Performance and cost analysis of a Geom/G/1 (G, SV) system

In this paper, a gate service Geom/G/1 queue system with single vacation is analyzed. Firstly, a gate service policy and single vacation are introduced in detail, the expected number of customers in the system at the beginning instant of a service period is obtained. Next, the Probability Generating Function (P.G.F.) of the stationary queue length is obtained by using regeneration cycle approach. Then the P.G.F. of the waiting time is derived under the condition of the independence between the arrival process and the waiting time. Moreover, the P.G.Fs. of the service period, the vacation period and the idle period are obtained, and the mean length of three periods are also obtained. Cost model is formulated to determine the optimal expected vacation length. Finally, based on numerical results, the relations of the performance measures and traffic intensity are discussed.     

一般限量服务M/G/1型单重休假排队系统的随机分解

本文研究了M／G／1型一般限量服务单重休假排队系统,根据稳态下M／G／1型非空竭服务休假排队系统的队长的随机分解的一般结构,提出简便算法一结构分析法,并且在先到先服务规则下(FCFS)求得稳态下队长的随机分解的概率母函数(GFP)和服务时间的随机分解的Laplace变换(LST),还充分地讨论了系统平稳的条件及其概率含义。     

When should we superpose M/G/1 queues?

Conventional wisdom suggests that a fast single-server queue is preferable to a number of slower single-server queues. The rationale is that if the service rate of the superposed queue is the sum of the service rates of the individual queues, then the average waiting time for the superposed queue is smaller than the weighted average waiting time of the individual queues. This argument, however, assumes that the service time of the customers is perfectly scalable. Unfortunately, this assumption rarely holds in reality. In this paper we analyze the superposition of M/G/1 queues. We assume that there are scalable and unscalable components in the service time. On the basis of the criteria of average waiting time and average response time, we derive necessary and sufficient conditions for whether the superposition of queues is preferable. Examples of magnetic rotational disks are given to illustrate our findings.     

On extreme values of orbit lengths in M/G/1 queues with constant retrial rate

In the design of waiting facilities for the units in a retrial queue, it is of interest to know probability distributions of extreme values of the orbit length. The purpose of this paper is to investigate the asymptotic behavior of the maximum orbit length in the queue with constant retrial rate, as the time interval increases. From the classical extreme value theory, we observe that, under standard linear normalizations, the maximum orbit length up to the nth time the positive recurrent queue becomes empty does not have a limit distribution. However, by allowing the parameters to vary with n, we prove the convergence of maximum orbit lengths to three possible limit distributions when the traffic intensity approaches 1 from below and n approaches infinity.

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Received: October 7, 1999 / Accepted: November 21, 2000     

Optimally maintaining an M/G/1-type production system

The concept of more maintenance effort (or training) implying higher system reliability (or efficiency) is well accepted. In this paper we study an optimal maintenance (training) program based on this concept for a simple production system using the M/G/l vacation queue model. We discuss how to determine the optimal maintenance (training) time to minimize the average time spent by a job in the system.     

On the analysis of single server finite queue with state dependent arrival and service processes: M(n)/G(n)/1/K

This paper discusses the analysis of a single server finite queue with Poisson arrival and arbitrary service time distribution, wherein the arrival rates are state dependent which are all distinct or all equal, service times are conditioned on the system length at the moment of service initiation. The analytic analysis of the queue is carried out and the final results have been presented in the form of recursive equations which can be easily programmed on any PC to obtain the distributions of number of customers in the system at arbitrary, departure and pre-arrival epochs. It is shown that the method works for all service time distributions including the non-phase type and also for low and high values of the model parameters. Some performance measures, and relations among the state probabilities at arbitrary, departure and pre-arrival epochs are also discussed. Furthermore, it is shown that results for a number of queueing models can be obtained from this model as special cases. To demonstrate the effectiveness of our method some numerical examples have been presented.     

启动-关闭型多级适应性休假Mx/G/1可修排队的可靠性分析

考虑带启动时间和关闭时间(延迟休假)的多级适应性休假Mx／G(M／G)／1可修排队系统,在假定启动时间、关闭时间、服务台的修理时间和休假时间都服从一般分布的情况下,通过引入服务台的“广义忙期”,采用一种较简单分析方法讨论了有关服务台的许多感兴趣的可靠性指标,得到了一些重要的可靠性结果。     

启动一关闭型多级适应性休假M^x／G／1可修排队的可靠性分析

考虑带启动时间和关闭时间（延迟休假）的多级适应性休假M^x／G（M/G）／1可修排队系统,在假定启动时间、关闭时间、服务台的修理时间和休假时间都服从一般分布的情况下,通过引入服务台的“广义忙期”,采用一种较简单分析方法讨论了有关服务台的许多感兴趣的可靠性指标,得到了一些重要的可靠性结果。．     

