THE M/M/c QUEUE WITH （e,d） SETUP TIME

The authors present a new queueing model with （e, d） setup time. Using the quasi-birth-and-death process and matrix-geometric method, the authors obtain the stationary distribution of queue length and the LST of waiting time of a customer in the system. Furthermore, the conditional stochastic decomposition results of queue length and waiting time are given.     

在一个二阶排队网络中对两类不同顾客的模糊控制

研究一个二阶的串联排队网络 ,其中第一阶有一个服务员 ,第二阶有两个并联服务员 ,每个服务员都有其自己的具有无限存储空间的排队队列 .在第一阶的存储空间中包含有足够多的两类顾客 .两类不同的顾客需要首先在第一阶的服务员上接受服务然后按顺序先后进入第二阶分别排在和两类顾客相对应的服务员前 .不管什么时侯只要第一阶的服务员服务完一个顾客 ,系统就得到一个收益 ,与此同时 ,每个在第二阶排队中的顾客都需要付一笔占位费 .系统的控制目标是根据系统的状态动态地选择在第一阶中下一个被服务的顾客的种类 ,以保证系统在一个无限长的时间里平均收益最大 .这个控制模型可以应用在生产和通讯网络中 .在这篇论文里 ,一个模糊控制的新方法被提出用以解决这个复杂问题 .计算机模拟的结果显示这种新方法是非常有效并极富研究潜力的 .     

N-策略与Min(N,V)-策略的M/G/1/∞排队系统等待时间的随机分解结构

在一些关于N-策略休假的M/G/1/∞排队模型研究中,由于顾客的等待时间与该顾客到达时刻以后的输入间隔时间不再独立,因此对顾客的稳态等待时间分布的讨论较为困难,更多是集中在系统的稳态队长和附加队长的讨论上,很少有文献讨论顾客的稳态等待时间及其随机分解.本文首先考虑经典N-策略休假的M/G/1/∞排队系统,讨论了顾客的稳态等待时间分布,给出了顾客的稳态等待时间的随机分解结果和顾客的附加延迟时间分布的显式表达式,同时,指出了已有结果的错误.其次,我们考虑在多重休假和单重休假下具有Min(N,V)-策略控制的M/G/1/∞排队系统,给出了顾客的稳态等待时间的随机分解结果,获得了顾客的平均稳态等待时间和平均附加延迟时间表达式.特别地,通过本文可直接获得一些特殊排队系统的相应结果.     

负顾客、带启动期和备用服务员的M/M/1休假排队系统

考虑一类有正、负顾客, 带启动期和有备用服务员的M/M/1休假排队系统. 负顾客一对一抵消队尾的正顾客(若有), 若系统中无正顾客, 到达的负顾客自动消失, 负顾客不接受服务.系统中两个服务员, 其中一个在岗工作时另外一个备用.上岗服务员若因为某种原因休假, 备用服务员立即替换上岗.当系统变空时, 系统关闭.用拟生灭过程和矩阵几何解方法, 得到了稳态队长的分布, 此外, 证明了稳态条件下队长的条件随机分解并得到了附加队长的分布. 最后, 通过两个数值例子说明该模型可以较好的模拟一些实际问题.     

具有强占优先权的不耐烦顾客的M/M/m/k排队模型

首先研究只有一类不耐烦顾客的M/M/m排队模型,其中顾客到达服从相互独立的泊松分布,服务时间服从相互独立的指数分布,到达率与服务率随着系统中的顾客数而发生变化。顾客的耐心等待时间(截止到服务开始前)服从指数分布。在此基础上进一步研究两类顾客到达的M/M/m/k排队系统。其中第一类顾客对于第二类顾客有强占优先权,两类顾客的到达率与服务率随着系统中顾客人数而发生变化。采用矩阵分析的方法得到了两类顾客各自的稳态分布,并有相应的性能分析,为系统的优化设计提供了依据。     

带负顾客和不耐烦顾客的离散时间Geo/G/1重试排队

考虑了一个带负顾客和不耐烦顾客且重试时间为一般分布的离散时间Geo/G/1重试排队系统. 负顾客带走一个正在服务的顾客, 而对重试组中的顾客无影响.正顾客到达系统若遇服务器忙则可能进入重试组也可能离开系统.通过对此排队系统的嵌入马氏链进行分析, 得到了重试组队长和系统队长的概率母函数. 进而得到了一系列重要的排队指标. 此外, 还推导出了系统的稳态存在条件. 以及对无负顾客和不耐烦顾客时的特例进行了分析. 最后通过几个具体的数值实例演示了一些参数对系统关键性能指标的影响.     

