Borel-Tanner Distribution Research Articles

A Borel-Tanher, bistribution and its approximation to the negative binomial distribution

In the sense of an unrestricted choice of a parameter in the Borel-Tanner distribution a distribution known as a generalized Poisson is deswibed. Approximations to the negative binomial sums using this distribution are given. The approximate values are more accurate than t,he ones obtained by using the incomplete gamma functions. The accuracy so obtained suggests the use of one model for another.

Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions

Considering $g( t )$ and $f( t )$ as two probability generating functions defined on nonnegative integers such that $g( 0 ) \ne 0$, we use Lagrange’s expansion, together with the transformation $t = u \cdot g( t )$, to define families of discrete generalized probability distributions by the name of Lagrange distributions as \[ \begin{gathered} \Pr \,[ {X = 0} ] = L( {g;f;0} ) = f( 0 ), \hfill \Pr \,[ {X = x} ] = L( {g;f;x} ) = \frac{1}{{x!}}\frac{{d^{x - 1} }}{{dt^{x - 1} }}\{ {( {g( t )} )^x \cdot f'( t )} \}|_{t = 0} \hfill \end{gathered} \]for $x = 1,2,3, \cdots $, where the different families are generated by assigning different values to $g( t )$ and $f( t )$. General formulas for writing down the central moments of Lagrange distributions are obtained and it is shown that they satisfy the convolution property. The double binomial family of Lagrange distributions is studied in greater detail as it gives a large number of discrete distributions, including Borel-Tanner distribution, Haight’s distr...

Delays to road traffic at an intersection

A model for road traffic delays at intersections is considered where vehicles arriving, possibly in bunches, in a Poisson process in a one way minor road yield right of way to traffic, which forms alternate bunches and gaps, in a major road. The gap acceptance times are random variables, and depend on whether or not a minor road vehicle is immediately following another minor road vehicle into the intersection or not.The transforms of the stationary waiting time and queue size distributions, and the mean stationary delay, for minor road vehicles are obtained by substitution of determined service time distributions into results for a generalisation of the M/G/1 queueing system. Some numerical results are given to illustrate the increase in the mean delay for variable gap acceptance times for a Borel-Tanner distribution of major road traffic, and a partial solution is given for a two way major road.

Delays to road traffic at an intersection

A model for road traffic delays at intersections is considered where vehicles arriving, possibly in bunches, in a Poisson process in a one way minor road yield right of way to traffic, which forms alternate bunches and gaps, in a major road. The gap acceptance times are random variables, and depend on whether or not a minor road vehicle is immediately following another minor road vehicle into the intersection or not. The transforms of the stationary waiting time and queue size distributions, and the mean stationary delay, for minor road vehicles are obtained by substitution of determined service time distributions into results for a generalisation of the M/G/1 queueing system. Some numerical results are given to illustrate the increase in the mean delay for variable gap acceptance times for a Borel-Tanner distribution of major road traffic, and a partial solution is given for a two way major road.

A Derivation of the Borel Distribution

A Derivation of the Borel Distribution

A Queueing Model for Road Traffic Flow

SUMMARY It is proposed that on roads which are uninterrupted by traffic signals, intersections, etc., vehicles should be considered as travelling in random queues. Criteria for determining the queues in actual traffic are found. A crude model is then used to study the formation of these queues in an attempt to derive the Borel–Tanner distribution of queue lengths. The random queues model is then used to study waiting times for pedestrians (or vehicles) wishing to cross one lane of traffic.

A derivation of the Borel distribution

A derivation of the Borel distribution

A PROBLEM OF INTERFERENCE BETWEEN TWO QUEUES

A two-lane road is assumed to be occupied mainly by platoons of cars travelling along the two opposite directions at constant speeds. A single car is assumed to try to move at a speed higher than that of the other cars moving in its direction. It can pass only when there is enough room for it without breaking up a platoon in either direction. The sizes of the platoons are supposed to be given by a borel distribution (e. Borel, Comptes Rendus Acad. Sci. Paris, vol. 214, 1942, pp. 452-456). An interesting result is that when the traffic flows increase beyond a certain level, appreciably below the theoretical capacity of the road, the fast vehicle cannot move any faster than the other vehicles. It spends most of its time waiting for an opportunity to pass, and the mean waiting time until passing goes to infinity. The average speed was tabulated after numerical computations corresponding to a wide range of the constants of the model. These numerical results lead to two general conclusions: First, that increasing the acceleration capability even at the expense of reducing the maximum speed, (e.G., by lowering the top gear ratio), would result in increased journey speeds for most modern cars under typical traffic conditions. And second, that reducing the safety margins in passing would not normally provide any worthwhile increase in mean journey speed. Language: en