Probability Generating Function Research Articles

Cost Optimization in Sintering Process on the Basis of Bulk Queueing System with Diverse Services Modes and Vacation

This research investigated a single bulk server queuing model where service modes and server vacations are dependent on the number of clients. The server operates in three different service modes: single service, fixed batch service, and variable batch service. Modes will be determined by queue length. The service starts only when the minimum number of customers, say ‘a’, has accumulated in the queue. At this point, the server selects one of three service modes. Transitions between duty modes are permitted only at the beginning of a duty period. At the end of the service, the server can go on vacation if the queue length drops below ‘a’. When returning from vacation, if threshold ‘a’ is not reached, the server will remain inactive until it is reached. A special technique called the Supplementary Variables Technique (SVT) was used to determine the probability-generating function when estimating the queue size at a given time. Appropriate numerical examples exemplify the method developed in the paper. An optimal cost analysis was performed to set the threshold values for different server modes with the intention of minimizing the aggregate average cost.

Transient and Steady-State Analysis of an M/PH2/1 Queue with Catastrophes

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

A generalized ASIP with arrivals to all sites and particle movements in all directions

AbstractA generalized n-site Asymmetric Simple Inclusion Process (ASIP) network is studied, where gate-opening instants are determined by a renewal process and arrivals occur to all sites. Various types of batch particle movements between sites are analyzed: (i) unidirectional probabilistic forward movements; (ii) probabilistic forward movements combined with feedback to the first site; and (iii) general probabilistic multidirectional movements. In contrast to the tedious successive substitution method used in previous ASIP studies, an efficient matrix approach is applied to derive the multidimensional probability generating function (PGF) of site occupancies right after gate opening instants. The complexity of the ASIP processes allows us to obtain explicit PGF results for small-size networks only, while for larger networks, a formula to calculate the mean site occupancies is derived for all types of movements. In movement case (i) the means are directly and explicitly calculated. For movement case (ii), where the network is homogeneous with equal probabilities of forward movements from site i to downstream sites $$j \ge i$$ j ≥ i , we show that the ratio between the mean occupancies of consecutive sites approaches a constant when the network becomes large, and calculate this ratio. Finally, we investigate an n-site network where at gate opening instants all gates open simultaneously, and particles move in all directions. Numerical examples are presented.

Unreliable M[X]/G(P1,P2)/1 feedback retrial queues with combined working vacation

The study examines M[X]/G(P1,P2)/1 feedback retrial queues coupled with starting failure, repair, delay to repair, working vacation, and general retrial times. Also, it explores how different batch sizes affect performance and how bulk arrival affects system behavior. When the server is not in use, a single customer initiates the system while the remaining customers transition to a state of orbit. A new customer must turn on the server to provide two phases of mandatory service at any time. The server could have starting issues. If the service is successfully started (with likelihood α), the customer receives service immediately. In the absence of that, the likelihood of starting failure happens (with likelihood 1−α=α¯). The server was taken for repair with some delay and that customer was transported to an orbital location. When the server was busy or unavailable, the arriving customers queued by FCFS in the orbit. We also discuss the idea of reworking with probability p, and restarting unsuccessful service attempts to improve customer happiness and service efficiency. We also introduce the concept of working vacation, which permits servers to temporarily stop providing services, affecting system performance and availability at both peak and off-peak times. A supplementary variable technique was adopted for the system's and orbit size's probability-generating function. Various performance measurements were provided with appropriate numerical examples.

A new family of copulas based on probability generating functions

Abstract We propose a method to obtain a new class of copulas using a probability generating function (PGF) of positive-integer valued random variable. Some existing copulas in the literature are sub-families of the proposed copulas. Various dependence measures and invariant property of the tail dependence coefficient under PGF transformation are also discussed. We propose an algorithm for generating random numbers from the PGF copula. The bivariate concavity properties, such as Schur concavity and quasi-concavity, associated with the PGF copula are studied. Two new generalized FGM copulas are introduced using PGFs of geometric and discrete Mittag-Leffler distributions. The proposed two copulas improved the Spearman’s rho of FGM copula by (−0.3333, 0.4751) and (−0.3333, 0.9573). Finally, we analyse a real dataset to illustrate the practical application of the proposed copulas.

