Probability Generating Function Research Articles

Unreliable two-server queueing system with variable repair rates

This article considers an infinite-capacity two-server queue system, where the servers are subject to failure at any time. A repairman is responsible for repair the failure server and the repair rate is changeable depending on the number of failure servers. The repair rate of only one server is failure and that of two servers are all failure may be different. The steady-state reliability indexes and the stability condition of queue are derived through the transfer rate matrix. The expected queue length and some related steady-state performances are obtained by the probability-generating function transform method. Numerical examples are provided for illustration purpose.

Spread of competing viruses on heterogeneous networks.

In this paper, we propose a model where two strains compete with each other at the expense of common susceptible individuals on heterogeneous networks by using pair-wise approximation closed by the probability-generating function (PGF). All of the strains obey the susceptible–infected–recovered (SIR) mechanism. From a special perspective, we first study the dynamical behaviour of an SIR model closed by the PGF, and obtain the basic reproduction number via two methods. Then we build a model to study the spreading dynamics of competing viruses and discuss the conditions for the local stability of equilibria, which is different from the condition obtained by using the heterogeneous mean-field approach. Finally, we perform numerical simulations on Barabási–Albert networks to complement our theoretical research, and show some dynamical properties of the model with competing viruses.This article is part of the themed issue ‘Mathematical methods in medicine: neuroscience, cardiology and pathology’.

Анализ системы массового обслуживания с рекуррентным обслуживанием и полным обновлением

We investigate the queueing system in which the losses of incoming orders due to the introduction of a special renovation mechanism are possible. The introduced queueing system consists of server with a general distribution of service time and a buffer of unlimited capacity. The incoming flow of tasks is a Poisson one. The renovation mechanism is that at the end of its service the task on the server may with some probability empty the buffer and leave the system, or with an additional probability may just leave the system. In order to study the characteristics of the system the Markov chain embedded upon the end of service times is introduced. Under the assumption of the existence of a stationary regime for the embedded Markov chain the formula for the probability generation function is obtained. With the help of the probability generation function such system characteristics as the probability of the system being empty, the average number of customers in the system, the probability of a task not to be dropped, the distribution of the service waiting time for non-dropped tasks, the average service waiting time for non-dropped requests are derived.

Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains

Abstract The 2D probability-generating function technique is a powerful method for modeling bivariate distributions of polymer properties. It is based on the transformation of bivariate population balance equations using 2D probability generating functions (pgf) followed by a recovery of the distributions from the transform domain by numerical inversion. A key step of this method is the inversion of the pgf transforms. Available numerical inversion methods yield excellent results for pgf transforms of distributions with independent dimensions with similar orders of magnitude, for example bivariate molecular weight distributions in copolymerization systems. However, numerical problems are found for 2D distributions in which the independent dimensions have very different ranges of values, such as the molecular weight distribution-branching distribution in branched polymers. In this work, two new 2D pgf inversion methods are developed, which regard the pgf as a complex variable. The superior accuracy of these innovative methods makes them suitable for recovering any type of bivariate distribution. This enhances the capabilities of the 2D pgf modeling technique for simulation and optimization of polymer processes. An application example of the technique in a polymeric system of industrial interest is presented.

A Novel Model of Stock Data Mining with M/G/1 Queue for Evaluation of Stock Crash

Data mining is the process of searching the information from a large amount of data. In order to evaluate the stock crash this paper proposes general decrementing service M/G/1 queue system with multiple adaptive vacations to find information related to stock crash in data about Shanghai Composite Index. We use the probability generating function (P.G.F.) of stationary queue length and LST of waiting time, and their stochastic decomposition to calculate Existing money flow. Existing Money flow calculation model is improved based on the stationary queue length and LST of waiting time. We program to achieve the stock of existing money flow algorithm, and get the number of existing money flow. The improved algorithm can early warn the stock market crash. The empirical result shows that: There will be a rise in price before the Stock Market Crash, and the stock of existing money inflow begin to decrease. The stock market crash fell for at least six months. The stock market crash fell by at least fifty-five percent. Most of the stock market crash fell by over seventy-percent. The stock market crash down time is inversely proportional to the magnitude of the decline. If the down time is short, the magnitude of the decline is large. If the down time is long, the magnitude of the decline is small. The stock market crash is great harm to investors.

