VGAM (version 0.9-1)

borel.tanner: Borel-Tanner Distribution Family Function

Description

Estimates the parameter of a Borel-Tanner distribution by maximum likelihood estimation.

Usage

borel.tanner(Qsize = 1, link = "logit", imethod = 1)

Arguments

Qsize

A positive integer. It is called $Q$ below and is the initial queue size.

link

Link function for the parameter; see Links for more choices and for general information.

imethod

See CommonVGAMffArguments. Valid values are 1, 2, 3 or 4.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Details

The Borel-Tanner distribution (Tanner, 1953) describes the distribution of the total number of customers served before a queue vanishes given a single queue with random arrival times of customers (at a constant rate $r$ per unit time, and each customer taking a constant time $b$ to be served). Initially the queue has $Q$ people and the first one starts to be served. The two parameters appear in the density only in the form of the product $rb$, therefore we use $a=rb$, say, to denote the single parameter to be estimated. The density function is $$f(y;a) = \frac{ Q! }{(y-Q)!} y^{y-Q-1} a^{y-Q} \exp(-ay)$$ where $y=Q,Q+1,Q+2,\ldots$. The case $Q=1$ corresponds to the Borel distribution (Borel, 1942). For the $Q=1$ case it is necessary for $0 < a < 1$ for the distribution to be proper. The Borel distribution is a basic Lagrangian distribution of the first kind. The Borel-Tanner distribution is an $Q$-fold convolution of the Borel distribution.

The mean is $Q/(1-a)$ (returned as the fitted values) and the variance is $Q a / (1-a)^3$. The distribution has a very long tail unless $a$ is small. Fisher scoring is implemented.

References

Tanner, J. C. (1953) A problem of interference between two queues. Biometrika, 40, 58--69.

Borel, E. (1942) Sur l'emploi du theoreme de Bernoulli pour faciliter le calcul d'une infinite de coefficients. Application au probleme de l'attente a un guichet. Comptes Rendus, Academie des Sciences, Paris, Series A, 214, 452--456.

Page 328 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

Consul, P. C. and Famoye, F. (2006) Lagrangian Probability Distributions, Boston: Birkhauser.

See Also

rbort, poissonff, felix.

Examples

bdata <- data.frame(y = rbort(n <- 200))
fit <- vglm(y ~ 1, borel.tanner, bdata, trace = TRUE, crit = "c")
coef(fit, matrix = TRUE)
Coef(fit)
summary(fit)