COMPREHENSIVE DISCUSSION OF THE REPAIRABLE SINGLE SERVER CATASTROPHE AND MULTIPLE VACATION QUEUEING MODEL

The queueing system with two types of vacation policies and disasters occurring in every state are examined in this study. As soon as the system is empty, the server goes on a short-term vacation. If there are no active clients in the system when the server returns from a short-term vacation, then the server goes on a long-term vacation. Upon seeing at least one client in the queue during a short or long-term vacation, the server will turn on right away. Catastrophes can happen during short or long vacations and also when the server is active. All current clients are removed from the system when a disaster occurs. When the system has been fixed, it starts to work properly once again. After repair, the server immediately switches to working mode(model 1) or takes a short-term vacation (model 2). A steady-state solution is determined by using the PGF (Probability-Generating Function). The above two categories of after-repair will be examined in this paper. Additionally, numerical examples related to performance measures, cost models, and the steady analysis of the provided models are discussed. Management for networking disaster recovery gives suggestions for restarting regular operations and network services following a disaster.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a K-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed K-dimensional cone, resulting in a K-dimensional SSC. The dimension~K represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with K=1 corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

Non-Markovian Feedback Retrial Queue with Two Types of Customers and Delayed Repair Under Bernoulli Working Vacation

This paper deals with an M/G/1 feedback retrial G-queue and delayed repair incorporating Bernoulli working vacation. Arrivals of both favorable and unfavorable consumers are two independent Poisson processes. The server is subjected to breakdown during the regular busy period due to the arrival of unfavorable customers and then the server will be down for a short period of time. Further, the concept of delay time is also discussed. After the fulfillment of regular service, the dissatisfied consumer may re-join the orbit to receive another service as a feedback consumer. During the working vacation period, the server provides the service to consumers at a reduced rate. By applying the supplementary variable technique (SVT), we have analyzed the steady-state probability generating function (PGF) for the system size and orbit size. In addition, we have presented the system performance, reliability indicators, and stochastic decomposition property. Finally, we provided numerical examples to illustrate our model’s mathematical results.

Probing scrambling and operator size distributions using random mixed states and local measurements

The dynamical spreading of quantum information through a many-body system, typically called scrambling, is a complex process that has proven to be essential to describe many properties of out-of-equilibrium quantum systems. Scrambling can, in principle, be fully characterized via the use of out-of-time-ordered correlation functions, which are notoriously hard to access experimentally. In this work, we put forward an alternative toolbox of measurement protocols to experimentally probe scrambling by accessing properties of the operator size probability distribution, which tracks the size of the support of observables in a many-body system over time. Our measurement protocols require the preparation of separable mixed states together with local operations and measurements, and combine the tools of randomized operations, a modern development of near-term quantum algorithms, with the use of mixed states, a standard tool in NMR experiments. We demonstrate how to efficiently probe the probability-generating function of the operator distribution and discuss the challenges associated with obtaining the moments of the operator distribution. We further show that manipulating the initial state of the protocol allows us to directly obtain the individual elements of the distribution for small system sizes. Published by the American Physical Society 2024

Performance Analysis of Two Different Types of Waiting Queues with Working Vacations

This work examines a new class of working vacation queueing models that contain regular (original) and retrial waiting queues. Upon arrival, a customer either starts their service instantly if the server is available, or they join the regular queue if the server is occupied. When it is empty, the server departs the system to take a working vacation (WV). The server provides services more slowly during the WV period. New customers join the retry queue (orbit), if the server is on vacation. The supplementary variable technique (SVT) examines the steady-state probability generating functions (PGFs) of queue size for different server states. Several system performances are numerically displayed, including system state probabilities, mean busy cycles, mean queue lengths, sensitivity analysis, and cost optimization values. The motivation for this model in a pandemic situation is to analyze new healthcare service systems and reflect the characteristics of patient services.

Integer-Valued Split-BREAK Process with a General Family of Innovations and Application to Accident Count Data Modeling

This paper presents a novel count time-series model, named integer-valued Split-BREAK process of the first order, abbr. INSB(1) model. This process is examined in terms of its basic stochastic properties, such as stationarity, mean, variance and correlation structure. In addition, the marginal distribution, over-dispersion and zero-inflation properties of the INSB(1) process are also examined. To estimate the unknown parameters of the INSB(1) process, an estimation procedure based on probability generating functions (PGFs) is proposed. For the obtained estimators, their asymptotic properties, as well as the appropriate simulation study, are examined. Finally, the INSB(1) process is applied in the dynamic analysis of some real-world series, namely, the numbers of serious traffic accidents in Serbia and forest fires in Greece.