A moment-convergence method for stochastic analysis of biochemical reaction networks.

Traditional moment-closure methods need to assume that high-order cumulants of a probability distribution approximate to zero. However, this strong assumption is not satisfied for many biochemical reaction networks. Here, we introduce convergent moments (defined in mathematics as the coefficients in the Taylor expansion of the probability-generating function at some point) to overcome this drawback of the moment-closure methods. As such, we develop a new analysis method for stochastic chemical kinetics. This method provides an accurate approximation for the master probability equation (MPE). In particular, the connection between low-order convergent moments and rate constants can be more easily derived in terms of explicit and analytical forms, allowing insights that would be difficult to obtain through direct simulation or manipulation of the MPE. In addition, it provides an accurate and efficient way to compute steady-state or transient probability distribution, avoiding the algorithmic difficulty associated with stiffness of the MPE due to large differences in sizes of rate constants. Applications of the method to several systems reveal nontrivial stochastic mechanisms of gene expression dynamics, e.g., intrinsic fluctuations can induce transient bimodality and amplify transient signals, and slow switching between promoter states can increase fluctuations in spatially heterogeneous signals. The overall approach has broad applications in modeling, analysis, and computation of complex biochemical networks with intrinsic noise.

On the modes of the negative binomial distribution of order k

ABSTRACTIn this paper, the modes of the negative binomial distribution of order k are studied. Firstly, the method of transition probability flow graphs is introduced to deal with the probability-generating function of the geometric distribution of order k, which is a special case of the negative binomial distribution of the same order. And then, the general negative binomial distribution of order k is investigated. By means of probability distribution function, the mode of the geometric distribution of order k is derived, i.e. . Based on the Fibonacci sequence and Poly-nacci sequence, the modes of the negative binomial distribution of order k in some cases are obtained: (1) and , for p=0.5; (2) for p=0.5. Finally, an application of negative binomial distribution of order k in continuous sampling plans is given.

On the Discrete-TimeGeo/G/1Queue with Vacations in Random Environment

A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.

Optimal launch of a new generation of technology: a multi attribute approach discrete time diffusion process

The art of critical decision making in the presence of conflicting objectives has given rise to multi attribute utility theory (MAUT). For any organisation to gain competitive edge in the market it's imperative to optimise launch time of their new product. Too quick a launch impacts the sales of their current generations while delay in launch may lead to customer's disinterest. In order to optimise launch time, customer's adoption behaviour and cost is considered in the proposed study. To analyse the simultaneous impact of both these attributes we make use of MAUT. A discrete Bass model, which is a discrete analogue of the famous Bass model, is proposed. Bass model is developed in continuous time, which limits its application on many real life applications having discrete time data. Therefore, in this paper a discrete innovation diffusion model is proposed using a discrete logistic function and an exact solution is obtained using probability generating function (PGF). The proposed model is validated on three real life datasets and is optimised to obtain optimal launch time using MAUT.

Research on the Critical Value of Traffic Congestion Propagation Based on Coordination Game

In order to analysis the process of congestion diffusion in traffic network and obtain the critical value of traffic congestion, this paper presented a coordination game model for traffic congestion diffusion. Through the coordination game between network individuals and the decision-making behavior of travelers in traffic congestion, this paper described the process of congestion diffusion in traffic network. And according to the process of pass through the behavior of adjacent nodes in network, built a traffic network congestion diffusion model, derived the critical condition of traffic congestion diffusion with the method of probability generation function. Finally, this paper built a simulation model to analysis the influence on the traffic congestion imposed by the road network structure and the distribution of node degree. When the results of simulation is same with the results of analytical analysis, and it can reflect the dynamic information of traffic congestion, then it is found that the influence of the congested node on other adjacent nodes is the key point to find the critical condition of traffic congestion diffusion, when the influence reaches to a certain extent, a massive traffic congestion will be generated.

