Showing papers on "Probability-generating function published in 2013"

Choice probability generating functions

Abstract: This paper establishes that every random utility discrete choice model (RUM) has a representation that can be characterized by a choice-probability generating function (CPGF) with specific properties, and that every function with these specific properties is consistent with a RUM. The choice probabilities from the RUM are obtained from the gradient of the CPGF. Mixtures of RUM are characterized by logarithmic mixtures of their associated CPGF. The paper relates CPGF to multivariate extreme value distributions, and reviews and extends methods for constructing generating functions for applications. The choice probabilities of any ARUM may be approximated by a cross-nested logit model. The results for ARUM are extended to competing risk survival models.

Automated design of probability distributions as mutation operators for evolutionary programming using genetic programming

Abstract: The mutation operator is the only source of variation in Evolutionary Programming. In the past these have been human nominated and included the Gaussian, Cauchy, and the Levy distributions. We automatically design mutation operators (probability distributions) using Genetic Programming. This is done by using a standard Gaussian random number generator as the terminal set and and basic arithmetic operators as the function set. In other words, an arbitrary random number generator is a function of a randomly (Gaussian) generated number passed through an arbitrary function generated by Genetic Programming. Rather than engaging in the futile attempt to develop mutation operators for arbitrary benchmark functions (which is a consequence of the No Free Lunch theorems), we consider tailoring mutation operators for particular function classes. We draw functions from a function class (a probability distribution over a set of functions). The mutation probability distribution is trained on a set of function instances drawn from a given function class. It is then tested on a separate independent test set of function instances to confirm that the evolved probability distribution has indeed generalized to the function class. Initial results are highly encouraging: on each of the ten function classes the probability distributions generated using Genetic Programming outperform both the Gaussian and Cauchy distributions.

Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy

Abstract: In this paper, we consider an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy, where we examine the case that customer impatience is due to the servers' vacation. Whenever a system becomes empty, the server takes a vacation. However, the server is allowed to take a maximum number $K$ of vacations if the system remains empty after the end of a vacation. This vacation policy includes both a single vacation and multiple vacations as special cases. We derive the probability generating functions of the steady-state probabilities and obtain the closed-form expressions of the system sizes when the server is in different states. We further make comparisons between the mean system sizes under the variant vacation policy and the mean system sizes under the single vacation policy or the multiple vacation policy. In addition, we obtain the closed-form expressions for other important performance measures and discuss their monotonicity with respect $K$. Finally, we present some numerical results to show the effects of some parameters on some performance measures.

Exact results for fixation probability of bithermal evolutionary graphs.

Abstract: One of the most fundamental concepts of evolutionary dynamics is the “fixation” probability, i.e. the probability that a mutant spreads through the whole population. Most natural communities are geographically structured into habitats exchanging individuals among each other and can be modeled by an evolutionary graph (EG), where directed links weight the probability for the offspring of one individual to replace another individual in the community. EGs have recently spurred huge interest, as it has been shown that some topology can amplify or suppress the effect of beneficial mutations. Very few exact analytical results however are known for EGs. In this article we show that the use of a new technique, the fixed point of probability generating function, allows us to compute the exact fixation probability for a large subset of bithermal graphs. We also show by numerical simulations that the computed solution holds for all bithermal graphs. Moreover, the analytical solution allows us to clarify the opposing consequences of birth–death versus death–birth processes as amplifier or suppressor of beneficial mutations for the same bithermal topology.

Mathematical Modeling of Bivariate Distributions of Polymer Properties Using 2D Probability Generating Functions. Part II: Transformation of Population Mass Balances of Polymer Processes

Abstract: This is the second of two works presenting a new mathematical method for modeling bivariate distributions of polymer properties. It is based on the transformation of population balances using 2D probability generating functions (pgf) and a posteriori recovery of the distribution from the transform domain by numerical inversion. Part I of this work was devoted to the numerical inversion step. Here the transformation of the population balances to the pgf domain is analyzed. A 2D pgf transform table is developed, which allows a simple transformation of any typical polymer balance equation. Three copolymerization examples are used to show the application of the complete procedure of this modeling technique.

Queues in tandem with customer deadlines and retrials

Abstract: We study queues in tandem with customer deadlines and retrials. We first consider a 2-queue Markovian system with blocking at the second queue, analyze it, and derive its stability condition. We then study a non-Markovian setting and derive the stability condition for an approximating diffusion, showing its similarity to the former condition. In the Markovian setting, we use probability generating functions and matrix analytic techniques. In the diffusion setting, we consider expectations of the first hitting times of compact sets.

The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times

Abstract: In this paper, we analyse the behaviour of a discrete-time single-server queueing system with general service times, equipped with the $NT$-policy. This is a threshold policy designed to reduce the number of service unit activation/deactivation cycles, whilst ensuring an acceptable delay trade-off. Once the server is deactivated, reactivation will be postponed until either $N$ customers have accumulated in the queue or the first customer has been in the queue for $T$ slots, whichever happens first. Due to this modus operandi, the system circulates between three phases: empty, accumulating and serving. We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.

