Showing papers on "Probability-generating function published in 2015"

Generating discrete analogues of continuous probability distributions-A survey of methods and constructions

Abstract: In this paper a comprehensive survey of the different methods of generating discrete probability distributions as analogues of continuous probability distributions is presented along with their applications in construction of new discrete distributions. The methods are classified based on different criterion of discretization.

General and exact approach to percolation on random graphs.

Abstract: We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative equations that solve the exact distribution of the size and composition of components in finite-size quenched or random multitype graphs. (ii) We define a very general random graph ensemble that encompasses most of the models published to this day and also makes it possible to model structural properties not yet included in a theoretical framework. Site and bond percolation on this ensemble is solved exactly in the infinite-size limit using probability generating functions [i.e., the percolation threshold, the size, and the composition of the giant (extensive) and small components]. Several examples and applications are also provided. (iii) Our approach can be adapted to model interdependent graphs-whose most striking feature is the emergence of an extensive component via a discontinuous phase transition-in an equally general fashion. We show how a graph can successively undergo a continuous then a discontinuous phase transition, and preliminary results suggest that clustering increases the amplitude of the discontinuity at the transition.

Analysis of M[X]/G/1 retrial queue with two phase service under Bernoulli vacation schedule and random breakdown

Abstract: This paper investigates the steady state behaviour of an M[x]/G/1 retrial queue with two phases of service under Bernoulli vacation schedule and breakdown. Any arriving batch finding the server busy, breakdown or on vacation enters an orbit. Otherwise one customer from the arriving batch enters the service immediately while the rest join the orbit. After completion of each two phases of service, the server either goes for a vacation with probability p or may wait for serving the next customer with probability (1-p). While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. We construct the mathematical model and derive the probability generating functions of number of customers in the system when it is idle, busy, on vacation and under repair. Some system performances are obtained.

Universal methods for generating random variables with a given characteristic function

Abstract: Universal generators for absolutely-continuous and integer-valued random variables are introduced. The proposal is based on a generalization of the rejection technique proposed by Devroye [The computer generation of random variables with a given characteristic function. Computers and Mathematics with Applications. 1981;7:547–552]. The method involves a dominating function solely requiring the evaluation of integrals which depend on the characteristic function of the underlying random variable. The proposal gives rise to simple algorithms which may be implemented in a few code lines and which may show noticeable performance even if some classical families of distributions are considered.

Limit distribution of the coefficients of polynomials with only unit roots

Abstract: We consider sequences of random variables whose probability generating functions have only roots on the unit circle, which has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth central and normalized by the standard deviation moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance is unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,707-738, 2015

On the Harris extended family of distributions

Abstract: A new method for generating new classes of distributions based on the probability-generating function is presented in Aly and Benkherouf [A new family of distributions based on probability generating functions. Sankhya B. 2011;73:70–80]. In particular, they focused their interest to the so-called Harris extended family of distributions. In this paper, we provide several general results regarding the Harris extended models such as the general behaviour of the failure rate function. We also derive a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution. Additionally, some stochastic order relations are established and limiting distributions of sample extremes are also considered for this model. These general results are illustrated in several special Harris extended models. Finally, we discuss estimation of the model parameters by the method of maximum likelihood and provide an application to real da...

Detection of Changes in INAR Models

Abstract: In the present paper we develop on-line procedures for detecting changes in the parameters of integer valued autoregressive models of order one. Tests statistics based on probability generating functions are constructed and studied. The asymptotic behavior of the tests under the null hypothesis as well as under certain alternatives is derived.

Model to Increase the Number of Output States of a Random Variable Using a Histogram Based PDF

Abstract: This paper is devoted to a pseudo-random number generator that generates numbers according to a known probability density function. This function has its origins in a histogram that contains the observed properties of a monitored random variable X. Using the recorded PDF and an ON/OFF model we are able to generate output states of a random variable Y, which is nearly alike distributed as the random variable X. Because the used PDF has its origins in a normed histogram function, the resulting number of all possible output states of variable Y is the same as the number of classes that are used to form the histogram. We propose this model that generates practically an unlimited set of possible output states of the random variable Y. At first a cumulative distribution function is formed from the input density function. Then the ON/OFF model driven by a random number generator in (0, 1) is applied to address classes that are defined in the distribution function. To address these classes a search function is applied. Then when a class is chosen, we propose to use a second uniform random number generator that generates numbers within the range of the selected class. The resulting output state of the variable Y is the output of the second random number generator. In this paper we apply this model on a packet generator that generates IPTV inter-departure times of an MPEG2-TS H.264 VBR based video source.

