Probability Generating Function Approach Research Articles

Preemptive Priority Markovian Queue Subject to Server Breakdown with Imperfect Coverage and Working Vacation Interruption

This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization.

ANFIS and metaheuristic optimization for strategic joining policy with re-attempt and vacation

This research work deals with the queueing analysis and strategic behavior of primitive users (PUs) and ordinary users (OUs) in cognitive radio system (CRS). Whenever fusion center (FC) is occupied, the ordinary users must decide whether to join a retrial orbit and make re-attempts or depart from the system. The OUs residing in the retrial orbit try to access the spectrum through FC. The FC becomes inactive and considered on vacation in case if there are no users in the CRS. Returning back from the vacation, the FC renders service to the primitive users. The service of ordinary users (OUs) may be interrupted due to breakdown of FC. The probability generating function approach is used to derive the performance indices related to the queueing characterization. The joining strategic behavior of the primitive as well as ordinary users has been investigated by considering a linear reward cost function. The social welfare by formulating the cooperative strategy is studied. The hybrid soft computing technique viz. adaptive network fuzzy inference system (ANFIS) is employed for the numerical simulation and sensitivity analysis of various system metrics. The optimal service rates are determined by minimizing the total cost involved in various activities using the quasi-Newton method (QNM) and metaheuristic algorithms.

Analysis of a Markovian queueing system with single working vacation and impatience of customers

In this paper, we study an infinite capacity single server Markovian queue with a single working vacation and reneging of impatient customers in the queue during working vacation period. Customers arrive to the system following a Poisson distribution. The server goes to vacation when the system is empty and stay in vacation for a random period of time that is exponentially distributed. During the working vacation period, the server still continue providing service with a slow service rate. After the completion of the vacation, the server returns back to the regular service period and continue providing service with the regular busy period service rate if there are one or more customers in the system or it will stay idle until a new customer arrives to the system. During working vacation, customers in the queue get impatient and renege from the system and the reneging time is assumed to follow an exponential distribution. The system is modeled as a quasi-birth-death process and the stationary probabilities of the model are obtained using probability generating function approach. Some numerical analysis is also carried out to show the effect of some of the parameters on some selected performance measures of the system.

A Stochastic Model to Analyze and Predict Transmission Dynamics of Tuberculosis in Ede Kingdom of Osun State

In this study the stochastic process model for estimating the incidence of tuberculosis (TB) infection in Ede kingdom (Ede North and Ede South Local Government Areas) of Osun State was carried out. The probability generating function approach was used to solve the associated birth process model to obtain the estimate of TB incidence. Also time series analysis was carried out using JMulti software to predict future incidence rate of the disease in the study area. Based on Autoregressive Integrated Moving Average (ARIMA) model, the autocorrelation and partial autocorrelation methods and a suitable model to forecast TB infection was obtained. The goodness of fit was measured using the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC). Having satisfied all the model assumptions ARIMA (0,1,1) model with standard error, 6.37086 was found to be the best model for the forecast. It was observed that the forecasted series were close to the actual data series

Theoretical solutions for degree distribution of decreasing random birth-and-death networks

In this paper, theoretical solutions for degree distribution of decreasing random birth-and-death networks [Formula: see text] are provided. First, we prove that the degree distribution has the form of Poisson summation, for which degree distribution equations under steady state and probability generating function approach are employed. Then, based on the form of Poisson summation, we further confirm the tail characteristic of degree distribution is Poisson tail. Finally, simulations are carried out to verify these results by comparing the theoretical solutions with computer simulations.

A discrete-time queueing system with changes in the vacation times

Abstract This paper considers a discrete-time queueing system in which an arriving customer can decide to follow a last come first served (LCFS) service discipline or to become a negative customer that eliminates the one at service, if any. After service completion, the server can opt for a vacation time or it can remain on duty. Changes in the vacation times as well as their associated distribution are thoroughly studied. An extensive analysis of the system is carried out and, using a probability generating function approach, steady-state performance measures such as the first moments of the busy period of the queue content and of customers delay are obtained. Finally, some numerical examples to show the influence of the parameters on several performance characteristics are given.

Prevention of infectious diseases by public vaccination and individual protection.