Bayesian EWMA control charts based on Exponential and transformed Exponential distributions

In this paper, we proposed the Bayesian exponentially weighted moving average (EWMA) control charts for mean under the nonnormal life time distributions. We used the time between events data which follow the Exponential distribution and proposed the Bayesian EWMA control charts for Exponential distribution and transformed Exponential distributions into Inverse Rayleigh and Weibull distributions. In order to develop the control charts, we used a uniform prior under five different symmetric and asymmetric loss functions (LFs), namely, squared error loss function (SELF), precautionary loss function (PLF), general entropy loss function (GELF), entropy loss function (ELF), and weighted balance loss function (WBLF). The average run length (ARL) and the standard deviation of run length (SDRL) are used to check the performance of the proposed Bayesian EWMA control charts for Exponential and transformed Exponential distributions. An extensive simulation study is conducted to evaluate the proposed Bayesian EWMA control chart for nonnormal distributions. It is observed from the results that the proposed control chart with the Weibull distribution produces the best results among the considered distributions under different LFs. A real data example is presented for implementation purposes.     

An analytical method for the calculation of the number of units at the arrival instant in a discrete time G/G/1-queueing system with batch arrivals

In this paper, we analyze a G/G/1-queueing system with batch arrivals in the discrete time domain. On the basis of the waiting time distribution, we present an analytical approach for the calculation of the number of units in the queue at the arrival instant. This approach is exact within an ε-environment. In addition, the computing times are low such that a huge set of computations can be performed in a short time. We close our paper with a set of numerical experiments. Our research is motivated through various applications in the field of logistics, where it is of important interest to know the distribution of the number of units waiting to be proceeded at the moment of a new arrival.     

A closed form solution for the G/M/r machine interference model

This paper solves the machine interference problem in which N different machines are looked after by a team of r operatives. The run time of each machine is assumed to have a general distribution, different for each machine and the repair times are assumed to have a negative exponential distribution with different means for the different machines. An explicit expression for the probability that a particular group of machines is found running in the steady state is derived. From this other useful measures for the system can be obtained. It is shown that these depend on the run time distributions only through the means of those distributions.     

Time‐between‐events control charts for an exponentiated class of distributions of the renewal process

Time‐between‐events control charts are commonly used to monitor high‐quality processes and have several advantages over the ordinary control charts. In this article, we present some new control charts based on the renewal process, where a class of absolutely continuous exponentiated distributions is assumed for the time between events. This class includes the generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions. Although we discuss the design structure for all the mentioned distributions, our main focus will be on the generalized exponential distribution due to its practical relevance and popularity. Since the generalized exponential distribution is a generalization of the traditional exponential distribution, the new control chart is more flexible than the existing exponential time‐between‐events charts. The control chart performance is evaluated in terms of some useful measures, including the average run length (ARL), the expected quadratic loss, continuous ranked probability, and the relative ARL. The effect of parameter estimation using the maximum likelihood and Bayesian methods on the ARL is also discussed in this article. The study also presents an illustrative example and 4 case studies to highlight the practical relevance of the proposal.     

多服务速率多重休假M/G(M/G)/1可修排队

本文通过对系统中顾客数设置门限N,考虑研究了服务台的服务速度依门限发生变化且当服务台未服务顾客(休假或故障)时到达顾客仅以概牢p进入系统的多重休假可修M／G(M／G)／1排队系统。通过L-变换、母函数以及补充变量方法得到了队长分布的瞬态解、稳态解及一些可靠性结果,并指出当两个服务速度相等时该模型与前人研究的M／G／1(E,MV）排队一致。     

Mean-variance interactions in process improvement and capacity design

We investigate mean-variance interactions of processing time as applied to process improvement and capacity design. For general capacity cost and flowcost functions, we demonstrate that production processes fall into one of six regions on the mean-variance interaction plane, each with its own policy implications. The general model is specialized to the case of an M/G/1 queue with linear and separable mean and variance costs, and with flowcosts proportional to mean queue length. Optimal solutions for processing-time mean and variance are derived, and easily obtained operating parameters are used to identify appropriate process improvement policies. A simulation example of a production network taken from industry verifies the efficacy of the linear M/G/1 model in a more general setting. We conclude that intelligent management of both processing capacity (i.e. mean processing time) and processing-time variances can be powerful tools for both capacity design and process improvement.     