A batch arrival retrial queue with starting failures,feedback and admission control

This paper is concerned with the analysis of a feedback M^[X]/G/1 retrial queue with starting failures and general retrial times. In a batch, each individual customer is subject to a control admission policy upon arrival. If the server is idle, one of the customers admitted to the system may start its service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn-on the server, which takes negligible time. If the server is started successfully (with a certain probability), the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. The Markov chain underlying the considered queueing system is studied and the necessary and sufficient condition for the system to be stable is presented. Explicit formulae for the stationary distribution and some performance measures of the system in steady-state are obtained. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.     

Computational analysis of （MAP1, MAP2）/（PH1, PH2）/N queues with finite buffer in wireless cellular networks

This paper studies a queueing model with the finite buffer of capacity K in wireless cellular networks,which has two types of arriving calls-handoff and originating calls,both of which follow the Markov arriving process with different rates.The channel holding times of the two types of calls follow different phase-type distributions.Firstly,the joint distribution of two queue lengths is derived,and then the dropping and blocking probabilities,the mean queue length and the mean waiting time from the joint distribution are gotten.Finally,numerical examples show the impact of different call arrival rates on the performance measures.     

Transient Queue Size Distribution Solution of Geom / G / 1 Queue with Feedback-A Recursive Method

This paper considers the Geom / G / 1 queueing model with feedback according to a late arrival system with delayed access (LASDA). Using recursive method, this paper studies the transient property of the queue size from the initial state N(0⁺) = i. Some new results about the recursive expression of the transient queue size distribution at any epoch n ⁺ and the recursive formulae of the equilibrium distribution are obtained. Furthermore, the recursive formulae of the equilibrium queue size distribution at epoch n ⁻, and n are obtained, too. The important relations between stationary queue size distributions at different epochs are discovered (being different from the relations given in M / G / 1 queueing system). The model discussed in this paper can be widely applied in all kinds of communications and computer network. This research is supported by the National Natural Science Foundation of China under Grant No. 70871084, the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 200806360001, and the Scientific Research Fund of Southwestern University of Finance and Economics.     

分析M／G／1排队系统队长分布的方法注记

利用系统忙期和ＬＳ变换，本文获得从任意初始状态出发的Ｍ／Ｇ／１排队系统队长瞬态解的ＬＳ变换表达式以及平稳队长分布的表达式，并在Ｍ／Ｍ／１排队系统中验证了其正确性．值得注意的是本文提出的方法简洁、直观．     

An M/G/1 Queueing System with Delay Server Vacations

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Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy.It is assumed that the server, after each service completion,begins a process of search in order to find the following customer to be served with a certain probability,or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition.The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle,busy or on vacation.Finally,the author gives two stochastic decomposition laws,and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.     

An Approximation of the Customer Waiting Time for Online Restaurants Owning Delivery System

Online restaurants, which receive online orders and deliver food directly to the customer's residence, are becoming increasingly popular. To be successful, online restaurants need to provide reliable and prompt deliveries. Careful design of the meal preparation and order delivery systems is needed to avoid excessive customer waiting time between ordering and delivery. This paper considers the meal preparation and delivery processes simultaneously to approximate average customer waiting time for deliveries. The authors first discuss the system performance with one cook and unit-capacity delivery vehicles, using an M/G/1 queue and a GI/G/1 queue. Numerical experiments show that our approximation can adequately describe real waiting times. Then, series queues with multiple cooks and multi-capacity delivery vehicles, e.g., an M/G/n queue and a GI/G~n/1 queue, are examined.Results show that except for situations with a large meal preparation time and a small vehicle capacity,compared with the result of simulation, the approximation in this paper is acceptable with a deviation of less than 20%. The marginal decrease in waiting time associated with hiring more vehicles is estimated under different meal preparation speeds, sizes of service area and vehicle capacities.     

延迟N-策略Geo/G/1排队系统的队长分布及数值计算

考虑延迟N-策略离散时间Geo/G/1排队系统,使用全概率分解技术,从任意初始状态出发,研究了队长的瞬态和稳态性质,导出了在任意时刻n瞬态队长分布的z-变换的递推表达式和稳态队长分布的递推表达式,以及稳态队长的随机分解.最后,通过数值实例, 讨论了稳态队长分布对系统参数的敏感性,并阐述了获得便于计算的稳态队长分布的表达式在系统容量设计中有重要的价值.     

