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Volume 34 - Issue 2 - December 2009
• The Mayor's dilemma
F. W. Steutel
A simple sorting problem is considered that has been informally discussed in The Netherlands for some time (see Steutel (2008)). The context is as follows. The Mayor of Amsterdam wishes to meet the tallest inhabitant of his city, and to this end he asks the inhabitants to form a long line, where the people are numbered from one to a million, say. Some of the people will have to be paid for their trouble, so the Mayor will have to carry some money. He would hate to have too little, but he would rather not have much more than is necessary. The question then is: how much money should the Mayor carry? This leads to rather surprising answers, and to some not entirely trivial mathematics.
• Euler and the functional equation for the zeta function
Thomas J. Osler
Euler (1768) (see also Willis and Osler (2006)) conjectured an equation which is equivalent to the functional equation for the zeta function. This work by Euler was forgotten for one hundred years until Riemann rediscovered this functional equation and provided a rigorous proof. All modern proofs of the functional equation involve mathematical tools that were unavailable to Euler, and it is remarkable that he was nevertheless able to predict its structure. In this paper we show how Euler used divergent series to discover the functional equation for the zeta function.
• Surface accumulation of spermatozoa: a fluid dynamic phenomenon
D. J. Smith, J. R. Blake
Recent mathematical fluid dynamics models have shed light on an outstanding problem in reproductive biology: why do spermatozoa cells show a `preference' for swimming near to surfaces? In this paper we review quantitative approaches to the problem, originating with the classic paper by Lord Rothschild in 1963. A recent `boundary integral/slender body theory' mathematical model for the fluid dynamics is described, and we discuss how it gives insight into the mechanisms that may be responsible for the surface accumulation behaviour. We use the simulation model to explore these mechanisms in more detail, and discuss whether simplified models can capture the behaviour of sperm cells. The far-field decay of the fluid flow around the cell is calculated, and compared with a stresslet model. Finally, we present some new findings showing how, despite having a relatively small hydrodynamic drag, the sperm cell `head' has very significant effects on surface accumulation and trajectory.
• The θ-logistic and Gompertz birth–death processes
Randall J. Swift
In this article a birth–death formulation of the θ-logistic population model is presented and shown to converge in distribution to a birth--death process with Gompertz mean population size.
• A generalization of the Monty Hall problem
Gérard C. Nihous
The Monty Hall problem is generalized by extending all game parameters to arbitrary integers (number of boxes n, number of prizes k, number of boxes p initially selected by the contestant, number of boxes m opened by the game show host, number of prizes r revealed in this move). It is found that the player has a greater chance of winning if he switches boxes instead of holding to his initial choice as long as the proportion of prizes in the boxes opened by the host is smaller than in the initial set (r/m < k/n).
• The duration of play in games of chance with win-or-lose outcomes and
Marc Estafanous, S. N. Ethier
The duration of play formulae of De Moivre are generalized, using restricted Pascal triangles, to games in which the gambler wins μ units or loses ν units at each play, where μ and ν are positive integers. We treat the cases of both one and two barriers.
• Stochastic and deterministic characteristics of recursively defined
G. S. Tsitsiashvili
The calculation of stochastic and deterministic characteristics of networks is of great interest in both reliability theory (see Barlow and Proschan (1965), Riabinin (2007), Ushakov (1985)) and the theory of transportation networks (see Belov, Vorobiev and Shatalov (1976), Kormen, Leizerson and Rivest (2004)). Some of them have exponential complexity. These problems are important in applied probability and informatics. In this paper we construct some recursive definitions, calculate different characteristics by recursive formulae, and derive linear bounds for the numbers of arithmetic operations for our algorithms. The results are applied to calculations of reliability and to the solution of a salesman problem.
• The transient solution to a time-dependent single-server queue with
R. O. Al-Seedy, A. A. El-Sherbiny, S. A. El-Shehawy, S. I. Ammar
Kumar, Parthasarathy and Sharafali (1993) introduced a new form of M/M/1 queue with balking, where balking behavior occurs when the number of customers exceeds a threshold value k. In this paper we extend this system to a more general form in which the arrival, departures, and balking rates are allowed to vary with time. We derive an integral equation where the transient probabilities of an M(t)/M(t)/1 queue are expressed in terms of a Volterra equation of the second kind. The method is a straightforward application of generating functions.