The analysis of MX/M/1 queue with two-stage vacations policy

Abstract

In this article, we consider a batch arrival M^X/M/1 queue with two-stage vacations policy that comprises of single working vacation and multiple vacations, denoted by M^X/M/1/SWV?+?MV. Using the matrix analytic method, we derive the probability generating function (PGF) of the stationary system size and investigate the stochastic decomposition structure of stationary system size. Further, we obtain the Laplace–Stieltjes transform (LST) of stationary sojourn time of a customer by the first passage time analysis. At last, we illustrate the effects of various parameters on the performance measures numerically and graphically by some numerical examples.     

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT

We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.     

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).     

Non-ergodic Markov decision processes with a constraint on the asymptotic failure rate: general class of policies

In this paper, we introduce a Markov decision model with absorbing states and a constraint on the asymptotic failure rate. The objective is to find a policy which maximizes the infinite horizon expected average reward, given that the system never fails. First, we show that it is sufficient to consider markovian policies. Second, for solving the problem, we restrict ourselves to find a stationary policy. Finally, we give sufficient conditions for optimality in the Markovian policies class.     

A Spectral Method for a Nonpreemptive Priority BMAP/G/1 QUEUE

Abstract

In this paper we consider a nonpreemptive priority queue with two priority classes of customers. Customers arrive according to a batch Markovian arrival process (BMAP). In order to calculate the boundary vectors we propose a spectral method based on zeros of the determinant of a matrix function and the corresponding eigenvectors. It is proved that there are M zeros in a set Ω, where M is the size of the state space of the underlying Markov process. The zeros are calculated by the Durand-Kerner method, and the stationary joint probability of the numbers of customers of classes 1 and 2 at departures is derived by the inversion of the two-dimensional Fourier transform. For a numerical example, the stationary probability is calculated.     

Admission policies for a two class loss system

We consider the problem of dynamic admission control in a Markovian loss queueing system with two classes of customers with different service rates and revenues. We show that under certain conditions, customers of one class, which we call a preferred class, are always admitted to the system. Moreover, the optimal policy is of threshold type, and we establish that the thresholds are monotone under very restrictive conditions.     

Analysis of an unreliable MX/G1G2/1/repeated service queue with delayed repair under randomized vacation policy

Abstract

In this article we consider an unreliable M^X/G/1 queue with two types of general heterogeneous service and optional repeated service subject to server’s break down and delayed repair under randomized vacation policy. We assume that customer arrive to the system according to a compound Poisson process. The server provides two types of general heterogeneous service and a customer can choose either type of service before its service start. After the completion of either type of service, the customer has the further option to repeat the same type of service once again. While the server is working with any types of service or repeated service, it may breakdown at any instant. Further the concept of randomized vacation is also introduced. For this model, we first derive the joint distribution of state of the server and queue size by considering both elapsed and remaining time, which is one of the objective of this article. Next, we derive Laplace Stieltjes transform of busy period distribution. Finally, we obtain some important performance measure and reliability indices of this model.     

A surplus process involving a compound Poisson counting process and applications

Abstract

In this paper, we introduce a surplus process involving a compound Poisson counting process, which is a generalization of the classical ruin model where the claim-counting process is a homogeneous Poisson process. The incentive is to model batch arrival of claims using a counting process that is based on a compound distribution. This reduces the difficulty of modeling claim amounts and is consistent with industrial data. Recursive formula, some properties and relevant main ruin theory results are provided. Further, we consider applications involving zero-truncated negative binomial and zero-truncated binomial batch arrivals when the claim amounts follow exponential or Erlang distribution.     

An optimal age-replacement policy for a simple repairable system with delayed repair

In this article, a simple repairable system (i.e., a repairable system consisting of one component and one repairman) with delayed repair is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, using the geometric process repair model, we consider a replacement policy T based on system age under which the system is replaced when the system age reaches T. Our problem is to determine an optimal replacement policy T*, such that the average cost rate (i.e., the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy T* can be determined by minimizing the average cost rate of the system. Finally, a numerical example is given to illustrate some theoretical results and the model's applicability.     

A geometric process repair model for a cold standby repairable system with imperfect delay repair and priority in use

In this article, a two-dissimilar-component cold standby repairable system with one repairman is studied. Assume that the repair after failure for each component is delayed or undelayed. Component 2 after repair is “as good as new” while Component 1 after repair is not, but Component 1 has priority in use. Under these assumptions, using a geometric process, we consider a replacement policy N based on the failure number of Component 1. An optimal replacement policy N* is determined by minimizing the average cost rate C(N) of the system. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Optimal acceptance sampling policy considering Bayesian risks

In this paper, we propose a sampling policy considering Bayesian risks. Various definitions of producer's risk and consumer's risk have been made. Bayesian risks for both producer and consumer are proven to give better information to decision-makers than classical definitions of the risks. So considering the Bayesian risk constraints, we seek to find optimal acceptance sampling policy by minimizing total cost, including the cost of rejecting the batch, the cost of inspection, and the cost of defective items detected during the operation. Proper distributions to construct the objective function of the model are specified. In order to demonstrate the application of the proposed model, we illustrate a numerical example. Furthermore, the results of the sensitivity analysis show that lot size, the cost of inspection, and the cost of one defective item are key factors in sampling policies. The acceptable quality level, the lot tolerance proportion defective, and Bayesian risks also affect the sampling policy, but variations of acceptable quality level and producer Bayesian risks, for values more than a specified value, cause no changes in sampling policy.     

