Analysis of the M/G/1 queue with discriminatory random order service policy

We consider an M/G/1 queue with different classes of customers and discriminatory random order service (DROS) discipline. The DROS discipline generalizes the random order service (ROS) discipline: when the server selects a customer to serve, all customers waiting in the system have the same selection probability under ROS discipline, whereas customers belonging to different classes may have different selection probabilities under DROS discipline. For the M/G/1 queue with DROS discipline, we derive equations for the joint queue length distributions and for the waiting time distributions of each class. We also obtain the moments of the queue lengths and the waiting time of each class. Numerical results are given to illustrate our results.     

基于两次丢包策略的主动队列管理算法研究

针对网络拥塞现象,基于两次丢包方法建立了一种新的主动队列管理算法TDPQW。该算法利用M/G/1排队模型推导了实际队列长度和等待时间的数学表达式,以此提出在队列头部和队中随机位置进行丢包的策略。同时,通过仿真实验对比分析了该算法与RED、DROP-TAIL算法的性能,结果表明TDPQW具有较好的适应性。     

M/G/1排队网络系统性能研究

本论文以单一路由器(服务器)为例,从最原始的队列理论出发,探讨具有容量C的M/G/1队列模型的系统平均时延、系统稳态下的报文(用户)平均值、以及时延等问题。并对报文(用户)的服务质量需求作了详尽的数学推导。     

On the M/Ck/1 queue

We obtain the exact analytic expression of the probability distribution of the number of units in a single server queue with Poisson arrivals and Coxian service time distribution (notated as M/C_k/1). A recursive procedure for calculating this probability distribution is given. The well-known queues M/E_k/1 and M/D/1 are re-derived as special cases of the M/C_k/1 queue. Finally, the cases of M/C₂/1 and M/C₃/1 are fully worked out.     

Analysis of the discrete-time Geo/G/1 working vacation queue and its application to network scheduling

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.     

A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution

The queue of a single server is considered with independent and identically distributed interarrivai and service times and an infinite (GI/G/1) or finite (GI/G/1/N) waiting room. The queue discipline is non-preemptive and independent of the service times.

A discrete time version of the system is analyzed, using a two-component state model at the arrival and departure instants of customers. The equilibrium equations are solved by a polynomial factorization method. The steady state distribution of the queue size is then represented as a linear combination of geometrical series, whose parameters are evaluated by closed formulae depending on the roots of a characteristic polynomial.

Considering modified boundary constraints, systems with finite waiting room or with an exceptional first service in each busy period are included.     

基于M/G/1/K排队理论的IEEE 802.15.4网络吞吐量分析

针对IEEE 802.15.4时隙载波侦听多址接入与碰撞避免（CSMA/CA）算法,利用二维Markov链分析方法提出了一个网络分析模型。该模型特别考虑了IEEE 802.15.4协议的休眠模式以及退避窗口先于退避阶数(NB)达到最大值的情况。在此基础上,结合M/G/1/K排队理论推导得到了吞吐量的表达式,进而分析了网络在非饱和状态下数据包到达率对吞吐量的影响,利用模拟平台NS2进行了仿真。实验结果显示理论分析结果与仿真结果可以较好地拟合,并能准确描述网络吞吐量的变化,验证了分析模型的有效性。     