Erlang消失系统的离散时间建模分析

针对数字通信中信源的特点,应用离散时间排队Geom/Geom/c/c对经典Erlang消失系统进行建模分析。要假设离散时间点上有多个到达和离去的基础之上,通过给出在稳态条件下系统中顾客数的母函数满足的函数方程,推导出系统稳态的消失概率(Loss Probability)和平均队长的表达式。并且设计出计算稳态队长概率和损失概率的递推算法,从而获得了系统重要的稳态性能指标。最后通过数值例子说明了顾客损失率与系统各参数之间的关系。     

Recursive Solution of Queue Length Distribution for Geo/G/1 Queue with Delayed Min(N,D)-Policy

In this paper we consider a discrete-time Geo/G/1 queue with delayed Min(N, D)-policy.Using renewal process theory, total probability decomposition technique and z-transform, we study the transient and equilibrium properties of the queue length from an arbitrary initial state, and obtain both the recursive expressions of the transient state queue length distribution and the steady state queue length distribution at arbitrary time epoch n~+. Furthermore, we derive the important relations between equilibrium queue length distributions at different time epochs n~-, n and n~+. Finally, we give some numerical examples about capacity decision in queueing systems to demonstrate the application of the analytical results reported in this paper.     

基于M/M/1/SV排队的均衡门限策略

考虑单重休假M/M/1排队, 在部分可视的前提下, 研究顾客的均衡门限策略, 首次将单重休假机制引入到连续时间排队经济学模型中. 系统的决策主体是顾客, 突破了以往只注重服务机构单方面行为的局限. 基于“收入-支出”结构, 利用马尔可夫过程理论, 通过求解差分方程, 分析了系统的稳态行为, 得到了顾客的平均逗留时间; 进而构造适当的函数, 给出了寻找 均衡纯门限策略, 均衡混合门限策略的具体方法并证明之; 而后在不同的策略下, 得出了系统的稳态分布和均衡社会收益; 最后, 通过数值实验分析了均衡行为的各指标对系统参数的敏感性. 研究结果为顾客决策提供了优化建议, 同时为管理者研究系统中的定价问题提供了理论参考.     

The Queue Length Distribution for M/G/1 Queue with Delay Single Server Vacation

1　IntroductionDuringrecentseveraldecadesmanyauthorsstudiedM/G/1queueswithdifferentservervacationregimes(seeRefs.[1～6]).Theynotonlystudiedthestochasticdecompositionpropertiesofthequeuelengthandwaitingtimewhenthesystemisinequilibrium,butalsostudiedthetransientandequilibriumdistributionsofthequeuelength.InRef.[6]theau-thorsstudiedM/G/1queuewithdelaymultipleservervacationsinwhichtheserverhasthreestates:vacation-preparation,vacationandbusyperiod.InthispaperwefirstdiscusstheM/G/1queuewithdelay…     

一对多交互式在线客户服务排队模型

以聊天工具为基础的在线客服大量取代了电话形式的传统客服.与电话客服不同,一个在线客服人员能同时为多个顾客提供交互式的服务,而传统的排队公式都是以一对一服务为基础推导的,无法直接用来计算在线客服的排队指标.分析了一对多交互模式下的排队特点,提出了该模式下的服务水平指标,构建了包含顾客层和消息层的双层排队模型,推导了系统平...     

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M^[x]G/1 queueing system with an unreliable server and a delayed repair,in which the server operates a randomized vacation policy with multiple vacations.Whenever the system is empty,the server immediately takes a vacation.If there is at least one customer found waiting in the queue upon returning from a vacation,the server will be immediately activated for service.Otherwise,if no customers are waiting for service at the end of a vacation,the server either remains idle with probability p or leaves for another vacation with probability 1-p.Whenever one or more customers arrive when the server is idle,the server immediately starts providing service for the arrivals.The server may also meet an unpredictable breakdown and the repair may be delayed.For such a system the authors derive the distributions of some important system characteristics,such as the system size distribution at a random epoch and at a departure epoch,the system size distribution at the busy period initiation epoch,and the distribution of the idle period and the busy period.The authors perform a numerical analysis for changes in the system characteristics,along with changes in specific values of the system parameters.A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.