具有位相型修理的离散时间可修排队系统

本文研究了具有一般独立输入，位相型修理的离散时间可修排队系统，假定服务台对顾客的服务时间和服务台寿命服从几何分布，运用矩阵解析方法我们给出系统嵌入在到达时刻的稳态队长分布和等待时间分布，并证明这些分布均为离散位相型分布．我们也得到在广义服务时间内服务台发生故障次数的分布，证明它服从一个修正的几何分布．我们对离散时间可修排队与连续时间可修排队进行了比较，说明这两种排队系统在一些性能指标方面的区别之处．最后我们通过一些数值例子说明在这类系统中顾客的到达过程、服务时间和服务台的故障率之间的关系．     

服务台可修的PH／PH（PH／PH）／1排队系统

本文利用准生灭过程理论系统地研究了服务台可修的ＰＨ／ＰＨ（ＰＨ／ＰＨ）／１排队系统的随机结构和性态。首先证明了在平稳状态下可修排除系统ＰＨ／ＰＨ（ＰＨ／ＰＨ）／１从排除论的角度可转化为一个等价的通常排队模型ＰＨ／ＳＭ／１，然后给出了服务台的所有可靠性指标。     

离散时间排队MAP/PH/3

本文研究具有马尔可夫到达过程的离散时间排队MAP/PH/3,系统中有三个服务台,每个服务台对顾客的服务时间均服从位相型分布。运用矩阵几何解的理论,我们给出了系统平稳的充要条件和系统的稳态队长分布。同时我们也给出了到达顾客所见队长分布和平均等待时间。     

相依修理的可修排队系统 MAP/PH(M/PH)/2

系统地研究了两个不同并行服务台的可修排队系统MAP/PH(M/PH)/2,其中两个不同的服务台拥有一个修理工.若其中一台处于修理状态,则另一台失效后就处于待修状态.利用拟生灭过程理论,我们首先讨论了两个服务台的广义服务时间的相依性,然后给出了系统的稳态可用度和稳态故障度,最后得到了系统首次失效前的时间分布及其均值.     

服务台可修的GI／M（M／PH）／1排队系统

本文首次讨论一个到达间隔为一般分布的可修排队系统。假定服务时间、忙期服务台寿命都服从指疏分布，修复时间是ＰＨ变量。首先证明该系统可转化为一个经典的ＧＩ／￣ＰＨ／１排队模型，然后给出系统在稳态下的各种排队论指标和可靠性指标。     

服务台可修的GI/PH/1排队系统

本文讨论服务台可修的GI/PH/1排队,其中服务台寿命和修复时间也是PH变量。首先证明系统在稳态下可转化为一个等价的经典GI/PH/1模型,然后给出系统的各种稳态指标。此外,对修复后重新服务和累积服务两种不同模型,我们给出了统一的处理。     

PH/M/c可修排队系统

研究了一个修理工和c个服务台的可修排队系统.假设顾客的到达过程为PH更新过程,服务台在忙时与闲时具有不同的故障率.顾客的服务时间、服务台的寿命以及服务台的修理时间均服从指数分布.通过建立系统的拟生灭过程,得到了系统稳态分布存在的充要条件.利用矩阵几何解方法,给出了系统的稳态队长.在此基础上,得到了系统的某些排队论和可靠性指标.     

Analysis of MAP/PH1,PH2/1 queue with vacations and optional secondary services

In this paper we study a queueing model in which the customers arrive according to a Markovian arrival process (MAP). There is a single server who offers services on a first-come-first-served basis. With a certain probability a customer may require an optional secondary service. The secondary service is provided by the same server either immediately (if no one is waiting to receive service in the first stage) or waits until the number waiting for such services hits a pre-determined threshold. The model is studied as a QBD-process using matrix-analytic methods and some illustrative examples are discussed.     

The MAP/(PH/PH)/1 queue with self-generation of priorities and non-preemptive service

Customers arriving according to a Markovian arrival process are served at a single server facility. Waiting customers generate priority at a constant rate γ$\gamma$; such a customer waits in a waiting space of capacity 1 if this waiting space is not already occupied by a priority generated customer; else it leaves the system. A customer in service will be completely served before the priority generated customer is taken for service (non-preemptive service discipline). Only one priority generated customer can wait at a time and a customer generating into priority at that time will have to leave the system in search of emergency service elsewhere. The service times of ordinary and priority generated customers follow PH-distributions. The matrix analytic method is used to compute the steady state distribution. Performance measures such as the probability of n consecutive services of priority generated customers, the probability of the same for ordinary customers, and the mean waiting time of a tagged customer are found by approximating them by their corresponding values in a truncated system. All these results are supported numerically.     

