两类负顾客M/GI/1系统的统计平衡条件

负顾客排队模型由于其灵活模拟各种复杂随机现象的广阔的应用前景，当前正越来越受到各类高性能通讯网络研究多方面的广泛关注．由于负顾客的抵消作用这类系统可以容许在顾客到达率大于服务率的情况下，进入平稳状态．本文用马尔可夫更新理论和Foster负偏移准则，研究了两类M／GI／1负顾客排队模型进入平稳状态的充要条件，首次得到了负顾客更新到达情况下，带负顾客抵消队列头部正顾客和队列尾部正顾客两种策略下的M／GI／1（FCFS）系统的统计平衡条件．当负顾客到达取更新过程的特例一泊松过程时，这一结果与Harrison＆Pital（1996）中所得结果完全一致．     

Maximum Entropy Condition in Queueing Theory

The main results in queueing theory are obtained when the queueing system is in a steady-state condition and if the requirements of a birth-and-death stochastic process are satisfied. The aim of this paper is to obtain a probabilistic model when the queueing system is in a maximum entropy condition. For applying the entropic approach, the only information required is represented by mean values (mean arrival rates, mean service rates, the mean number of customers in the system). For some one-server queueing systems, when the expected number of customers is given, the maximum entropy condition gives the same probability distribution of the possible states of the system as the birth-and-death process applied to an M/M/1 system in a steady-state condition. For other queueing systems, as M/G/1 for instance, the entropic approach gives a simple probability distribution of possible states, while no close expression for such a probability distribution is known in the general framework of a birth-and-death process.     

An M/G/1 Retrial Queueing System with Two-Phase Service and Preemptive Resume

An M/G/1 retrial queueing system with additional phase of service and possible preemptive resume service discipline is considered. For an arbitrarily distributed retrial time distribution, the necessary and sufficient condition for the system stability is obtained, assuming that only the customer at the head of the orbit has priority access to the server. The steady-state distributions of the server state and the number of customers in the orbit are obtained along with other performance measures. The effects of various parameters on the system performance are analysed numerically. A general decomposition law for this retrial queueing system is established.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

A single-server discrete-time queue with correlated positive and negative customer arrivals

An MMBP/Geo/1 queue with correlated positive and negative customer arrivals is studied. In the infinite-capacity queueing system, positive customers and negative customers are generated by a Bernoulli bursty source with two correlated geometrically distributed periods. I.e., positive and negative customers arrive to the system according to two different geometrical arrival processes. Under the late arrival scheme (LAS), two removal disciplines caused by negative customers are investigated in the paper. In individual removal scheme, a negative customer removes a positive customer in service if any, while in disaster model, a negative customer removes all positive customers in the system if any. The negative customer arrival has no effect on the system if it finds the system empty. We analyze the Markov chains underlying the queueing systems and evaluate the performance of two systems based on generating functions technique. Some explicit solutions of the system, such as the average buffer content and the stationary probabilities are obtained. Finally, the effect of several parameters on the system performance is shown numerically.     

高负荷下带重尾服务强占优先排队的扩散逼近

考虑的排队系统是单服务台,顾客的初始到来是依泊松过程来到服务台,顾客的服务时间是重尾分布,服务的原则是强占优先服务.在高负荷条件下对此模型进行研究,获得了系统中的负荷过程,离去过程和队长过程的扩散逼近.     

Equilibrium threshold strategies in observable queueing systems under single vacation policy

This paper studies the equilibrium behavior of customers in continuous/discrete time queueing systems under single vacation policy. In the single vacation queueing system, the server can only take exactly one vacation when no customers exist in the system. This scheme is more practical under many specific circumstances. Based on the reward–cost structure, equilibrium behavior is considered in the fully observable and almost observable cases. The threshold strategies in equilibrium are obtained and the stationary system behavior is analyzed under the corresponding strategies. Finally, we illustrate the effect of the information level as well as several parameters on the equilibrium thresholds and social benefits via numerical experiments. The research results could instruct the customers to take optimal strategies and provide the managers with reference information on the pricing problem in the queueing system.     

System with priority queues and unreliable server

A single server queueing system with an unreliable server and priority customers is considered. The limit distribution of the number of ordinary customers in the system is obtained.     

具有第二次多选择服务的M[X]/G/1排队系统

本文研究成批到达的具有第二次多选择服务的单服务员排队系统．顾客的到达形成一广义泊松过程，不同批的顾客按先到先服务的规则，而同一批的顾客按随机次序接受服务．两次服务的服务时间都是一般分布且相互独立．本文采用补充变量法，求得在瞬态和稳态情况下系统队长的概率母函数，然后又计算出顾客的平均队长和平均等待时间．     

Exact tail asymptotics in a priority queue—characterizations of the preemptive model

In this paper, we consider the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. For this system, we carry out a detailed analysis on exact tail asymptotics for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and for the distribution of the total number of customers in the system, respectively. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through analysis of generating functions. This characterization has never before been completed. It is interesting to note that the exact tail asymptotics along the high-priority queue direction is of a new form that does not fall within the three types of exact tail asymptotics characterized by various methods for this type of two-dimensional system reported in the literature. We expect that the method employed in this paper can also be applied to the exact tail asymptotic analysis for the non-preemptive priority queueing model, among other possibilities.     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.     

