Cost-minimization analysis of a working vacation queue with N-policy and server breakdowns

This paper investigates the N-policy M/M/1 queueing system with working vacation and server breakdowns. As soon as the system becomes empty, the server begins a working vacation. The server works at a lower service rate rather than completely stopping service during a vacation period. The server may break down with different breakdown rates during the idle, working vacation, and normal busy periods. It is assumed that service times, vacation times, and repair times are all exponentially distributed. We analyze this queueing model as a quasi-birth–death process. Furthermore, the equilibrium condition of the system is derived for the steady state. Using the matrix-geometric method, we find the matrix-form expressions for the stationary probability distribution of the number of customers in the system and system performance measures. The expected cost function per unit time is constructed to determine the optimal values of the system decision variables, including the threshold N and mean service rates. We employ the particle swarm optimization algorithm to solve the optimization problem. Finally, numerical results are provided, and an application example is given to demonstrate the applicability of the queueing model.     

Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process

In this paper, we consider a discrete-time queuing system with head-of-line non-preemptive priority scheduling and a single server subjected to server interruptions. We model the server interruptions by a correlated Markovian on/off process with geometrically distributed on and off periods. Two classes of traffic are considered, namely high-priority and low-priority traffic. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form. This allows us to derive the joint probability generating function (pgf) of the system contents. From this pgf, closed-form expressions for various performance measures, such as mean and variance of system contents and customer delay can be derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation in the interruption process on the performance of both classes. Some numerical results illustrating the impact of second-order characteristics of the arrival process on mean delays are also presented. The proposed approach which is purely based on pgfs is entirely analytical and enables the derivation of not only steady-state but transient performance measures, as well. The paper presents new insights into the performance analysis of discrete-time queues with service interruption and it also covers some previously published results as a special case.

Scope and purpose

In this contribution, we consider a practical queuing model, with HOL priority scheduling, two classes of traffic, and a server which is subjected to a correlated Markovian interruption process. We first derive a non-linear functional equation relating the joint pgf of the system state vector between two consecutive slots. Then we outline a solution technique to solve for this functional equation. This allows us to derive the joint pgf of the system contents of both classes, from which various performance measures related to mean system contents and customer delays are derived. We also demonstrate how the proposed approach allows for derivation of transient performance measures, as well. It should be noted that detailed coverage of the transient analysis of the system is beyond the scope of this paper.To our best knowledge, this is the first initiative that aims to explore the performance of queuing systems with priority scheduling when the shared server is subjected to service interruption. The paper also generalizes the results of Walraevens et al. (Analysis of a single-server ATM queue with priority scheduling, Computers & Operations Research 2003;30(12):1807–30) by incorporating service interruption into their original queuing model. By means of numerical results, the paper also demonstrates the effect of correlation in the service interruption process on the performance of both classes of customers. The impact of second-order characteristics of the arrival process on mean delays is also investigated.     

IEEE 802.16e节能类型III的性能分析

基于IEEE 802.16e节能类型III的工作原理,考虑用户触发数据的无后效性,建立一个带有多重休假的Geom/G/1排队模型。利用嵌入式马尔可夫链方法推导出该排队系统的稳态指标,相应给出数据帧平均响应时间、能量节省率等性能指标的表达式。构造一个综合考虑能量节省与系统响应时间的成本函数,通过数值例子说明系统性能指标对休眠窗口长度的依赖关系,给出不同负载下休眠窗口长度的最优值。     

移动宽带城域网节能类型Ⅱ的性能分析

为了有效分析移动网络中移动站点的节能效率,研究了应用于移动宽带城域网的IEEE802.16e节能类型Ⅱ的工作原理,建立了一个具有两种休假机制,且在一种休假机制内可以传输少量数据帧的多重休假排队模型;构建了由能量节省率、系统切换率和数据帧平均响应时间等组成的指标体系,并给出了各个性能指标的解析表达式.通过数值示例,刻画了系统性能与系统参数的依赖关系,表明了该模型的可行性与有效性.     

Grid resource management policies for load-balancing and energy-saving by vacation queuing theory

The resource management is the central component of grid system. The analysis of the workload log file of LCG including the job arrival and the resource utilization daily cycle shows that the idle sites in the Grid are the source of load imbalance and energy waste. Here we focus on these two issues: balancing the workload by transferring jobs to idle sites at prime time to minimize the response time and maximize the resource utilization; power management by switch the idle sites to sleeping mode at non-prime time to minimize the energy consume. We form the M/G/1 queue model with server vacations, startup and closedown to analysis the performance metrics to instruct the design of load-balancing and energy-saving policies. We provide our Adaptive Receiver Initiated (ARI) load-balancing strategy and power-management policy for energy-saving. The simulation experiments prove the accuracy of our analysis and the comparisons results indicate our policies are largely suitable for large-scale heterogeneous grid environment.     

带有门限的IBP+MMBP/Geo/1/K休假排队系统

建立了基于门限策略的双队列单服务台有限缓存包级休假排队模型IBP+MMBP/Geo/1/K。针对到达分布不同的两种业务各自的特点来定义休假策略,将H2H(Human to human)业务的休假期等效为M2M(Machine to machine)业务的服务期。为了寻求统一的标准来评价休假策略,定义了一个评价业务QoS的效能函数E。同时,研究了不同条件下的M2M业务对休假策略的影响。最后,通过仿真给出各种情况下系统各项性能指标以及效能函数的变化趋势,该研究结果可以用来指导无线资源分配和业务配置等问题。     

带负顾客的N策略启动时间的Geo/Geo/1工作休假排队

研究了正、负两类顾客的离散时间Geo/Geo/1多重工作休假排队模型,并加入N策略和启动时间,负顾客不接受服务,并只起一对一抵消队首正接受服务的顾客的作用,即服从RCH(Remove customer from head)策略。使用拟生灭过程及矩阵几何解方法得到队长的稳态分布,进一步得出了系统队长的随机分解的结果。     

On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.     

The analysis of the M/M/1 queue with two vacation policies (M/M/1/SWV+MV)

We consider an M/M/1 queue with two vacation policies which comprise single working vacation and multiple vacations, denoted by M/M/1/SMV+MV. Using two methods (called R-matrix method and G-matrix method), we obtain the stationary distribution of queue length (including the customer being in service) and make further analysis on the stationary numbers of customers in the working vacation and vacation period, respectively. The stochastic decomposition results of stationary queue length and the sojourn time of a customer are also derived. Meanwhile, we show that a simple and direct method of decomposition developed in Liu et al. [Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 (2007), pp. 595–600] is also applicable to our model. Furthermore, busy period is analysed by the limiting theorem of alternative renewal process. Finally, some boundary properties and numerical analysis on performance measures are presented.     

修理工可单重休假带有一个冷贮备部件并联系统的适定性

本文运用线性算子半群理论研究修理工可单重休假的带有一个冷贮备部件的Gaver并联可修系统的适定性问题．文中假定部件的工作时间服从指数分布,修理时间和修理工的休假时间均服从一般连续分布．通过对描述该系统行为的偏微分方程组的规范化,并引入系统的状态空间,主算子及其定义域,我们将该系统方程转化成Banach空间中的抽象的Cauchy问题．然后,运用泛函分析中的Hille-Yosida定理、Phillips定理与Fattorini定理,我们证明了该系统存在唯一的、满足概率性质的正时间依赖解．