具有插队和止步行为的M/M/c 排队系统

研究了具有插队和止步行为的M/M/c排队系统. 将到达顾客分为常规顾客和插队顾客, 常规顾客在队尾排队等待服务, 插队顾客总是尽可能的靠近队首插队等待服务. 插队行为由到达顾客的插队概率和队列中等待顾客对插队行为的容忍来描述. 利用负指数分布的性质、Laplace-Stieltjes变换和全概率公式, 给出了处于等待队列位置n的顾客、任意一个常规顾客和任意一个插队顾客的等待时间的表达式. 在此基础上, 讨论了系统相关指标随系统参数的变化情况.     

带有不耐烦顾客的M/M/m排队系统的顾客损失率

研究了带有不耐烦顾客的M/M/m排队系统的顾客损失率问题,其中顾客的不耐烦时间服从负指数分布.首先利用数值分析的方法,验证了Boots和Tims(1999,Management Science,45(3):444-448)提出的顾客损失率计算公式在研究的排队系统中不成立.其次进一步分析了公式对实际的顾客损失率的近似效果.最后通过数值试验得到了一个近似效果更好的近似公式.     

具有插队和止步行为的M/M/1/m+1排队系统的等待时间分析

本文研究具有插队和止步行为的M/M/1/m+1排队系统中顾客的等待时间问题.首先,将顾客分为两类,一类顾客到达系统在队尾排队等待服务,称为常规顾客;另一类顾客总是尽可能的靠近队首插队,称为插队顾客.利用负指数分布、Laplace-Stieltjes变换、全概率公式,本文给出处于等待队列位置n的顾客、常规顾客、插队顾客的等待时间的表达式,并在此基础上,给出了相关指标随系统参数的变化情况.     

多类顾客共享排队系统的信息理论

多类顾客的共享排队系统是排队论中一个既重要又困难的研究方向,它在计算机网络、生产制造系统与交通网络等领域中有着许多重要的实际应用.近年来,国外学者对多类顾客的共享排队系统已经开展了一些关键性的研究工作,给出了稳态联合队长的母函数,由此可以得到稳态联合队长的一阶矩和二阶矩.然而,由这个母函数反演来提供多类顾客共享排队系统的稳态联合队长的直接表达式却是一个多年来的困难问题.基于此,本文利用信息论中的最大熵原理,提供了一个高精度的近似表达式,其中这个近似表达式与它的精确表达式能够保证前三阶矩是相同的.另一方面,针对这个近似表达式,本文实现了它的有效数值计算,并通过数值算例分析了这个近似表达式中的重要因子是如何依赖于系统的原始参数.因此这个近似表达式对于推进多类顾客共享排队系统的实际应用具有重要的理论意义,同时本文的方法与结果不仅为研究多类顾客的共享排队系统提供了一条新的重要途径,而且为如何将信息理论应用于排队系统、排队网络以及更一般的随机模型研究提供了理论依据与技术支撑.     

多服务台排队系统下提高顾客等待满意度的两类排队管理策略北大核心CSCD

文章在周文慧,等(2014)研究的基础上提出了等待因子t_r,得出了在排队系统M/M/C(C≥2)中顾客处于第n位时到接受服务的等待时间,用全概率公式得到的顾客在系统中的等待时间t的概率密度函数,进而建立了在M/M/C(C≥2)排队系统下提高顾客等待满意度的两类排队管理策略的优化模型.并进一步将两类排队管理策略推广到了M/M/C(C≥2)排队系统,给银行、超市等服务类企业提供了如何选择两类排队管理策略的科学依据.     

多服务台排队系统下提高顾客等待满意度的两类排队管理策略

文章在周文慧,等(2014)研究的基础上提出了等待因子t_r,得出了在排队系统M/M/C(C≥2)中顾客处于第n位时到接受服务的等待时间,用全概率公式得到的顾客在系统中的等待时间t的概率密度函数,进而建立了在M/M/C(C≥2)排队系统下提高顾客等待满意度的两类排队管理策略的优化模型.并进一步将两类排队管理策略推广到了M/M/C(C≥2)排队系统,给银行、超市等服务类企业提供了如何选择两类排队管理策略的科学依据.     

