Showing papers on "Probability-generating function published in 2014"

A retrial system with two input streams and two orbit queues

Abstract: Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented.

Choice Probability Generating Functions

Abstract: This paper considers discrete choice, with choice probabilities coming from maximization of preferences from a random utility field perturbed by additive location shifters (ARUM). Any ARUM can be characterized by a choice-probability generating function (CPGF) whose gradient gives the choice probabilities, and every CPGF is consistent with an ARUM. We relate CPGF to multivariate extreme value distributions, and review and extend methods for constructing CPGF for applications.

Determining the First Probability Density Function of Linear Random Initial Value Problems by the Random Variable Transformation (RVT) Technique: A Comprehensive Study

Abstract: Deterministic differential equations are useful tools for mathematical modelling. The consideration of uncertainty into their formulation leads to random differential equations. Solving a random differential equation means computing not only its solution stochastic process but also its main statistical functions such as the expectation and standard deviation. The determination of its first probability density function provides a more complete probabilistic description of the solution stochastic process in each time instant. In this paper, one presents a comprehensive study to determinate the first probability density function to the solution of linear random initial value problems taking advantage of the so-called random variable transformation method. For the sake of clarity, the study has been split into thirteen cases depending on the way that randomness enters into the linear model. In most cases, the analysis includes the specification of the domain of the first probability density function of the solution stochastic process whose determination is a delicate issue. A strong point of the study is the presentation of a wide range of examples, at least one of each of the thirteen casuistries, where both standard and nonstandard probabilistic distributions are considered.

The Fokker-Planck equation for power system stability probability density function evolution

Abstract: This paper presents an analysis of the evolution of the probability density function of the dynamic trajectories of a single machine infinite bus power system. The probability density function can be used to determine the impact of random (stochastic) load perturbations on system stability. The evolution of the state probability density function over time leads to several interesting observations regarding stability regions as a function of damping parameter. The Fokker-Planck equation (FPE) is used to describe the evolution of the probability density of the states. The FPE is solved numerically using PDE solvers (such as finite difference method). Based on the results, the qualitative changes of the stationary density produce peak-like, ridge-like and other complicated shapes. Lastly, the numerical FPE solution combined with SMIB equivalent techniques lay the framework extended to the multimachine system.

Batch Arrival Queueing System with Two Stages of Service

Abstract: This Paper studies batch arrival queue with two stages of service. Random breakdowns and Bernoulli schedule server vacations have been considered here .After a service completion, the server has the option to leave the system or to continue serving customers by staying in the system. It is assumed that customers arrive to the system in batches of variable size, but served one by one. After completion of first stage of service, the server must provide the second stage of service to the customers. Vacation time follows general distribution, while we consider exponential distribution for repair time.We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, average number of customers ,and the average waiting time in the queue.

The Sherrington–Kirkpatrick spin glass model in the presence of a random field with a joint Gaussian probability density function for the exchange interactions and random fields

Abstract: The magnetic systems with disorder form an important class of systems, which are under intensive studies, since they reflect real systems. Such a class of systems is the spin glass one, which combines randomness and frustration. The Sherrington–Kirkpatrick Ising spin glass with random couplings in the presence of a random magnetic field is investigated in detail within the framework of the replica method. The two random variables (exchange integral interaction and random magnetic field) are drawn from a joint Gaussian probability density function characterized by a correlation coefficient ρ . The thermodynamic properties and phase diagrams are studied with respect to the natural parameters of both random components of the system contained in the probability density. The de Almeida–Thouless line is explored as a function of temperature, ρ and other system parameters. The entropy for zero temperature as well as for non zero temperatures is partly negative or positive, acquiring positive branches as h 0 increases.

On total progeny of multitype Galton-Watson process and the first passage time of random walk on lattice

Abstract: In this paper, we form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the first passage time of corresponding random walks. Especially, for recurrent random walk which could stay at its position with probability 0 < r < 1, we show that the tail probability of the first passage time decays as \(\frac{2} {{\sqrt {\pi (1 - r)} }}\frac{1} {{\sqrt n }} \) when n → ∞.

Tail generating functions for Markov branching processes

Abstract: We give a concise self-contained presentation of known and new limit theorems for the one-type Markov branching processes with continuous time. The new streamlined proofs are based on what we call, the tail generating function approach. Our analysis focuses on the singularity points of the master integral equation for the probability generating functions of the current population size.