Algorithms for generating random variables with a rational probability-generating function

Abstract: Two algorithms for generating random variables with a rational probability-generating function are presented. One of them implements the recently developed general range reduction method, and the other is an extension of the alias method designed for generating discrete finite-valued random variables to the case where the generated random variable is infinite-valued. An example of a random variable which was efficiently generated by random number generators implementing the presented algorithms is given. Possible ways of improving the complexity of the presented algorithms are discussed.

Discrete-time Retrial Queue with Bernoulli Vacation, Preemptive Resume and Feedback Customers

Abstract: Purpose: We consider a discrete-time Geo/G/1 retrial queue where the retrial time follows a general distribution, the server subject to Bernoulli vacation policy and the customer has preemptive resume priority, Bernoulli feedback strategy. The main purpose of this paper is to derive the generating functions of the stationary distribution of the system state, the orbit size and some important performance measures. Design/methodology: Using probability generating function technique, some valuable and interesting performance measures of the system are obtained. We also investigate two stochastic decomposition laws and present some numerical results. Findings: We obtain the probability generating functions of the system state distribution as well as those of the orbit size and the system size distributions. We also obtain some analytical expressions for various performance measures such as idle and busy probabilities, mean orbit and system sizes. Originality/value: The analysis of discrete-time retrial queues with Bernoulli vacation, preemptive resume and feedback customers is interesting and to the best of our knowledge, no other scientific journal paper has dealt with this question. This fact gives the reason why efforts should be taken to plug this gap.

A non-markovian batch arrival queue with service interruption and extended server vacation

Abstract: A single server provides service to all arriving customers with service time following general distribution. After every service completion the server has the option to leave for phase one vacation of random length with probability p or continue to stay in the system with probability 1 p. As soon as the completion of phase one vacation, the server may take phase two vacation with probability q or to remain in the system with probability 1 q, after phase two vacation again the server has the option to take phase three vacation with probability r or to remain in the system with probability 1 r. The vacation times are assumed to be general. The server is interrupted at random and the duration of attending interruption follows exponential distribution. Also we assume, the customer whose service is interrupted goes back to the head of the queue where the arrivals are Poisson. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and system and the waiting time in the queue and system are also derived. Particular cases and numerical results are discussed.

Performance comparison of methods for design of experiments for analysis of tasks involving random variables

Abstract: Nowadays, computer experiments are often used for evaluation of functions with random inputs. Instead of accessing a complex function and calculating the probability distribution function of the result, a set of computer simulations may be processed and the information concerning the properties of the resulting variable/random vector can be estimated. The quality of such an estimate depends on the number of simulations and also on the distribution of points in the sampling space. Since the time required for the calculation is proportional to the number of the design points, it is desirable to reduce this number while keeping the quality of the design. This can be achieved by appropriate selection of the design points (specific combinations of values of input variables out of the permissible range of values given by the probability distribution of each variable).This paper describes several frequently used methods for generation and optimization of the designs and compares them using several basic function...

On Some Compound Random Variables Motivated by Bulk Queues

Abstract: We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group ( ) and its size ( ) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variable which expresses the number of customers that arrive in this bulk queue during any considered period . Notice that can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributions concerning certain pairs of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases when and/or are some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

An existence theorem for bounds on the expectation of a random variable. Its opportunities for utility theories. V. 2

Abstract: An existence theorem is proven for the case of a discrete random variable that can take on only a finite set of possible values. If the random variable takes on values in a finite interval and there is a lower non-zero bound on the modulus of (at least one) its central moment, then non-zero bounds on its expectation exist near the borders of the interval. The revealed bounds can be considered as “forbidden zones” for the expectation. They can be useful, e.g., in utility theories.

Path integral representation for stochastic jump processes with boundaries

Abstract: We propose a formalism to analyze discrete stochastic processes with finite-state-level N. By using an (N+1)-dimensional representation of su(2) Lie algebra, we re-express the master equation to a time-evolution equation for the state vector corresponding to the probability generating function. We found that the generating function of the system can be expressed as a propagator in the spin coherent state representation. The generating function has a path integral representation in terms of the spin coherent state. We apply our formalism to a linear Susceptible-Infected-Susceptible (SIS) epidemic model with time-dependent transition probabilities. The probability generating function of the system is calculated concisely using an algebraic property of the system or a path integral representation. Our results indicate that the method of analysis developed in the field of quantum mechanics is applicable to discrete stochastic processes with finite-state-level.