The discrete-time MAP/PH/1 queue with multiple working vacations

We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed.     

On bulk-service MAP/PH L,N /1/N G-Queues with repeated attempts

We consider a retrial queue with a finite buffer of size N, with arrivals of ordinary units and of negative units (which cancel one ordinary unit), both assumed to be Markovian arrival processes. The service requirements are of phase type. In addition, a PH^L,N bulk service discipline is assumed. This means that the units are served in groups of size at least L, where 1≤ L≤ N. If at the completion of a service fewer than L units are present at the buffer, the server switches off and waits until the buffer length reaches the threshold L. Then it switches on and initiates service for such a group of units. On the contrary, if at the completion of a service L or more units are present at the buffer, all units enter service as a group. Units arriving when the buffer is full are not lost, but they join a group of unsatisfied units called “orbit”. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a level dependent quasi-birth-and-death process. We start by analyzing a simplified version of our queueing model, which is amenable to numerical calculation and is based on spatially homogeneous quasi-birth-and-death processes. This leads to modified matrix-geometric formulas that reveal the basic qualitative properties of our algorithmic approach for computing performance measures. AMS Subject Classification: Primary 60K25 Secondary 68M20 90B22.     

Analysis of the MAP/PH/1/K queue with service control

We consider a finite capacity queue with Markovian arrivals, in which the service rates are controlled by two pre-determined thresholds, M and N. The service rate is increased when the buffer size exceeds N and then brought back to normal service rate when the buffer size drops to M. The normal and fast service times are both assumed to be of phase type with representations (β, S), and β θS), respectively, where θ>1. For this queueing model, steady state analysis is performed. The server duration in normal as well as fast periods is shown to be of phase type. The departure process is modelled as a MAP and the parameter matrices of the MAP are identified. Efficient algorithms for computing system performance measures are presented. We also discuss an optimization problem and present an efficient algorithm for arriving at an optimal solution. Some numerical examples are discussed.     

Two finite-difference methods for solving MAP(t)/PH(t)/1/K queueing models

In this paper two solution methods to the MAP(t)/PH(t)/1/K queueing model are introduced, one based on the Backwards Euler Method and the other on the Uniformization Method. Both methods use finite-differencing with a discretized, adaptive time-mesh to obtain time-dependent values for the entire state probability vector. From this vector, most performance parameters such as expected waiting time and expected number in the system can be computed. Also presented is a technique to compute the entire waiting (sojourn) time distribution as a function of transient time. With these two solution methods one can examine any transient associated with the MAP(t)/PH(t)/1/K model including time-varying arrival and/or service patterns. Four test cases are used to demonstrate the effectiveness of these methods. Results from these cases indicate that both methods provide fast and accurate solutions to a wide range of transient scenarios. This revised version was published online in June 2006 with corrections to the Cover Date.     

MAP/(PH/PH)/c Queue with Self-Generation of Priorities and Non-Preemptive Service

Abstract

Customers arriving according to a Markovian arrival process are served at a c server facility. Waiting customers generate into priority while waiting in the system (self-generation of priorities), at a constant rate γ; such a customer is immediately taken for service, if at least one of the servers is free. Else it waits at a waiting space of capacity c exclusively for priority generated customers, provided there is vacancy. A customer in service is not preempted to accommodate a priority generated customer. The service times of ordinary and priority generated customers follow distinct PH-distributions. It is proved that the system is always stable. We provide a numerical procedure to compute the optimal number of servers to be employed to minimize the loss to the system. Several performance measures are evaluated.     

服务台可修的M／SM（PH／SM）／1排队系统

本文研究服务台可修的Ｍ／ＳＭ（ＰＨ／ＳＭ）／１排队系统的随机结构和性态．先证明这个可修排队系统在平稳状态下可转化为一个等价的通常排队模型，然后给出服务台的所有稳态可靠性指标及其相关的结果．