Probability Generating Function Research Articles

Random walks on a tree with applications

The various dynamics of many complex systems, both natural and artificial, can be well described as random walks on a newly proposed yet powerful object known as a complex network. Here we consider random walks on a special kind of network, a tree. Using the methods of probability-generating functions, we derive a close connection between the widely studied parameters mean first-passage time M and mean shortest path length A on trees. This suggests that for a given tree, the analytic value for M can be easily obtained by calculating the value A when determining the latter is more convenient and vice versa. As a result, the well-known T graph is selected as one of various applications of our methods, and we then obtain an exact solution to its quantity M. On the one hand, the result addressed here is in perfect agreement with previous ones. On the other hand, our method is easier to manipulate than most preexisting ones, for instance, methods from spectral graph theory, since no complicated techniques are involved.

Stable Polynomials and Sums of Dependent Bernoulli Random Variables: Application to Hoeffding Inequalities

We give, in this paper, a characterization of the independent representation in law for a sum of dependent Bernoulli random variables. This characterization is related to the stability property of the probability-generating function of this sum, which is a polynomial with positive coefficients. As an application, we give a Hoeffding inequality for a sum of dependent Bernoulli random variables when its probability-generating function has all its roots with negative real parts. Some sufficient conditions on the law of the sum of dependent Bernoulli random variables guaranteeing the negativity of the real parts of the roots are discussed. This paper generalizes some results in Liggett (Stoch Process Appl 119:1–15, 2009).

On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation

This article considers a single server, infinite buffer, bulk service Poisson queue with single and multiple vacation. The customers are served in batches following ‘general bulk service’ (GBS) rule. The customers are arriving according to the Poisson process, and the service time of the batches follows an exponential distribution. Using bivariate probability generating function (PGF) method the steady-state joint distributions of the queue content and server content (when server is busy), and joint distribution of the queue content and type of the vacation taken by the server (when server is in vacation) have been obtained. Here by the ‘type of the vacation’ we mean the queue length at vacation initiation epoch. The information about these joint distributions may help in increasing the system performance. Finally, several numerical examples are carried out using MAPLE software to verify the analytical results.

On some properties of the hazard rate function for compound distributions

The hazard rate function plays a crucial role to extract some informations of several random life phenomena. In this paper, we investigate important properties satisfied by the hazard rate function for a general power series class of distributions, as monotonicity properties, sharp bounds and convexity properties. In particular, we highlight how these properties are related to those of the baseline distribution and the corresponding probability-generating function of discrete power series distributions.

Renewal Redundant Systems Under the Marshall–Olkin Failure Model. A Probability Analysis

In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements.

Network percolation of the disease transmission based on bipartite networks

In this paper, considered heterogeneous infectivity and susceptibility, a general stochastic Susceptible-Infectious-Removed (SIR) epidemic model with the cumulative distribution functions (CDFS) of the infectious contact rate and the infectious period based on bipartite networks is discussed. It is isomorphic to a semidirected random network called the bipartite epidemic percolation network. The epidemic threshold corresponds to the phase transition where a giant strongly connected component appears. It is obtained by using the method of the probability generation function. We show that the critical value of the transmissibility predicted by the bond percolation model is larger than that predicted by the epidemic percolation network. We analyze the influences of the network structure and individual heterogeneity on the epidemic threshold by numerical simulations.

EXPLICIT SOLUTIONS FOR CONTINUOUS-TIME QBD PROCESSES BY USING RELATIONS BETWEEN MATRIX GEOMETRIC ANALYSIS AND THE PROBABILITY GENERATING FUNCTIONS METHOD

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equationA0+RA1+R2A2= 0. PGFs involve a row vector$\vec{G}(z)$of unknown generating functions satisfying$H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$where the row vector$\vec{b}(z)$contains unknown “boundary” probabilities calculated as functions of roots of the matrixH(z). We show that: (a)H(z) and$\vec{b}(z)$can be explicitly expressed in terms of the tripleA0,A1, andA2; (b) when each matrix of the triple is lower (or upper) triangular, then (i)Rcan be explicitly expressed in terms of roots of$\det [H(z)]$; and (ii) the stability condition is readily extracted.

Performance prediction and ANFIS computing for unreliable retrial queue with delayed repair under modified vacation policy

This paper deals with the analysis of MX/G/1 retrial queue with impatient customers, modified vacation policy and Bernoulli feedback. When the incoming customer finds the server busy, on vacation or in the state of breakdown, he joins the virtual queue called retrial orbit, otherwise the service is provided to the customer who is at the head of the queue. The service is provided in l(0 ≤ i ≤ l) phases where first is compulsory service and remaining services are optional. When the system becomes empty, server leaves for the vacation of arbitrary length and can take at most J number of vacations. When server comes back from the vacation and finds at least one customer in the queue, he starts providing service to the customer. Supplementary variable technique (SVT) and probability generating function (PGF) method is used to derive the system size distribution and other performance indices. We have also approximated the analytical results using adaptive neuro-fuzzy interface system (ANFIS) soft computing technique, which can identify parameters using supervised learning methods.