基于Geo/Geo/1(E,SV)排队系统的均衡止步策略

基于单重休假Geo/Geo/1排队系统,研究顾客的均衡止步策略,首次将休假服务机制引入到离散时间排队经济学模型中. 顾客基于“收入--支出”结构,自主决定去留. 利用拟生灭过程理论,运用差分方程求解技巧,对系统进行了稳态分析,得到了顾客的平均逗留时间;进而构造适当的函数,给出了寻找均衡止步策略的具体方法并证明之;而后分析了在均衡策略下, 系统的稳态行为和社会收益;最后通过数值实验讨论了系统参数对均衡行为的影响.     

具有负顾客和特殊类顾客的串联排队系统分析

讨论了带有普通类顾客、负顾客和特殊类顾客的M/M/1→M/M/1两级串联排队系统模型,负顾客在一级服务系统中,一对一抵消队尾的普通类顾客(若有),若一级服务系统无普通类顾客,负顾客自动消失,负顾客不进入二级服务系统.特殊类顾客不经过一级服务系统,直接进入二级服务系统等待接受服务.用拟生灭过程和矩阵几何解方法,得到了系统稳态队长的分布,以及系统忙期的分布和顾客逗留时间的分布等相关指标.     

到达不确定呼叫中心人员配置的鲁棒优化模型

针对呼叫中心实际运营中顾客到达不确定的特点,采用鲁棒离散优化方法,建立呼叫中心人员配置的鲁棒模型。利用对偶原理将鲁棒模型转换易于求解的线性鲁棒对等式,通过调节模型中的鲁棒参数来权衡鲁棒解的保守性与最优性之间的关系,计算模型中约束违背概率上限来表示鲁棒解的可靠性。通过现实呼叫中心数据算例,验证了模型的有效性,分析了不同鲁棒水平下各时间段服务人员配置规律,以及系统最小成本与违背概率之间的权衡关系。最后,对到达扰动系数进行了敏感性分析。     

带有止步和中途退出的成批到达的M~x/M/R/N同步休假排队系统的性能分析

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,R个服务员立即进行同步多重休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.     

Diffusion approximation for an overloaded X model via a stochastic averaging principle

In previous papers we developed a deterministic fluid approximation for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. The control aims to keep the two queues at a pre-specified fixed ratio. We supported the fluid approximation by establishing a functional weak law of large numbers involving a stochastic averaging principle. In this paper we develop a refined diffusion approximation for the same model based on a many-server heavy-traffic functional central limit theorem.     

The QNET method for two-moment analysis of open queueing networks

Consider an open network of single-server stations, each with a first-in-first-out discipline. The network may be populated by various customer types, each with its own routing and service requirements. Routing may be either deterministic or stochastic, and the interarrival and service time distributions may be arbitrary. In this paper a general method for steady-state performance analysis is described and illustrated. This analytical method, called QNET, uses both first and second moment information, and it is motivated by heavy traffic theory. However, our numerical examples show that QNET compares favorably with W. Whitt's Queueing Network Analyzer (QNA) and with other approximation schemes, even under conditions of light or moderate loading. In the QNET method one first replaces the original queueing network by what we call an approximating Brownian system model, and then one computes the stationary distribution of the Brownian model. The second step amounts to solving a certain highly structured partial differential equation problem; a promising general approach to the numerical solution of that PDE problem is described by Harrison and Dai [8] in a companion paper. Thus far the numerical solution technique has been implemented only for two-station networks, and it is clear that the computational burden will grow rapidly as the number of stations increases. Thus we also describe and investigate a cruder approach to two-moment network analysis, called ΠNET, which is based on a product form approximation, or decomposition approximation, to the stationary distribution of the Brownian system model. In very broad terms, ΠNET is comparable to QNA in its level of sophistication, whereas QNET captures more subtle system interactions. In our numerical examples the performance of ΠNET and QNA is similar; the performance of QNET is generally better, sometimes much better.     

