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This paper provides the solution to the problem of the distribution of busy periods of the single-channel queue characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. The notation is chosen in such a fashion that the main body of computations and mathematical analysis is identical with what was required in the solution of the general queuing equations considered in a previous paper which should be read first [4]. The solution of the problem for winch the traffic intensity is constant and the holding time distribution is Pearson type III is derived in closed form in terms of the Ink functions introduced previously. [ABSTRACT FROM AUTHOR]

This paper sets forth mathematical means for optimizing the simultaneous apportionments of reliability and maintainability by means of two techniques Lagrange multipliers and dynamic programming. Since one of the constraint equations revolves allocating both variables, this simultaneous apportionment problem differs from the usual two-dimensional dynamic programming problems. Two cost problems are treated the first minimizes cost while satisfying availability requirements under the assumption that an unlimited ceiling exists for the cost, and the second maximizes reliability and maintainability given a fixed budget. Although these problems appear closely related, step functions usually preclude considering them together The paper also considers two numerical examples, an interesting cost model for maintainability, and using computers with its models. [ABSTRACT FROM AUTHOR]

The main object of the paper is to demonstrate a method by which a number of problems arising in the study of impatient customers in queues can be solved. The main difficulty in analyzing such queuing systems is the presence of variable coefficients in the difference-differential equations describing the processes; the usual generating function method fails in such cases. When there is a finite number of states, it has been shown that at least formal solutions can be obtained through matrix algebra, which yields what are called 'discrete transforms.' The results are explicitly given when the balking and reneging coefficients are linear in n. An alternative method in which the spectral resolution of matrices is used is also given for solution of the set of equations. In the special case of a M/G/1 queue with finite waiting room, both the methods fail (as the underlying matrix is in the Jordan form), and the solution is obtained by a simple manipulation of the underlying matrix. The methods presented here lend themselves readily to computer solutions. This paper demonstrates the generality of the basic M/G/1 model by relating it to problems arising in other branches of operations research, such as reliability theory, inventory control, etc. Finally, the methods used for the M/G/1 case are shown to hold for a similar class of problems in the GI/M/1 case. As very little work has been reported on impatient customer phenomena in GI/M/1, systems, possibly because of the difficulty in solving equations with variable coefficients, this method may prove to be useful. [ABSTRACT FROM AUTHOR]

The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, 'condensed' flow. [ABSTRACT FROM AUTHOR]

The article reports on the discussion of the study concerning the learning of mathematics. The discussion contains arguments regarding the paper written by professor Stone about functions. One of the arguments that is given attention to, is the concept of function being connected with the primitive concepts of pure logic, nearly unanalyzable and must be grasped by a psychological act of direct perception. This argument is also related to the child's development as the notion of grasping and perception are described to occur at some point in the development.

The causal relationships in a two variable system are examined, using a simultaneous linear differential equation model. Emphasis is placed on the necessity for considering mutual interaction instead of looking for unidirectional causality. To demonstrate the usefulness of this approach an example is taken from the management science literature concerning the relationship between labour turnover and overtime. It is shown that, on certain assumptions, the data disconfirm a hypothesis previously thought to be confirmed. Certain informal observations made in the original paper were shown to receive confirmation from the data. [ABSTRACT FROM AUTHOR]

A theory is given for circumnutations in plants, especially hypocotyls of Helianthus annuus, which were used as experimental material. The theory is based on the lateral auxin transport, which arises when a gravitational force component acts on the plant. With a suitable time delay between stimulus and response, oscillations or circumnutations should arise. It is possible to describe these oscillation phenomena by the solutions of a differential equation, derived in this paper. The time delay has a central role in this equation. The time delay is assumed to be identical with the geotropic reaction time for the hypocotyls. The ratio between the periodic time for the circumnutations and the reaction time for geotropic curvatures was found to be approximately constant in the temperature region investigated (namely 15-40°C), which supports the theory. Different methods of recording the circumnutations were used, 8 mm film camera technique being the most frequently employed. The introduction of a weighting function for describing the plants "memory" of the stimulation makes it possible to relate the periodic time of the circumnutations to the reaction time for geotropic curvatures. The necessity of this weighting function as well as of the time delay in the equations is emphasized. An explanation of the "Fünfphasenbewegungen", reported in the literature, is presented. [ABSTRACT FROM AUTHOR]