Showing papers on "Probability-generating function published in 1987"
TL;DR: In this article, the Moolgavkar-Venzon-Knudson two-stage model is extended to nonhomogeneous cases and the probability generating functions (PGF) of initiated cells and tumors are derived under very general conditions.
20 citations
TL;DR: A discrete buffered system with infinite buffer size, one single output channel, and periodic opportunities for service (synchronous transmission) is considered in a two-state environment with random sojourn times with "mixture of geometrics"-type distributions.
Abstract: A discrete buffered system with infinite buffer size, one single output channel, and periodic opportunities for service (synchronous transmission) is considered in a two-state environment. The output channel is subjected to a random interruption process, which is characterized by a Bernoulli sequence of independent random variables, with probabilities dependent on the environment state. The environment states have random sojourn times with "mixture of geometrics"-type distributions. The arrival process is dependent on the environment state, but arbitrary. For this system, expressions are derived for the probability generating functions of the number of messages in the buffer at various time instants. A number of special cases and possible applications of the model are discussed, and an extended example is given as an illustration of the study.
13 citations
TL;DR: In this article, the authors extended this approach to cover a number of bulk-service queues discussed by Chaudhry et al. and discussed in the present paper, and extended it to cover the bulk-arrival queues discussed in this paper.
Abstract: Queueing theorists have presented, as solutions to many queueing models, probability generating functions in which state probabilities are expressed as functions of the roots of characteristic equations, evaluation of the roots in particular cases being left to the reader. Many users have complained that such solutions are inadequate. Some queueing theorists, in particular Neuts [6], rather than use Rouche's theorem to count roots and an equation-solver to find them, have developed new algorithms to solve queueing problems numerically, without explicit calculation of roots. Powell [7] has shown that in many bulk service queues arising in transportation models, characteristic equations can be solved and state probabilities can be found without serious difficulty, even when the number of roots to be found is large. We have slightly modified Powell's method, and have extended his work to cover a number of bulk-service queues discussed by Chaudhry et al. [1] and a number of bulk-arrival queues discussed in the present paper.
11 citations
TL;DR: In this paper, the authors considered a scheme of random assignment of n particles in an infinite sequence of cells, where each particle is assigned to the k-th cell with probability pk ≤ pk+1 and pk > 0 for each k.
Abstract: One considers the following scheme of random assignment of n particles in an infinite sequence of cells. Each particle is. assigned to the k-th cell with probability pk
and one assumes that pk⩾ pk+1 and pk > 0 for each k. Let Xk(n) be the number of particles in the k-th cell and let f1(x), f2(x), ... be a sequence of real-valued functions defined for x=0, 1,2, ... Under certain conditions on the distribution of the probabilities
and on the sequence f1(x), f2(x), ..., one investigates the asymptotic normality of the random variable
. (The random variable Zn is proper since
)
2 citations
TL;DR: In this paper, the authors considered the prototypical case of a lattice that is homogeneous in the large but inhomogeneous in small, and obtained the generating function for the probability distribution of being at position x after N steps.
Abstract: We consider the prototypical case of a lattice that is homogeneous in the large but inhomogeneous in the small—a one-dimensional random walk with alternating homogeneous lattice fragments and appropriate boundary probabilities.The generating function for the probability distribution of being at position x after N steps is obtained. We also find various asymptotic forms and limiting distributions as a function of the step probability and the lattice fragments.
1 citations