In the face of serious infectious diseases, governments endeavour to implement containment measures such as public vaccination at a macroscopic level. Meanwhile, individuals tend to protect themselves by avoiding contacts with infections at a microscopic level. However, a comprehensive understanding of how such combined strategy influences epidemic dynamics is still lacking. We study a susceptible-infected-susceptible epidemic model with imperfect vaccination on dynamic contact networks, where the macroscopic intervention is represented by random vaccination of the population and the microscopic protection is characterised by susceptible individuals rewiring contacts from infective neighbours. In particular, the model is formulated both in populations without and then with demographic effects (births, deaths, and migration). Using the pairwise approximation and the probability generating function approach, we investigate both dynamics of the epidemic and the underlying network. For populations without demography, the emerging degree correlations, bistable states, and oscillations demonstrate the combined effects of the public vaccination program and individual protective behavior. Compared to either strategy in isolation, the combination of public vaccination and individual protection is more effective in preventing and controlling the spread of infectious diseases by increasing both the invasion threshold and the persistence threshold. For populations with additional demographic factors, we investigate temporal evolution of infected individuals and infectious contacts, as well as degree distributions of nodes in each class. It is found that the disease spreads faster but is more restricted in scale-free networks than in the Erdös-Rényi ones. The integration between vaccination intervention and individual rewiring may promote epidemic spreading due to the birth effect. Moreover, the degree distributions of both networks in the steady state is closely related to the degree distribution of newborns, which leads to uncorrelated connectivity. All the results demonstrate the importance of both local protection and global intervention, as well as the demographic effects. Our work thus offers a more comprehensive description of disease containment.

An Analytical Framework for Studying Small-Number Effects in Catalytic Reaction Networks: A Probability Generating Function Approach to Chemical Master Equations.

Cell activities primarily depend on chemical reactions, especially those mediated by enzymes, and this has led to these activities being modeled as catalytic reaction networks. Although deterministic ordinary differential equations of concentrations (rate equations) have been widely used for modeling purposes in the field of systems biology, it has been pointed out that these catalytic reaction networks may behave in a way that is qualitatively different from such deterministic representation when the number of molecules for certain chemical species in the system is small. Apart from this, representing these phenomena by simple binary (on/off) systems that omit the quantities would also not be feasible. As recent experiments have revealed the existence of rare chemical species in cells, the importance of being able to model potential small-number phenomena is being recognized. However, most preceding studies were based on numerical simulations, and theoretical frameworks to analyze these phenomena have not been sufficiently developed. Motivated by the small-number issue, this work aimed to develop an analytical framework for the chemical master equation describing the distributional behavior of catalytic reaction networks. For simplicity, we considered networks consisting of two-body catalytic reactions. We used the probability generating function method to obtain the steady-state solutions of the chemical master equation without specifying the parameters. We obtained the time evolution equations of the first- and second-order moments of concentrations, and the steady-state analytical solution of the chemical master equation under certain conditions. These results led to the rank conservation law, the connecting state to the winner-takes-all state, and analysis of 2-molecules M-species systems. A possible interpretation of the theoretical conclusion for actual biochemical pathways is also discussed.

A Novel Approach for Analyzing Single Buffer Queueing Systems with State-Dependent Vacation and Correlated Input Process under Four Different Service Disciplines

The author presents performance analysis of a single buffer multiple-queue system. Four different types of service disciplines (i.e., non-preemptive, pre-emptive repeat different, state dependent random polling and globally gated) are analyzed. His model includes correlated input process and three different types of non-productive time (i.e., switchover, vacation and idle time). Special cases of the model includes server with mixed multiple and single vacations, stopping server with delayed vacation and stopping server with alternating vacation and idle time. For each of the four service disciplines the key performance measures such as average customer waiting time, loss probability, and throughput are computed. The results permit a detailed discussion of how these performance measures depends on the customer arrival rate, the customer service time, the switchover time, the vacation time, and the idle time. Moreover, extensive numerical results are presented and the four service disciplines are compared with respect to the performance measure. Previous studies of the single buffer multiple-queue systems tend to provide separate analysis for the two cases of zero and nonzero switchover time. The author is able to provide a unified analysis for the two cases. His results generalize and improve a number of known results on single buffer multiple-queue systems. Furthermore, this method does not require differentiation while it is needed if one uses the probability generating function approach. Lastly, the author's approach works for all single buffer multiple-queue systems in which the next queue to be served is determines solely on the basis of the occupancy states at the end of the cycle time.