Impatient customers in Markovian queue with Bernoulli feedback and waiting server under variant working vacation policy

This paper deals with customers’ impatience behaviour for single server Markovian queueing system under K-variant working vacation policy, waiting server, Bernoulli feedback, balking, reneging, and retention of reneged customers. Using probability generating function (PGF) technique, we obtain the steady-state solution of the system. In addition, we prove the stochastic decomposition properties. Useful performance measures of the considered queueing system are derived. A cost model is developed. Then, the parameter optimisation is carried out numerically, using quadratic fit search method (QFSM). Finally, numerical examples are provided in order to visualize the analytical results.

Analysis of a Retrial Queue With Two-Type Breakdowns and Delayed Repairs

This article studies an M/G/1 retrial queue with two types of breakdowns. When the server is idle, it is subject to breakdowns according to a Poisson process with rate δ and it cannot be repaired immediately. While when the server is busy, it may break down according to a Poisson process with rate θ and can be immediately repaired. Firstly, based on embedded Markov chain technique and probability generating function (PGF) method, we present the necessary and sufficient condition for the system to be stable and the PGF of the orbit size at the departure epochs. Secondly, we give the steady-state joint queue length distribution by supplementary variable method, and present some important performance measures and reliability indices. Thirdly, we provide the analysis of sojourn time of an arbitrary customer in the system when the system is in stable state. Finally, some numerical examples are presented to illustrate the effect of the some system parameters on important performance measures and reliability indices.

Analysis of a bulk service queue with unreliable server, multiple vacation, overloading and stand-by server

We studied the behaviour of an M[X]/G(a, b)/1 queueing system with unreliable server, multiple vacation, overloading and stand-by server in this paper. The stand-by server is utilised only during main server's repair period. The probability generating function (PGF) of queue size and some important performance measures are derived. An extensive numerical result is illustrated.

Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities.

A pathogen can infect multiple hosts. For example, zoonotic diseases like rabies often colonize both humans and animals. Meanwhile, a single host can sometimes be infected with many pathogens, such as malaria and meningitis. Therefore, we studied two susceptible classes and , each of which can be infected when interacting with two different infectious groups and . The stochastic models were formulated through the continuous time Markov chain (CTMC) along with their deterministic analogues. The statistics for the developed model were studied using the multi-type branching process. Since each epidemic class was assumed to transmit only its own type of pathogen, two reproduction numbers were obtained, in addition to the probability-generating functions of offspring. Thus, these, together with the mean number of infections, were used to estimate the probability of extinction. The initial population of infectious classes can influence their probability of extinction. Understanding the disease extinctions and outbreaks could result in rapid intervention by the management for effective control measures.

Solution of an M/G/1 queueing model with k sequential heterogeneous service steps and vacations using the Tauberian property

This article studies a transient queueing model with k sequential heterogeneous service steps and vacations. Entrants arrive according to a Poisson process, and each arrival is serviced sequentially and consistently in k phases in the order of arrival. The service times follow a general distribution function. Upon completion of the service, the server could take a vacation with the probability θ and remains in the system to provide service to other customers with the probability For this model, first, we obtain the Laplace transform of the probability generating functions (PGFs) of system size, and then we obtain the PGFs for a special case in the transient state, and the corresponding steady state results explicitly. Also, we derive the system performance measures such as the means system size, waiting time, and busy period in a closed form.

Performance Analysis of IEEE 802.11p for Continuous Backoff Freezing in IoV

With the rapid development of cloud computing and big data, traditional Vehicular Ad hoc Networks (VANETs) are evolving into the Internet of Vehicles (IoV). As an important communication technology in IoV, IEEE 802.11p protocols have been studied by many experts and scholars. In IEEE 802.11p, a node’s backoff counter will be frozen when the channel is detected as busy. However, most studies did not consider the possibility of continuous backoff freezing when calculating delay. Thus, in this paper, we focus on the performance analysis of IEEE 802.11p for continuous backoff freezing. Specifically, we establish an analytical model to analyze the broadcast performance in the highway scene where vehicles can obtain traffic density from roadside units through Vehicle to Infrastructure (V2I) communications. We first calculate the relationship between vehicle density and the number of vehicles. Then, we derive the relationship between the number of vehicles and packet delay according to Markov chains. Next, we utilize the probability generating function (PGF) to transform traditional Markov chains into z domain under the situation of non-saturation. Finally, we employ the Mason formula to derive packet delay. As compared with the performance without considering the continuous backoff freezing, the simulation results have demonstrated that our analytical model is more reasonable.