Queue Length and Busy Period Analysis for the M/G/1 Queue with Negative Arrivals

In this paper, the queue lengths and the busy period lengths of the M/G/1 queueing systems with negative arrivals are analyzed. Two types of negative arrivals are considered. One type is negative customers and the other type is disasters. When a negative customer arrives to a system, one positive customer is removed if the number of positive customers is more than one. In particular, we assume the RCH (Removal of a Customer at the Head) type of negative customers which represent a kind of work-canceling signal to the positive customer in service. On the other hand, disasters get rid of all customers in the system. In this paper, the Probability Generating Function (PGF) of the stationary queue length and busy period length of M/G/1 queue with both negative customers and disasters are derived.

Markov-modulated infinite-server queues driven by a common background process

ABSTRACTThis paper studies a system with multiple infinite-server queues that are modulated by a common background process. If this background process, being modeled as a finite-state continuous-time Markov chain, is in state j, then the arrival rate into the i-th queue is λi, j, whereas the service times of customers present in this queue are exponentially distributed with mean μ− 1i, j; at each of the individual queues all customers present are served in parallel (thus reflecting their infinite-server nature).Three types of results are presented: in the first place (i) we derive differential equations for the probability-generating functions corresponding to the distributions of the transient and stationary numbers of customers (jointly in all queues), then (ii) we set up recursions for the (joint) moments, and finally (iii) we establish a central limit theorem in the asymptotic regime in which the arrival rates as well as the transition rates of the background process are simultaneously growing large.

Single passage in mouse organs enhances the survival and spread of Salmonella enterica.

Intravenous inoculation of Salmonella enterica serovar Typhimurium into mice is a prime experimental model of invasive salmonellosis. The use of wild-type isogenic tagged strains (WITS) in this system has revealed that bacteria undergo independent bottlenecks in the liver and spleen before establishing a systemic infection. We recently showed that those bacteria that survived the bottleneck exhibited enhanced growth when transferred to naive mice. In this study, we set out to disentangle the components of this in vivo adaptation by inoculating mice with WITS grown either in vitro or in vivo. We developed an original method to estimate the replication and killing rates of bacteria from experimental data, which involved solving the probability-generating function of a non-homogeneous birth–death–immigration process. This revealed a low initial mortality in bacteria obtained from a donor animal. Next, an analysis of WITS distributions in the livers and spleens of recipient animals indicated that in vivo-passaged bacteria started spreading between organs earlier than in vitro-grown bacteria. These results further our understanding of the influence of passage in a host on the fitness and virulence of Salmonella enterica and represent an advance in the power of investigation on the patterns and mechanisms of host–pathogen interactions.

Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random ``impatience timer. If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.

Modeling MAC Protocol Based on Frame Slotted Aloha for Low Energy Critical Infrastructure Sensor Networks

We propose and analyse a medium access control (MAC) protocol for low energy critical infrastructure monitoring (LECIM) networks. As the packet drop probability plays crucial role in LECIM applications, we propose framed slotted aloha based MAC for LECIM using linearly increased contention window size to reduce the packet drop probability. We present a mathematical model of our proposed MAC under both saturated and nonsaturated traffic scenarios. We use probabilistic approach to find the performance metrics such as collision probability, packet drop probability, throughput, and energy consumption. Also, we obtain the probability-generating function of the head-of-line (HoL) delay of packet. The analytical results match with simulations. Our results can be used in the design of a system by providing the optimum system parameters for endpoints satisfying the given quality of service requirements on packet drop probability, energy consumption, and delay.

A geometric analysis of fast-slow models for stochastic gene expression.