On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

AbstractWe consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$ . The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m<1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

A Discrete-Time Queueing Model of a Bottleneck with an Energy-Saving Mechanism Based on Setup and Shutdown Times

Producers are encouraged to reduce their energy consumption of manufacturing systems by applying less-energy-intensive modern technologies and advanced machine tools and operating methods at the system level. In the paper, organizational and analytical solutions are combined to model the sustainable production system. Managers can study the behavior of a production system organized using energy-saving rules by changing key parameters of the input model (arrival intensity, bottleneck service rate, buffer size, setup and shutdown time) to analyze the queue size of the production system and therefore performance. A discrete-time queueing model of a single-bottleneck production line with a finite input buffer capacity is proposed. Jobs occur according to a binomial process and are processed individually, one by one, according to the natural FIFO service discipline, with a general discrete-type cumulative distribution function. The total number of jobs present in the system is bounded by a non-random fixed value N. Every time the system becomes empty, an energy-saving mechanism is started: the processing machine (server) is turned off during a geometrically distributed shutdown time. Similarly, the first job arriving into the empty system initializes a geometrically distributed setup time. Identifying renewal moments in the evolution of the model, a system of difference equations is built for the transient queue-size distribution conditioned by the state of the system at the opening. The solution is obtained explicitly in terms of probability-generating functions. In addition, the Drum-Buffer-Rope concept is proposed to reduce the energy consumption of the production line. The throughput of the production system is maximized by adjusting the time between the order arrivals and the size of the input buffer to the capacity of the bottleneck. Turning off a machine under certain conditions and slowing down non-critical machines are strategies to reduce energy consumption. A detailed illustrating numerical and simulation study of the considered model is attached as well, in which the sensitivity of the queue-size behavior to changes of the key input model parameters is investigated.

A Mathematical Approach to Ullmann Reaction: ANFIS Computing of a Chemical Retrial Queue Model under Hybrid Vacation

Herein, we developed the first report on the chemical retrial queue and hybrid vacation process for the significant Ullmann coupling for enhanced high-throughput reaction discovery. The problem of controlling the waiting time for chemical retrials is addressed here. We incorporated the supplementary variables method (SVT) and generated the steady-state probability generating function (PGF) for system size and orbit size. Important cases are outlined, and a few key metrics are utilized to evaluate system performance. Additionally, the impact of changing certain system parameters has also been analyzed via numerical examples. One particularly interesting aspect of this research is the comparison of neuro-fuzzy technique results with validation results utilizing the "adaptive neuro-fuzzy inference system" (ANFIS).

A cost analysis of single-server discouraged arrivals with differentiated vacation queueing model

This paper investigates an M/M/1 queueing system with differentiated vacations and discouraged arrivals, focusing on two types of vacations. The server switches to type I vacation with rate γ1 when the system is empty during an active state. If no customers are waiting when it returns from a type I vacation, it then switches to a type II vacation with a rate of γ2. Both vacation times and service duration follow exponential distributions. The study utilises the Probability Generating Function (PGF) technique to derive steady-state solutions for both vacation policies. Furthermore, the research explores relevant performance metrics and provides numerical examples to illustrate the system's behaviour under various conditions. The cost analysis of the M/M/1 differentiated vacation system with discouraged arrival queueing and various aspects of the system's behaviour under different arrival rates (λ, λ1, λ2) are discussed.

Retrial G-queue with precipitous breakdown and non-tenacious customers

We perform an analysis on an M/G/1 system that has bulk arrival Poisson process as well as instantaneous service, Precipitous breakdown and random repair, Gqueue, and operation under an MBV (Modified Bernoulli vacation) policy. The positive outcome of a customer retrial is lost when an Anti-positive customer arrives during a positive (good) customer?s assistance period. If a new customer enters and notices that the server is currently undergoing repairs or on vacation, they may renege or balk from the system. To obtain a wide variety of additional outcomes, we use the details provided by the supplementary variable method regarding the rates at which different events occur and it is possible to determine the probability-generating functions of queue length distributions as well as the explicit formulations of important performance metrics. The numerical data confirm the analytical findings about the major performance metrics.