The MX/M/c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{X}/M/c$$\\end{document} Bernoulli feedback queue with variant multiple working vacations and impatient customers: performance and economic analysis

The present paper deals with an MX/M/c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^{X}/M/c$$\\end{document} Bernoulli feedback queueing system with variant multiple working vacations and impatience timers which depend on the states of the servers. Whenever a customer arrives at the system, he activates an random impatience timer. If his service has not been completed before his impatience timer expires, the customer may abandon the system. Using certain customer retention mechanism, the impatient customer can be retained in the system. After getting incomplete or unsatisfactory service, with some probability, each customer may comeback to the system as a Bernoulli feedback. Using the probability generating functions (PGFs), we derive the steady-state solution of the model. Then, we obtain useful performance measures. Moreover, we carry out an economic analysis. Finally, numerical study is performed to explore the effects of the model parameters on the behavior of the system.

Modeling the DQE(f) of photon-counting detectors: impact of the pixel sensitivity profile

Purpose. To find comprehensive equations for the frequency-dependent MTF and DQE of photon counting detectors including the effect that the combination of crosstalk with an energy threshold is changing the pixel sensitivity profile and to compare the results with measurements. Methods: The framework of probability-generating functions (PGF) is used to find a simple method to derive the MTF and the DQE directly from a Monte-Carlo model of the detection process. Results. In combination with realistic model parameters for the detector, the method is used to predict the MTF and the DQE for different pixel sizes and thresholds. Particularly for small pixels, the modification of the sensitivity profile due to crosstalk substantially affects the frequency dependence of both quantities. Conclusion. The phenomenon of the pixel sensitivity profile, i.e. the fact that the choice of the threshold is affecting the detector sharpness, may play a substantial role in exploiting the full potential of photon counting detectors. The model compares well with measurements: with only two model parameters, the model can predict the MTF(f) and the DQE(f) for a wide range of thresholds.

Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services

In this investigation, a novel sort of retrial queueing system with working breakdown services is introduced. Two distinct kinds of customers are considered, which are priority and ordinary customers. The normal busy server may become inadequate due to catastrophes at any time which cause the major server to fail. At a failure moment, the major server is sent to be fixed and the server functions at a lower speed (called the working breakdown period) during the repair period. The probability generating functions (PGF) of the system size is found using the concepts of the supplementary variable technique (SVT). The impact of parameters in system performance measures and cost optimization are examined numerically.

A Markov Modulated Dynamic Contagion Process with Application to Credit Risk

Self-exciting point processes are applied to various fields such as seismology, finance, neurophysiology, criminology, biology etc to model the clustering/contagion phenomenon and extreme risk events. This article proposes an analytically tractable point process, a generalization of the classical Hawkes process, with the intensity process following a Markov modulated mean-reverting affine jump-diffusion process with contagion effects. The proposed process has both self-exciting and externally-exciting jumps that represent the effects of endogenous and exogenous events. This article derives the closed-form expressions, for distributional properties such as the first order moment, probability generating function (PGF) of the point process and Laplace transform of the intensity process, which makes it computationally efficient. The application of the proposed process to price the synthetic collateralized debt obligations (CDOs) in a top-down framework is also presented.

LDPE Production in Tubular Reactors: Comprehensive Model for the Prediction of the Joint Molecular Weight-Short (Long) Chain Branching Distributions

Good control on molecular properties, and as a consequence on end-use properties, is very important for low-density polyethylene (LDPE) manufacturers. However, the connection between the architecture of polymer chains and the kinetic mechanism and polymerization conditions is still a subject of study. In this work, we present a comprehensive model of the polymerization of ethylene in high-pressure tubular reactors. In addition to the usual predictions of conversion, temperature profiles, and average molecular properties, this model also provides bivariate distributions such as molecular-weight–long-chain branching distribution and molecular-weight–short-chain branching distribution of LDPE produced under different operating conditions. The 2D probability-generating function technique is applied to obtain the bivariate distributions. This is a deterministic technique that allows the calculation of the distributions without any prior assumption of their shape.

Multistrain edge‐based compartmental model on networks

Multistrain diseases, which are infected through individual contacts, pose severe public health threat nowadays. In this paper, we build competitive and mutative two‐strain edge‐based compartmental models using probability generation function (PGF) and pair approximation (PA). Both of them are ordinary differential equations. Their basic reproduction numbers and final size formulas are explicitly derived. We show that the formula gives a unique positive final epidemic size when the reproduction number is larger than unity. We further consider competitive and mutative multistrain diseases spreading models and compute their basic reproduction numbers. We perform numerical simulations that show some dynamical properties of the competitive and mutative two‐strain models.