An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue

This paper studies a single server queueing system with multiple types of customers. The first part of the paper discusses some modeling issues associated with the Markov arrival processes with marked arrivals (MMAP[K], where K is an integer representing the number of types of customers). The usefulness of MMAP[K] in modeling point processes is shown by a number of interesting examples. The second part of the paper studies a single server queueing system with an MMAP[K] as its input process. The busy period, virtual waiting time, and actual waiting times are studied. The focus is on the actual waiting times of individual types of customers. Explicit formulas are obtained for the Laplace–Stieltjes transforms of these actual waiting times.     

Analysis of a finite buffer model with two servers and two nonpreemptive priority classes

In this paper, we analyze a finite buffer queueing model with two servers and two nonpreemptive priority service classes. The arrival streams are independent Poisson processes, and the service times of the two classes are exponentially distributed with different means. One of the two servers is reserved exclusively for one class with high priority and the other server serves the two classes according to a nonpreemptive priority service schedule. For the model, we describe its dynamic behavior by a four-dimensional continuous-time Markov process. Applying recursive approaches we present the explicit representation for the steady-state distribution of this Markov process. Then, we calculate the Laplace–Stieltjes Transform and the steady-state distribution of the actual waiting times of two classes of customers. We also give some numerical comparison results with other queueing models.     

Exact tail asymptotics in a priority queue—characterizations of the non-preemptive model

This is a companion paper to Li and Zhao (Queueing Syst. 63:355–381, 2009) recently published in Queueing Systems, in which the classical preemptive priority queueing system was considered. In the current paper we consider the classical non-preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server serving the two classes of customers at possibly different rates. A complete characterization of the regions of system parameters for exact tail asymptotics is obtained through an analysis of generating functions. This is done for the joint stationary distribution of the queue length of the two classes of customers, for the two marginal distributions and also for the distribution of the total number of customers in the system, respectively. This complete characterization is supplemental to the existing literature, which would be useful to researchers.     

基于排队的两类顾客易逝品库存控制策略研究

本文采用排队理论在面向两类顾客需求的环境下讨论易逝品库存排队系统控制策略问题。首先刻画面向两类顾客服务且具有马尔科夫结构的易逝品库存系统排队模型,获得库存水平状态的稳态概率分布以及作为库存成本控制的系统稳态性能指标。接下来,构建库存控制成本函数及考虑服务水平约束的库存控制优化模型,设计了改进的遗传算法。最后,数值算例揭示出系统参数的敏感性和相应的管理启示。     

Effect of global FCFS and relative load distribution in two-class queues with dedicated servers

In this paper, we investigate multi-class multi-server queueing systems with global FCFS policy, i.e., where customers requiring different types of service—provided by distinct servers—are accommodated in one common FCFS queue. In such scenarios, customers of one class (i.e., requiring a given type of service) may be hindered by customers of other classes. The purpose of this paper is twofold: to gain (qualitative and quantitative) insight into the impact of (i) the global FCFS policy and (ii) the relative distribution of the load amongst the customer classes, on the system performance. We therefore develop and analyze an appropriate discrete-time queueing model with general independent arrivals, two (independent) customer classes and two class-specific servers. We study the stability of the system and derive the system-content distribution at random slot boundaries; we also obtain mean values of the system content and the customer delay, both globally and for each class individually. We then extensively compare these results with those obtained for an analogous system without global FCFS policy (i.e., with individual queues for the two servers). We demonstrate that global FCFS, as well as the relative distribution of the load over the two customer classes, may have a major impact on the system performance.     

带有负顾客的M/M/c多重工作休假排队

考虑服务台在休假期间不是完全停止工作,而是以相对于正常服务期低些的服务率服务顾客的M/M/c工作休假排队模型.在此模型基础上,针对现实的M/M/c排队模型中可能出现的外来干扰因素,提出了带有负顾客的M/M/c工作休假排队这一新的模型.服务规则为先到先服务.工作休假策略为空竭服务异步多重工作休假.抵消原则为负顾客一对一抵消处于正常服务期的正顾客,若系统中无处于正常服务期的正顾客时,到达的负顾客自动消失,负顾客不接受服务.首先,由该多重休假模型得到其拟生灭过程及生成元矩阵,然后运用矩阵几何方法给出系统队长的稳态分布表达式和若干系统指标.     

有两个服务阶段、反馈、强占型的M/G/1重试排队

在假定重试区域中只有队首的顾客允许重试的条件下,重试时间是一般分布时,考虑具有两个服务阶段、反馈、强占型的M／G／1重试排队系统．得到了系统稳态的充要条件．求得稳态时系统队长和重试区域中队长分布及相关指标,并且得到了系统的随机分解性质．