Sparse variable optimization device, sparse variable optimization method, and sparse variable optimization program

Abstract: A gradient computation unit computes a gradient of an objective function in a variable to be optimized. An added variable selection unit adds a variable corresponding to a largest absolute value of the computed gradient from among variables included in a variable set, to a nonzero variable set. A variable optimization unit optimizes a value of the variable to be optimized, for each variable included in the nonzero variable set. A deleted variable selection unit deletes a variable that, when deleted, causes a smallest increase of the objective function from among variables included in the nonzero variable set, from the nonzero variable set. An objective function evaluation unit computes a value of the objective function for the variable to be optimized.

A discrete-time queue with finite-support service capacities

Abstract: In discrete-time queueing theory, the service process is traditionally modeled using the notion of service time, the time it takes the server to completely process one customer. In our research, we take a different approach and model the service process by means of two more basic quantities: service demands and service capacities. The service demands are independent and identically distributed (i.i.d.) random variables that describe the number of work units that each customer requires from the system, whereas the service capacities are i.i.d. random variables that describe the amount of work units that the server can process per timeslot. If a customer requires more work units than the server can provide in a slot, the service continues in the next slot. Conversely, if the service capacity in a slot is higher than the customer in service still requires, the remaining capacity is used for the next customer in line, and more than one customer might be served in that slot. This type of model has been studied in previous work, but with either the restriction that the service capacities follow a geometric distribution or that they are deterministically equal to a given constant. In our research, we analyze the system with the restriction that the service capacity distribution must have finite support. The numbers of customers arriving per slot and the service demands of the customers can be general i.i.d. random variables. The analysis is performed using probability generating functions (pgfs), and as a result we obtain expressions for the pgfs of the delay of a random customer, the amount of unfinished work and the number of customers in the system in a random slot. These pgfs are then used to derive expressions for the moments of these quantities, and to approximate the probability mass function of these quantities with much higher precision than simulations could provide.

The Properties Related to the Moment Generating Function of the Fuzzy Variable

Abstract: In the paper, some properties related to the moment generating function of a fuzzy variable are discussed based on uncertainty theory. And we obtain the result that the convergence of moment generating functions to an moment generating function implies convergence of credibility distribution functions. Thats, the moment generating function characterizes a credibility distribution.

Optimal Strategy Analysis of Two-Phase, N-Policy of M/EK/1 Queuing System with System Break -Downs and Balking

Abstract: This Paper deals with the economic behaviour of the M/Ek/1 queue with server start-up, two-phases of compulsory service, and server breakdowns during both batch and individual services with balking under Npolicy. The first phase of service is a batch service to all existing customers in the queue and the second phase of service is to each customer in the batch in ’ k’ independent and identically distributed exponential phases. Arriving customers may balk with a certain probability and may depart without getting service due to impatience. For this model the probability generating functions for the number of customers present in the system at various states of the server are derived and obtained the closed-form expressions for various performance measures of interest. Further a total expected cost model is formulated to determine the optimal threshold of N at a minimum cost. Finally, numerical examples are given.

A Batch Arrival Queue With a Second Optional Service Channel Under (m,N) Policy And Single Bernoulli Vacation

Abstract: This paper analyses Bi-level Threshold policy for a bulk arrival queueing system with second optional service facility under Bernoulli Schedule Single vacation. It is assumed that the server may take vacation whenever the server completes service to a customer.Also,whenever the system becomes empty the server may take a longer vacation with different probability following different distributions.We derive the Probability Generating Functions of the system using supplementary variable technique.Moreover,some important performance measures of this model are also presented with some numerical examples

• transient solution of m[x]/g/1 retrial queue with two stage heterogeneous service, extended vacation, non-persistent customers and setup time

Abstract: T his paper investigates a single server retrial queuing system with two stages heterogeneous serviceCustomers arrive in batches in accordance with compound Poisson processes After the completion of first stage service, the second stage service starts with probability 1 In addition to this, the server takes Bernoulli vacation and setup times We assume that the retrial time, the service time, the repair time, the vacation time and the setup time of the server are all arbitrarily distributed We obtain the time dependent probability generating functions in terms of their Laplace transforms and the corresponding steady state results explicitly Also we derive the average number of customers in the queue and the average waiting time in closed form with numerical illustration