Analysis of a Discrete-Time Queueing System with a Single Server and Heterogeneous Markovian Arrivals

We consider a discrete-time queueing system with a single deterministic server, heterogeneous Markovian arrivals and finite capacity. Most existing techniques model the queueing system using a direct bivariate Markov chain which requires a state space that grows rapidly as the number of customer types increases. In this paper, we define renewal cycles in terms of the input process and model the system occupancy level on each renewal cycle using a one-dimensional Markov chain. We derive the exact joint steady-state probability distribution of both states of input and system occupancy with a considerably reduced state space, which leads to the efficient calculation of overall/individual performance measures such as loss probability and average delay.     

Gaussian skewness approximation for dynamic rate multi-server queues with abandonment

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al., Queueing Syst 30:149–201 (1998) for Markovian service networks. The asymptotics used here reduce to the Halfin and Whitt, Oper Res 29:567–588 (1981) scaling for multi-server queues. The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Ko and Gautam, INFORMS J Comput, to appear (2012) found that a modified version of these differential equations yield better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that is based on estimating the mean, variance, and third cumulant moment. This improves on the previous approaches by fitting the distribution from a quadratic function of a Gaussian random variable.     

Dynamic rate Erlang-A queues

The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large-scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally when a manager updates a staffing schedule in response to a forecast of increased customer demand. Mathematically, this type of scaling ultimately gives us the fluid and diffusion limits as found in Mandelbaum et al. (Queueing Syst 30(1):149–201, 1998) for Markovian service networks. These asymptotics were inspired by the Halfin and Whitt (Oper Res 29(3):567–588, 1981) scaling for multi-server queues. In this paper, we provide a review and an in-depth analysis of the Erlang-A queueing model. We prove new results about cumulant moments of the Erlang-A queue, the transient behavior of the Erlang-A limit cycle, new fluid limits for the delay time of a virtual customer, and optimal static staffing policies for healthcare systems. We combine tools from queueing theory, ordinary differential equations, complex analysis, cumulant moments, orthogonal polynomials, and dynamic optimization to obtain new insights about this fundamental queueing model.     

System delay versus system content for discrete-time queueing systems subject to server interruptions

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.     

Delay analysis of extended rtPS for VoIP service in IEEE 802.16e by matrix analytic method

Extended real time polling service (ErtPS) is added to IEEE 802.16e-2005 standards in order for VoIP service to use uplink resources efficiently by considering on/off characteristic of voice source. Recently average queueing delay of ErtPS algorithm for VoIP service was investigated, and it was shown that ErtPS allows to admit more users than UGS algorithm. But we need the probability distribution of queueing delay rather than average queueing delay in order to provide a necessary information for QoS. In this paper we obtain the probability distribution of queueing delay of ErtPS for VoIP service by using the matrix analytic method for the GI/M/1 type and the M/G/1 type matrices in cases of the service time being exponential and deterministic respectively. By applying the results on deterministic service time we find the maximum allowable number of VoIP users with the required constraint on queueing delay. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

Tandem behavior of a telecommunication system with finite buffers and repeated calls

The tandem behavior of a telecommunication system with finite buffers and repeated calls is modeled by the performance of a finite capacityG/M/1 queueing system with general interarrival time distribution, exponentially distributed service time, the first-come-first-served queueing discipline and retrials. In this system a fraction of the units which on arrival at a node of the system find it busy, may retry to be processed, by merging with the incoming arrival units in that node, after a fixed delay time. The performance of this system in steady state is modeled by a queueing network and is approximated by a recursive algorithm based on the isolation method. The approximation outcomes are compared against those from a simulation study. Our numerical results indicate that in steady state the non-renewal superposition arrival process, the non-renewal overflow process, and the non-renewal departure process of the above system can be approximated with compatible renewal processes.     

M/G/1 queue with event-dependent arrival rates

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.     

Instability of LAS multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the least attained service (LAS) service protocol. It is a reentrant line with two servers and six customer classes. The customer interarrival times in our system are bounded below and have finite exponential moments, while the corresponding service times are deterministic. As a special case, we obtain a deterministic, strictly subcritical unstable LAS network.