A Tree-Structured Markovian Model of the Shipment Consolidation Process

This article studies the dispatch of consolidated shipments. Orders, following a batch Markovian arrival process, are received in discrete quantities by a depot at discrete time epochs. Instead of immediate dispatch, all outstanding orders are consolidated and shipped together at a later time. The decision of when to send out the consolidated shipment is made based on a “dispatch policy,” which is a function of the system state and/or the costs associated with that state. First, a tree structured Markov chain is constructed to record specific information about the consolidation process; the effectiveness of any dispatch policy can then be assessed by a set of long-run performance measures. Next, the effect on shipment consolidation of varying the order-arrival process is demonstrated through numerical examples and proved mathematically under some conditions. Finally, a heuristic algorithm is developed to determine a favorable parameter of a special set of dispatch policies, and the algorithm is proved to yield the overall optimal policy under certain conditions.     

A MARKOV MODULATED FLUID QUEUE WITH BATCH ARRIVALS AND PREEMPTIONS

We consider a preemptive priority fluid queue with two buffers for continuous fluid and batch fluid inputs. Those two types of fluids are governed by a Markov chain with a finite state space, called a background process, and the continuous fluid is preemptively processed over the batch fluid. The stationary joint distribution of the two buffer contents and the background state is obtained in terms of matrix transforms. Numerical computation algorithms are presented for various moments of the two buffer contents.     

Quality engineering in japan

For controlling quality of products, there are two things we have to consider; how to measure the quality and how to improve it. The former is discussed in section 2 by introducing the difference in quality levels between Japan and USA. After briefly introducing all countermeasures for improving quality of products, an on-line quality control system design is proposed as the one which is more economical than removing defec- tives after manufacturing by conducting inspection. It has been ma?y Japanese companies' policy to keep pro- cess conditions at best in order to produce uniform products as far as possible. This paper discusses the optimum feed-back process control system design by checking either quality characteristics or process pa- rameters after introducing overall quality countermea- sures starting with product design.     

Simulation-Based Inspection Policies for a One-Shot System in Storage Over a Finite Time Span

In this paper, we consider an inspection policy problem for a one-shot system with two types of units over a finite time span and want to determine inspection intervals optimally with given replacement points of Type 2 units. The interval availability and life cycle cost are used as optimization criteria and estimated by simulation. Two optimization models are proposed to find the optimal inspection intervals for the exponential and general distributions. A heuristic method and a genetic algorithm are proposed to find the near-optimal inspection intervals, to satisfy the target interval availability and minimize the life-cycle cost. We study numerical examples to compare the heuristic method with the genetic algorithm and investigate the effect of model parameters to the optimal solutions.     

Offenders as victims of crime?: an investigation into the relationship between criminal behaviour and victimization

Summary.  We consider the association between victimization and offending behaviour by using data from the Youth Lifestyles Survey. We consider the effect of violent and non-violent offending on the probability of being a victim of violent and non-violent crime and find a positive association between these by using univariate probit estimates. However, taking into account the endog- enous nature of offending and victimization via a bivariate probit model, we find that univariate estimates understate the association. We suggest that policy recommendations should only be informed by the bivariate analysis of the association between offending and victimization.     

Adaptive Sequential Preventive Maintenance Policy and Bayesian Consideration

This article proposes an adaptive sequential preventive maintenance (PM) policy for which an improvement factor is newly introduced to measure the PM effect at each PM. For this model, the PM actions are conducted at different time intervals so that an adaptive method needs to be utilized to determine the optimal PM times minimizing the expected cost rate per unit time. At each PM, the hazard rate is reduced by an amount affected by the improvement factor which depends on the number of PM's preceding the current one. We derive mathematical formulas to evaluate the expected cost rate per unit time by incorporating the PM cost, repair cost, and replacement cost. Assuming that the failure times follow a Weibull distribution, we propose an optimal sequential PM policy by minimizing the expected cost rate. Furthermore, we consider Bayesian aspects for the sequential PM policy to discuss its optimality. The effect of some parameters and the functional forms of improvement factor on the optimal PM policy is measured numerically by sensibility analysis and some numerical examples are presented for illustrative purposes.     

Nonparametric model checking for k-out-of-n systems

It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this paper, we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.     

Functionals of Markovian Branching D-BMAPS

Abstract

We consider a stochastic system in which Markovian customer attribute processes are initiated at customer arrivals in a discrete batch Markovian arrival process (D-BMAP). We call the aggregate a Markovian branching D-BMAP. Each customer attribute process is an absorbing discrete time Markov chain whose parameters depend both on the phase transition, of the driving D-BMAP, and the number of arrivals taking place at the customer's arrival instant. We investigate functionals of Markovian branching D-BMAPs that may be interpreted as cumulative rewards collected over time for the various customers that arrive to the system, in the transient and asymptotic cases. This is achieved through the derivation of recurrence relations for expected values and Laplace transforms in the former case, and Little's law in the latter case.     

央行沟通对于货币政策实际干预的预测能力研究

信息沟通有助于公众理解和预测货币政策的实际干预,进而提高货币政策有效性。研究中国人民银行的信息沟通对实际政策干预的预测能力需要解决沟通文本的测度问题,同时考虑实际政策干预多工具并用的复杂性。基于此,本文通过文本分析方法提取《货币政策执行报告》的信息,进而考察央行沟通对于货币政策实际干预的预测能力。研究表明,整体而言,市场对中国人民银行“听其言”有助于“观其行”。具体来看,央行沟通对于直接可控的基准利率和存款准备金率具有持续的预测能力,但对于市场利率的预测能力较差,对于M2增长率的预测甚至存在方向不一致的情况。