An analytical method for the calculation of the waiting time distribution of a discrete time G/G/1-queueing system with batch arrivals

This paper presents an analytical method to calculate the waiting time distribution for the G/G/1-queueing system with batch arrivals. Using the discrete time scale, it is possible to calculate the distribution of the waiting times and the idle times of a G/G/1-queueing system based on the Wiener–Hopf factorization. The influence of batch arrivals on the waiting time distribution is analyzed. The waiting time distribution is calculated for batch arrivals with both constant and stochastic batch sizes. The effect of stochastic batch sizes on the waiting process is highlighted. With the developed methods, it is possible to obtain congestion measures of high precision for logistic systems. The analytical results are evaluated by simulation. Several numerical examples are presented to emphasize the quality of the introduced methods.     

CUSUM Statistical Monitoring of M/M/1 Queues and Extensions

Many production and service systems can be modeled as queueing systems. Their operational efficiency and performance are often measured using queueing performance metrics (QPMs), such as average cycle time, average waiting length, and throughput rate. These metrics need to be quantitatively evaluated and monitored in real time to continuously improve the system performance. However, QPMs are often highly stochastic, and hence are difficult to monitor using existing methods. In this article, we propose the cumulative sum (CUSUM) schemes to efficiently monitor the performance of typical queueing systems based on different sampling schemes. We use M/M/1 queues to illustrate how to design the CUSUM chart and compare their performance with several alternative methods. We demonstrate that the performance of CUSUM is superior, responding faster to many shift patterns through extensive numerical studies. We also briefly discuss the extensions of CUSUM charts to more general queues, such as M/G/1, G/G/1, or M/M/c queues. We use case studies to demonstrate the applications of our approach. Supplementary materials for this article are available online.     

Optimal batching in a wafer fabrication facility using a multiproduct G/G/c model with batch processing

Wafer fabrication, the first portion of semiconductor manufacturing, typically involves numerous batch-processing operations. These operations play an important role in determining how the system performs in terms of throughput, WIP and cycle time. In this paper, batch sizes that minimize the expected cycle time of batch-processing operations for a real-world semiconductor manufacturer are determined by a new approximated analytical model. This model, denoted by G / G ( bp ) , represents multiple products, multiple servers, batch-processing, incompatible products and unequal batch service size queues. Incompatible products mean that different products are not allowed to be in the same batch. Unequal batch service size means that batch service sizes depend upon products. Steady-state approximation formulas for cycle time and WIP of this queuing system are derived. These approximate performance measures are compared with those of discrete-event simulation. The results are reasonable and the approximation formulas much more computationally efficient than conducting the corresponding simulation studies. Finally, a batch-processing system with the goal of minimizing the total expected cycle times of items by determining the 'optimal' batch sizes is presented. Solutions are obtained using genetic algorithms.     

Performance evaluation of a transportation-type bulk queue with generally distributed inter-arrival times

This paper is motivated by the performance evaluation of circulating vertical conveyor systems (CVCSs). CVCSs are bulk queues of transportation type. These material handling systems feature generally distributed inter-arrival times, which can be longer than the bulk service time. This leads to interdependencies between the number of arrivals in consecutive service intervals and the number of loads in the queue. We propose a new discrete-time approach for the steady-state analysis of such bulk service queues of transportation type with general arrival and service processes and finite server and limited queue capacities. The approach is based on a finite Markov chain that generates complete probability distributions for the key performance measures, including the queue length, waiting time and departing batch size. The proposed approach is exact in the cases of discrete-time slots, e.g. as in communication systems. We investigate the discretisation error that arises if the approach is used as an approximation for the continuous time using a numerical comparison to a discrete-event simulation. Moreover, we examine the impact of arrival stream variability on the system performance and compare the positive effects of a higher frequency of server visits with the effects arising from larger pickup capacities.     

Integration of process monitoring,quality control and maintenance in an M/G/1 queue-like production system

For an unreliable production system, process monitoring, quality control, and maintenance are essential activities for assuring the proper operation of the production system and for managing the quality of the system's outputs. Although these activities are interrelated, they are seldom studied simultaneously. This paper interrelates and synthesizes the economic design of these activities and the management of an unreliable production system described in terms of an M/G/1 queue. A mathematical model is developed to determine the optimal policies of these activities when both the quality and quantity issues in the system are considered. Then, managerial insights are provided regarding the role and interrelationship of these activities. Finally, hypothetical data are used to illustrate the impact of these activities on the operating characteristics of the production system     

Estimating the throughput of a cyclic assembly system

We consider a production system consisting of several fabrication lines feeding an assembly station. The machines in the fabrication lines and at assembly are assumed to have general processing time distributions. Releases to the system are governed by the CONWIP protocol. We model this system as an assembly-like queue and develop approximations for the throughput of the system. Comparisons with simulations show that this approximation is robust over a wide range of conditions. Finally, we observe that throughput tends to be higher when machines with higher mean processing times and/or higher variances are in fabrication rather than assembly.