An existence theorem for bounds (restrictions) on the expectation of a random variable. Its opportunities for utility and prospect theories

Abstract: An existence theorem is proved for the case of a discrete random variable with finite support. If the random variable takes on values in a finite interval and there is a lower non-zero bound on its dispersion, then non-zero bounds (or non-zero “forbidden zones”) on its expectation exist near the borders of the interval. The theorem can be used in utility and prospect theories, in particular, in the analysis of Prelec’s probability weighting function.

Fuzzy Random Variable and the Dempster-Shafer Theory of Evidence

Abstract: This chapter presents the concept of uncertainty propagation in real world desicion problems, where some input parameters are stochastic while information about others is partial and is represented by fuzzy random variable. It also introduces fuzzy random variable and the Dempster-Shafer theory which provide mathematical background for such propagation.

Low-Priority Queue Fluctuations in Tandem of Queuing Systems Under Cyclic Control with Prolongations

Abstract: A tandem of queuing systems is considered. Each system has a high-priority input flow and a low-priority input flow which are conflicting. In the first system, the customers are serviced in the class of cyclic algorithms. The serviced high-priority customers are transferred from the first system to the second one with random delays and become the high-priority input flow of the second system. In the second system, customers are serviced in the class of cyclic algorithms with prolongations. Low-priority customers are serviced when their number exceeds a threshold. A mathematical model is constructed in form of a multidimensional denumerable discrete-time Markov chain. The recurrent relations for partial probability generating functions for the low-priority queue in the second system are found.

Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity

Abstract: This paper analyzes a single-server discrete-time queueing model with general independent arrivals, where the service process of the server is characterized in two steps. Specifically, the model assumes that (1) each customer represents a random, arbitrarily distributed, amount of work for the server, the service demand, and (2) the server disposes of a fixed number of work units that can be executed per slot, the service capacity. For this non-classical queueing model, we obtain explicit closed-form results for the probability generating functions (pgf’s) of the unfinished work in the system (expressed in work units) and the queueing delay of an arbitrary customer (expressed in time slots). Deriving the pgf of the number of customers in the system turns out to be hard, in general. Nevertheless, we derive this pgf explicitly in a number of special cases, i.e., either for geometrically distributed service demands, and/or for Bernoulli arrivals or geometric arrivals. The obtained results show that the tail distributions of the unfinished work, the customer delay and the system content all exhibit a geometric decay, with semi-analytic formulas for the decay rates available. Another interesting conclusion is that, for a given system load, the mean customer delay converges to constant limiting values when the service capacity per slot goes to infinity, and either the mean arrival rate or the mean service demand increases proportionally. Accurate approximative analytical expressions are available for these limiting values.

Some New Results on the Number of Paths

Abstract: Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by Anm. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.

Transient solution of /G/1 with Second Optional Service Subject to Reneging during Vacation and Breakdown time

Abstract: In this paper, we present a batch arrival non- Markovian queuing model with second optional service. Batches arrive in Poisson stream with mean arrival rate λ, such that all customers demand the first essential service, whereas only some of them demand the second ‘optional’ service. We consider reneging to occur when the server is unavailable during the system breakdown or vacations periods. The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly.

A New Weighted Information Generating Function for Discrete Probability Distributions

Abstract: The object of this paper is to introduce a new weighted information generating function whose derivative at point 1 gives some well known measures of information. Some properties and particular cases of the proposed generating function have also been studied.

Statistical characterization of the output of nonlinear power-cubic detection unit for ultrashort light pulse communication in the presence of Gaussian noise

Abstract: In this paper, an accurate model for the probability density function (pdf) of the random decision variable Y in an ultrafast digital lightwave communication system, utilizing power-cubic all-optical nonlinear preprocessor is presented. The proposed model can replace the prevalent Gaussian approximation, as the accuracy of the latter is discredited by Monte-Carlo simulation. The Log-Pearson type-3 probability density function (LP3 pdf) is shown to appropriately represents the random decision variable Y. Three characteristic parameters of the LP3 pdf are also obtained through the three moments of the decision variable Y. Finally, the system error probability is revisited using the obtained LP3 pdf of the decision variable, the result of which is in excellent consistency with rigorous Monte-Carlo simulation.