New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resource allocation, evaluation, and performance measurement. This paper focuses on a one-parameter discrete distribution obtained by compounding the Poisson and new X-Lindley distributions. The probability-generating function, moments, skewness, kurtosis, and other properties are derived in the closed form. The maximum likelihood method, method of moments, least squares method, and weighted least squares method are used for parameter estimation. A simulation study is carried out. The proposed distribution is applied as the innovation in an INAR(1) process. The importance of the proposed model is confirmed through the analysis of two real datasets.

Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation

In this paper, a bulk arrival and two-phase bulk service with active Bernoulli feedback, vacation, and breakdown is considered. The server provides service in two phases as mandatory according to the general bulk service rule, with minimum bulk size ′a′ and maximum bulk size ′b′. In the first essential service (FES) completion epoch, if the server fails, with probability ′δ′, then the renewal of the service station is considered. On the other hand, if there is no server failure, with a probability ′1−δ′, then the server switches to a second essential service (SES) in succession. A customer who requires further service as feedback is given priority, and they join the head of the queue with probability β. On the contrary, a customer who does not require feedback leaves the system with a probability ′1−β′. If the queue length is less than ′a′ after SES, the server may leave for a single vacation with probability ′1−β′. When the server finds an inadequate number of customers in the queue after vacation completion, the server becomes dormant. After vacation completion, the server requires some time to start service, which is attained by including setup time. The setup time is initiated only when the queue length is at least ′a′. Even after setup time completion, the service process begins only with a queue length ‘N’ (N > b). The novelty of this paper is that it introduces an essential two-phase bulk service, immediate Bernoulli feedback for customers, and renewal service time of the first essential service for the bulk arrival and bulk service queueing model. We aim to develop a model that investigates the probability-generating function of the queue size at any time. Additionally, we analyzed various performance characteristics using numerical examples to demonstrate the model’s effectiveness. An optimum cost analysis was also carried out to minimize the total average cost with appropriate practical applications in existing data transmission and data processing in LTE-A networks using the DRX mechanism.

Nonlinear metaheuristic cost optimization and ANFIS computing of feedback retrial queue with two dependent phases of service under Bernoulli working vacation

Today, with real-life problems, modeling is a primary step in organizing, analyzing and optimizing them. Queueing theory is a particular approach used to model this category of issues. Space constraints, feedback, service dependency, etc. are often inseparable from the issues they create. In light of this objective, this research presents a model and analysis of the steady-state behavior of an [Formula: see text] feedback retrial queue with two dependent phases of service under a Bernoulli vacation policy. The service times for the two stages are often independent in normal queueing frameworks. We presume that they are dependent random variables in this case. Indeed, this dependence is one-way (i.e., the service time of the second phase has no effect on the service time of the first phase). Yet, the first phase service time has an impact on the second phase service time. In order to determine the steady-state probabilities and probability-generating functions (PGF) for the different states, the supplementary variable technique (SVT) was utilized. Furthermore, a broad range of performance metrics had been established. The generated metrics are then envisioned and validated with the aid of graphs and tables. Additionally, a nonlinear cost function is constructed, which is subsequently minimized by distinct approaches like particle swarm optimization (PSO), artificial bee colony (ABC) and genetic algorithm (GA). Furthermore, we used certain figures to examine the convergence of these optimization methods. Finally, validation outcomes are compared with neuro-fuzzy results generated with the “adaptive neuro-fuzzy inference system” (ANFIS).

A single server Non-Markovian with non-compulsory re-service and balking under Modified Bernoulli Vacation

In this paper, a single-server queue with Modified Bernoulli Vacation(MBV) has been considered. The server provides two kinds of services, Compulsory Service (CR) and Non-Compulsory Re-service (NCR). Any new customer who arrives and finds the server engaged, on breakdown, or on vacation will be placed on an orbit; or else if he finds the server free, he enters the service immediately. The server takes a vacation as soon as the orbit is empty at the usual service completion moment. The dissatisfied customer may re-enter the orbit after usual service is complete to receive another service. We have used SVM (Supplementary Variable Method) to derive PGF (Probability Generating Function) for the system. In order to demonstrate the influence that the system parameters have, numerical examples are discussed.