Stochastic models for gene expression frequently exhibit dynamics on several different scales. One potential time-scale separation is caused by significant differences in the lifetimes of mRNA and protein; the ratio of the two degradation rates gives a natural small parameter in the resulting chemical master equation, allowing for the application of perturbation techniques. Here, we develop a framework for the analysis of a family of 'fast-slow' models for gene expression that is based on geometric singular perturbation theory. We illustrate our approach by giving a complete characterisation of a standard two-stage model which assumes transcription, translation, and degradation to be first-order reactions. In particular, we present a systematic expansion procedure for the probability-generating function that can in principle be taken to any order in the perturbation parameter, allowing for an approximation of the corresponding propagator probabilities to that same order. For illustrative purposes, we perform this expansion explicitly to first order, both on the fast and the slow time-scales; then, we combine the resulting asymptotics into a composite fast-slow expansion that is uniformly valid in time. In the process, we extend, and prove rigorously, results previously obtained by Shahrezaei and Swain (Proc Natl Acad Sci USA 105(45):17256-17261, 2008) and Bokes et al. (J Math Biol 64(5):829-854, 2012; J Math Biol 65(3):493-520, 2012). We verify our asymptotics by numerical simulation, and we explore its practical applicability and the effects of a variation in the system parameters and the time-scale separation. Focussing on biologically relevant parameter regimes that induce translational bursting, as well as those in which mRNA is frequently transcribed, we find that the first-order correction can significantly improve the steady-state probability distribution. Similarly, in the time-dependent scenario, inclusion of the first-order fast asymptotics results in a uniform approximation for the propagator probabilities that is superior to the slow dynamics alone. Finally, we discuss the generalisation of our geometric framework to models for regulated gene expression that involve additional stages.

Geo (λ)/ Geo (μ) +G/2 queues with heterogeneous servers operating under FCFS queue discipline

This article discusses the steady analysis of a discrete time queue of Geo/Geo+G/2 type. All arriving customers are served either by server-1 according to a geometrically distributed service time S1=k slots for k=1,2, …∞, with mass function f1(k)==Pr(S1=k) = μ(1- μ) k-1 with mean rate 0 or mean service rate μ2=1/β. Sequel to some objections raised on the use of the classical 'First Come First Served (FCFS)' queue discipline when the two heterogeneous servers operate as parallel service providers, an alternative queue discipline in a serial configuration of servers are considered in this work; the objective is that if, in a single-channel queue in equilibrium, the service rate suddenly increases and exceeds the present service capacity, install a new channel to work serially with the first channel as suggested by Krishnamoorthy (1968). Using the embedded method subject to different service time distributions we present an exact analysis for finding the ‘Probability generating Function (PGF)’ of steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. This work shows that one can obtain all stationery probabilities and other vital measures for this queue under certain additional and simple but realistic assumptions.

GI/G/1 대기행렬 대기시간 분포의 새로운 유도방법

This paper presents a heuristic approach to derive the Laplace-Stieltjes transform (LST) and the probability generating function (PGF) of the waiting time distributions of a continuous- and a discrete-time GI/G/1 queue, respectively. This is a new idea to derive the well-known results, the waiting time distribution of GI/G/1 queue, in a different way.

Admission control of bulk retrial feedback queue with K-optional vacations

In this study, we have made an attempt to provide a ready reckoner for solving many congestion problems of day–to–day life under techno–economic constraints which can be seen in many industrial organisations and computer and communication systems. This can be made possible by designing a control policy called Bernoulli Admission Control Policy (BACP) wherein one can control the arrival rate of the customers by allowing only some of them to enter the system. We have applied this policy to a bulk retrial queueing system with Bernoulli feedback under Bernoulli vacation schedule. For the proper utilisation of the server's resources, there is a provision for the server to go for a fixed number of optional vacations (say K) depending upon its choice. We have provided a numerical simulation for solving the problem dealing with local area networks (LAN) especially relating to mobile IP. The service time, repair time, vacation time and setup time are assumed to be generally distributed. We have used supplementary variable technique (SVT) and probability generating function (PGF) method to compute various queueing and reliability measures of interest.