of companies but existing conflict modelling has proven less than attractive to practitioners and their advisors. A network-based approach known as Powergraph is described in this study. It represents the corporate situation as a directed graph in which the nodes are the possible states of play. Connections between nodes are represented by logical expressions deriving from the capacities of the players individually or in coalition to bring about the transitions. By consideration of the ordinal preferences of the participants, trajectories of the situation can be described and these trajectories form the basis for management action and intelligence gathering. The method is illustrated by means of a real case study of three defence companies competing for two major contracts.     

Stability analysis and stabilization of linear symmetric matrix-valued continuous,discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.     

Integer comparisons across the grades: Students’ justifications and ways of reasoning

This study is an investigation of students’ reasoning about integer comparisons—a topic that is often counterintuitive for students because negative numbers of smaller absolute value are considered greater (e.g., −5 > − 6). We posed integer-comparison tasks to 40 students each in Grades 2, 4, and 7, as well as to 11th graders on a successful mathematics track. We coded for correctness and for students’ justifications, which we categorized in terms of 3 ways of reasoning: magnitude-based, order-based, and developmental/other. The 7th graders used order-based reasoning more often than did the younger students, and it more often led to correct answers; however, the college-track 11th graders, who responded correctly to almost every problem, used a more balanced distribution of order- and magnitude-based reasoning. We present a framework for students’ ways of reasoning about integer comparisons, report performance trends, rank integer-comparison tasks by relative difficulty, and discuss implications for integer instruction.     

Invariance analysis and reduction of discrete Painlevé equations

The construction and role of symmetries for difference equations have been established, relatively, recently. In this paper, a symmetry analysis and reductions of the discrete Painlevé equations are considered. We assume that the characteristics of the ‘vector fields’ have a particular dependence since the general form lead to cumbersome calculations. Where possible, these symmetries are used to construct exact solutions in some cases.     

Bivariate statistical analysis of TCP-flow sizes and durations

We approximate the distribution of the TCP-flow rate by deriving it from the joint bivariate distribution of the flow sizes and flow durations of a given access network. The latter distribution is represented by a bivariate extreme value distribution using the Pickand’s dependence A-function. We estimate the A-function to measure the dependencies of random pairs: TCP-flow size and duration, the rate of TCP-flow and size, as well as the rate and duration. We provide a method to test that the achieved estimate of A-function is good and perform the analysis with one concrete data example.     

A computation of the dimensions of an underground cylindrical tank—an undergraduate modelling exercise

This paper develops the solution of a ‘real’ problem which involves simple numerical techniques. The numerical solution is interesting because of possible divergence and round‐off error growth. The problem makes an ideal case study for undergraduates following a course in numerical mathematics.     

Mathematical analysis of a fluid—solid interaction problem

This paper analyzes a fluid—solid interaction model which describes the interaction between an inviscid fluid and an elastic solid In the model, the linear elastodynamic equations complemented with appropriate interface and boundary conditions are used to describe the wave propagation in the fluid and solid regions, and absorbing boundary conditions are used to minimize unphysical wave reflections. It is shown that the initial boundary value problem of the mathematical model posses a unique global (in time) quasi-strong solution. Regularity of the quasi-strong solution is also obtained under some reasonable assumptions on the data and on the domain.     

Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub’s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.