On The Renewal Input Batch-arrival Queue Under Single And Multiple Working Vacation Policy With Application To EPON

Abstract: This paper analyzes renewal input batch arrival queues with multiple and single working vacation policy. We obtain the steady state system-length distributions at pre-arrival and arbitrary epochs. The Laplace-Stieltjes transforms (LST) of the sojourn time distribution for the first customer and an arbitrary customer in an accepted batch have been derived. Also the probability density function of sojourn time distribution for the first customer in an accepted batch has been obtained. The proposed analysis is based on the roots of the characteristics equations formed using the probability generating functions (p.g.f.) of embedded pre-arrival epoch probabilities. Several performance measures such as the mean system-length and mean sojourn times of first and an arbitrary customer in an accepted batch are presented. The suggested queues have potential application in ethernet passive optical network (EPON).

Limit laws of the coefficients of polynomials with only unit roots

Abstract: We consider sequences of random variables whose probability generating functions are polynomials all of whose roots lie on the unit circle. The distribution of such random variables has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth normalized (by the standard deviation) central 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 goes 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.

Sum of Discrete i.i.d. Random Variables and Its Application to Cooperative Spectrum Sensing

Abstract: In this paper, we derive the probability mass function (PMF) and the cumulative mass function (CMF) of the sum of discrete independent and identically distributed random variables. As an application of the PMF and CMF, we analyze the missed-detection, false-alarm, and overall error probabilities of cooperative spectrum sensing with any numbers of quantization levels and any quantization thresholds in a closed form. Computer simulation results match the analysis perfectly. In addition, centralized and distributed threshold selections are discussed.

Probability density function filter design based on symmetric K-L distance

Abstract: Recently the filtering design based on probability density function become an important method to solve Non-gaussian filtering. However, the existing methods are difficult to use in practice because the performance of non-negative cannot be guaranteed or owning to the highly conservative. So this paper make symmetric K-L distance as a new performance index function and give constraint range of the weighting function. Then present the optimal iteration filtering method based on gradient search technique and numeric integration. Finally, a simulation example is used to illustrate the use of proposed algorithm and desired results have been obtained.

The Bivariate Confluent Hypergeometric Series Distribution and Some of Its Properties

Abstract: In this paper we develop a bivariate version of the confluent hypergeometric series distribution through its probability generating function and study some of its properties by deriving its probability mass function, factorial moments, probability generating functions of its marginal and conditional distributions and recursion formulae for probabilities, raw moments and factorial moments. Further certain mixtures and limiting cases of this distribution are also obtained.

Estimating end-to-end delays under changing conditions

Abstract: We consider the problem of estimating the end-to-end latency of intermittently connected paths in disruption/delay tolerant networks. This is useful when performing source routing, in which a complete path is chosen for a packet to travel from source to destination (when intermediate nodes are really low complexity devices that can only forward packets but cannot perform route computations), or in linear network topologies. While computing the time to traverse such a path may be straightforward in fixed, static networks, doing so becomes much more challenging in dynamic networks, in which the state of an edge in one timeslot (i.e., its presence or absence) is random, and may depend on its state in the previous timeslot. The traversal time is due to both time spent waiting for edges to appear and time spent crossing them once they become available. We compute the expected traversal time (ETT) for a dynamic path in a number of special cases of stochastic edge dynamics models, and for three different edge failure models, culminating in a surprisingly nontrivial yet realistic ``hybrid network" setting in which the initial configuration of edge states for the entire path is known. We show that the ETT for this "initial configuration" setting can be computed in quadratic time (as a function of path length), by an algorithm based on probability generating functions. We also give several linear-time upper and lower bounds on the ETT, which we evaluate, along with our ETT algorithm, using numerical simulations.

A numerical characteristic method for probability generating functions on stochastic first-order reaction networks

Abstract: We propose an efficient and accurate numerical scheme for solving probability generating functions arising in stochastic models of general first-order reaction networks by using the characteristic curves. A partial differential equation derived by a probability generating function is the transport equation with variable coefficients. We apply the idea of characteristics for the estimation of statistical measures, consisting of the mean, variance, and marginal probability. Estimation accuracy is obtained by the Newton formulas for the finite difference and time accuracy is obtained by applying the fourth order Runge–Kutta scheme for the characteristic curve and the Simpson method for the integration on the curve. We apply our proposed method to motivating biological examples and show the accuracy by comparing simulation results from the characteristic method with those from the stochastic simulation algorithm.