On generalized start-up demonstration tests

Start-up demonstration tests and various extensions have been discussed, in which a unit under the test is accepted or rejected according to some criteria. CSTF, CSCF, TSCF and TSTF are most well known start-up demonstration tests. In this paper, two kinds of more general start-up demonstration tests are introduced. CSTF, TSTF, TSCF and CSCF are all special situations of the new tests. For the new generalized start-up demonstration tests, under the assumption of independent and identically distributed trials for each test, the analytic expressions for the expectation, the probability mass function and the distribution of the test length, as well as the probability of acceptance or rejection of the unit are given. All the analyses are based on the finite Markov chain imbedding approach which avoids the complexities of the probability generating function approach and makes the results readily understood and easily extended to the non-i.i.d. cases. Furthermore, an optimal model for generalized start-up demonstration tests is proposed. Finally, a numerical example is presented to make our results more transparent, and it can demonstrate the advantages of the new tests.

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.

An analytic study of a scalable video buffer

In this contribution, we investigate the performance of the output buffer of an `on-demand' video streaming server. The server maintains a local database of stored video clips and movies which can be streamed to the users upon request. We assume that the stored video is encoded in a scalable way, which means that the data streams contain a base layer ensuring a minimum of guaranteed quality and a stack of additional enhancement layers progressively improving the quality of the video. For the purpose of performance analysis, we assume that a video stream is split up in logical units called frames. Every frame consists of a number of packets, each containing information of one layer only. When the output buffer gets congested, one may choose to drop the transmission of some of the layers in a frame, thus reducing the frame transmission time and expediting the restoration of the buffer size to normal levels. A discrete-time finite capacity queueing model with buffer size dependent transmission times is proposed. Using a probability generating function approach, we focus on the characteristics of idle and busy periods. We obtain performance measures such as the frame loss ratio and the average frame transmission time. The latter measure relates to the quality of the video stream. We conclude with some numerical examples, including a realistic case study.

The analysis and comparison of start-up demonstration tests

Two new start-up demonstration tests are introduced as alternatives to a previously studied one. The first test is based on total successes and total failures (TSTF) while the second is based on consecutive successes and consecutive failures (CSCF). The probabilistic analysis of these tests in the independent and identically distributed case uses a Markov chain approach to avoid the complexities of the probability generating function approach. We discuss advantages of these new tests in certain situations. Practical guidance on choosing tests, estimation, and comparisons of the various criteria are studied. The Markov chain approach can be easily extended and generalized to study the probabilistic analysis of the non-i.i.d. case.

The Discrete-Time Gated Vacation Queue Revisited

Abstract We consider the discrete-time gated multiple-vacation queue. Vacations are modeled as independent random variables with distributions depending on the number of the immediately preceding vacations. Using a probability generating function approach, we focus on various performance measures such as moments of queue contents and customer delay in equilibrium. These measures are functions of a constant value which we obtain numerically.

On truncated bivariate discrete distributions: A unified treatment

A unified treatment of three types of zero class truncation for bivariate discrete distributions is presented. Using the probability generating function approach, various properties of the truncated distributions are examined in association with the corresponding properties of the initial complete form of the distribution. Expressions for moments and conditional distributions are also obtained. Bivariate versions of the Thomas and the Intervened Poisson distributions are introduced and used as illustrative examples.

DISCRETE TIME MULTI-CLASS FEEDBACK QUEUE WITH VACATIONS AND CLOSE TIME UNDER RANDOM ORDER OF SERVICE DISCIPLINE

We study a discrete-time single server priority queueing model with vacations under random order of service discipline within each class. This model captures the behavior of the head-of-line request queues in large input-buffered ATM switches. The server takes vacations when the queue has been empty for a random number of slots. We presume a message consists of a geometrically distributed number of cells. To represent this aspect, we assume that once a message gets in turn for service, it is served for a constant time which corresponds to one-cell-time and it rejoins the queue after service with a given probability. We derive the joint probability distribution of the queue length and waiting time through probability generating function approach. Mean waiting times are obtained and their numerical results are shown.

Performance analysis of an input-buffered ATM switching fabric with independent window-access scheme

In this paper, an Independent Window-Access(IWA) scheme is proposed, and the performance of an input-buffered ATM switching fabric with the IWA scheme is analysed by means of a probability generating function approach, the closed formulas of the average cell delay and the maximum throughput are given, and results show that the IWA scheme makes the switching fabric have better performances than traditional window-access scheme. The computer simulation results are in good agreement with these analytical results.

A generalized stochastic model for systems of first-order reactions—probability generating function approach

A generalized stochastic model has been developed for systems of n-component first-order reactions by means of the probability generating function approach. A comparison of the stochastic model with the deterministic model is given.