System modeling and performance analysis of the power saving class type II in BWA networks

Abstract: For reducing the energy consumption of the Mobile Station in mobile Broadband Wireless Access networks, IEEE 802.16 offers three kinds of sleep mode operations called power saving classes type I, type II and type III. In order to investigate mathematically the inherent relationships between the performance measures and the system parameters, we propose in this paper a novel method for modeling the sleep mode with the power saving class type II in IEEE 802.16 and analyzing the performance of this sleep mode. Considering the attractive feature that some data frames can be transmitted during the listening state, we present a queueing model with two kinds of busy mechanisms to capture the working principle of the sleep mode operations with the power saving class type II. With the first and higher derivatives of the probability generating functions, we can give the averages and the standard deviations for the system performance using the diffusion approximation for the operating process of the system. We also propose methods for measuring the system performance in terms of the switching ratio, the energy saving ratio, and the average response time of data frames, as well as giving the expressions for these performance measures. Numerical results are provided with analysis and simulation to show the average performance measures, standard deviations and the cost function with different system loads. Moreover, we construct a cost function with the aim of determining the optimal time length of the sleep window to minimize the cost function.

Stochastic Derivation of an Integral Equation for Probability Generating Functions

Abstract: Functional, integral and differential equations of transformed probability generating functions are generally recognized as powerful analytical tools for establishing characterizations of discrete probability distributions. The present paper establishes a characterization of the distribution of an important integral part model by incorporating an integral equation based on three fundamental transformed probability generating functions. Interpretations of such a characterization in analyzing and implementing information risk frequency reduction operations are also established.

A reduction formula for the hypergeometric function using gamma random variables

Abstract: We show that the hypergeometric function p F q is the mean of 0 F q with its last argument multiplied by the product of independent gamma random variables. We use this to express for α>0 and β>0 in terms of 1 F 1, where I p is the modified Bessel function. A second derivation gives the moments of the non-central chi-square random variable. Some related results are derived, including an analog of Gauss's duplication formula for p F q .

INNER PRODUCT OF RANDOM VECTORS ON n S

Abstract: Probability distribution function of the inner product of random vectors uniformly distributed on the standard unit sphere n S is derived. The pdf (probability density function) of the inner product is useful for analyzing the autocorrelation of time series.

Moment-Based Approximations of Discrete Probability Distributions Using Rational Functions

Abstract: Provost et al. (2009) introduced an approximation to discrete probability mass functions using a base density function and a polynomial function multiplier. In this article, we examine a new estimator constructed by using a rational function multiplier. The new approximation requires fewer moments to calculate, and generally performs better. Several example cases will be presented.

Descent variation of samples of geometric random variables

Abstract: In this paper, we consider random words ω1ω2ω3⋯ωn of length n, where the letters ωi ∈ℕ are independently generated with a geometric probability such that Pωi=k=pqk-1 where p+q=1 . We have a descent at position i whenever ωi+1 < ωi. The size of such a descent is ωi-ωi+1 and the descent variation is the sum of all the descent sizes for that word. We study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities, and find the probability generating functions for descent variation of such words.

Transient Solution of an Batch arrival Queue with Two Types of Service, Multiple Vacation, Random breakdown and Restricted Admissibility

Abstract: In this paper we analyse a batch arrival queue with two types of service subject to random breakdowns having multiple server vacation. We assume that the server provides two types of service, type 1 with probability and type 2 with probability with the service times following general distribution and each arriving customer may choose either type of service. The server takes vacation each time the system becomes empty and the vacation period is assumed to be general. On returning from vacation if the server finds no customer waiting in the system, then the server again goes for vacation until he finds at least one customer in the system. The system may breakdown at random and repair time follows exponential distribution. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. 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. Average queue lenth and average system size are also computed.

Calculate the Moment Generating Function for Continuous Random Variable by Using Decomposition Method

Abstract: In this paper, decomposition method is applied to develop a computational method for the moment generating function of continuous random variable. The proposed method is easy to implement from a computational viewpoint and can be employed for finding moment generating function of continuous random variable without solving any integral. Sometimes, this integral cannot be solved in general and in this case, the moment generating function remains in integral form. Some examples are illustrative for demonstrating the advantage of the proposed

Research on Control of Passing Priority in Logistics Nodes Among the Distribution Centers

Abstract: In this article, we adopt the Markov Chain theory and Probability generating functions as analysis and study of the function methods. With complete service rules and limited rules constructed passing ability and control analysis model of logistics nodes of distribution center. From the mean waiting time of through passing logistics nodes to analyze out its control of the priority and presents the corresponding simulation experiment results, research and analysis of the conclusion.

Time Dependent Solution of Batch Arrival Queueing Systemwith Random Breakdowns and Second Optional Server Vacation

Abstract: This paper investigates a single server queue with Poisson arrivals, general (arbitrary) service time distributions and Bernoulli vacation subject to random breakdowns. However, after the completion of a service, the server will take Bernoulli vacation, that the server make take a vacation with probability θ or may continue to stay in the system with probability 1 − θ for serving the next customer, if any. In addition to this, the vacation period of the server has two phases in which first phase is compulsory followed by the second phase in a such way a that the server may choose second phase with probability p or may return back to the system with probability 1−p and the vacation time follows general (arbitrary) distribution. The system may breakdown at random with mean break down rate α and repair process starts immediately in which the repair time follows exponential distribution with mean repair rate β. 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. AMS subject classification: 60K25, 60K30

Computing Probability Generating Functions of Coin Flipping and Its Generalizations

Abstract: A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different letters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probability generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.