M/M/1-PS queue and size-aware task assignment

We consider a distributed server system in which heterogeneous servers operate under the processor sharing (PS) discipline. Exponentially distributed jobs arrive to a dispatcher, which assigns each task to one of the servers. In the so-called size-aware system, the dispatcher is assumed to know the remaining service requirements of some or all of the existing jobs in each server. The aim is to minimize the mean sojourn time, i.e., the mean response time. To this end, we first analyze an M/M/1-PS queue in the framework of Markov decision processes, and derive the so-called size-aware relative value of state, which sums up the deviation from the average rate at which sojourn times are accumulated in the infinite time horizon. This task turns out to be non-trivial. The exact analysis yields an infinite system of first order differential equations, for which an explicit solution is derived. The relative values are then utilized to develop efficient dispatching policies by means of the first policy iteration (FPI). Numerically, we show that for the exponentially distributed job sizes the myopic approach, ignoring the future arrivals, yields an efficient and robust policy when compared to other heuristics. However, in the case of highly asymmetric service rates, an FPI based policy outperforms it. Additionally, the size-aware relative value of an M/G/1-PS queue is shown to be sensitive with respect to the form of job size distribution, and indeed, the numerical experiments with constant job sizes confirm that the optimal decision depends on the job size distribution.     

Randomized control of T-policy for an M/G/1 system

We study an M/G/1 queueing system with a server that can be switched on and off. The server can take a vacation time T after the system becomes empty. In this paper, we investigate a randomized policy to control a server with which, when the system is empty, the server can be switched off with probability p and take a vacation or left on with probability (1 − p) and continue to serve the arriving customers. For this system, we consider the operating cost and the holding cost where the operating cost consists of the system running and switching costs (start up and shut down costs). We describe the structure and characteristics of this policy and solve a constrained problem to minimize the average operating cost per unit time under the constraint for the holding cost per unit time.     

基于CPN的空竭服务单重休假M/G/1型排队系统建模与分析

空竭服务单重休假M/G/1型排队系统是经典排队系统的推广,在许多领域有着广泛的应用.到目前为止对其的处理方法还都是建立在概率论和数理统计的基础上,运用马尔可夫随机过程求解,推导十分复杂,没有直观的模型描述.因此,利用着色Petri网对空竭服务单重休假M/G/1型排队系统进行建模,并对主要性能指标进行仿真分析是迫切以及可行地.仿真软件选用CPNTools[1],仿真结果证明该方法具有较高的精确度以及实用价值.     

The optimal service time allocation of a versatile server to queue jobs and stochastically available non-queue jobs of different types

In this paper, we consider a service system in which the server can process N+1$N+1$ different types of jobs. Jobs of type 0 are generated randomly according to a Poisson stream. Jobs of types 1 to N are non-queue types which may or may not be available for completion by the server. To optimally allocate the server's time to these jobs, we formulate a finite state semi-Markov decision process model to this environment. With this model, the optimal stationary policies can be numerically determined via a policy-iteration algorithm. We also discuss the practical applications of this model to tele-service and tele-marketing operations.     

A method for calculating successive approximate solutions for a class of block banded M/G/1 type Markovian models

This paper presents an efficient equilibrium solution algorithm for a class of infinite block banded M/G/1 type Markov chains. By re-blocking the states, these are a class of the so-called quasi-birth-and-death (QBD) type chains. The proposed algorithm is not based on an iterative approach, so that the exact solution can be computed in a known finite number of steps. The key point on which the algorithm is based is the identification of a linear dependence among variables. This dependence is expressed in terms of a companion matrix. The equilibrium solution of the Markov chain is obtained operating on this matrix.

An attractive feature of the algorithm is that it allows the computation of a succession of approximate solutions with growing accuracy, until the exact solution is obtained in a finite number of steps. The class of block-banded M/G/1 type Markov chains we consider requires that the lower diagonal block is invertible and that the chain is ergodic. However, many models arising from telecommunication systems satisfy this restriction. Results for a case study show that the proposed algorithm is efficient and quite accurate, even when providing approximate solutions.     

一类排队服务系统的最优控制策略研究

建立一个双阈值排队服务模型, 用来协调顾客的等待时间和服务系统的运行成本. 采用一种精细的概率分解方法获得瞬态和稳态队长的概率分布性质; 再根据系统稳态性能指标, 建立系统营运利润函数, 并设计一种针对双离散变量函数的全局优化收索算法来研究最优控制策略; 进一步的数值实验揭示出双重阈值策略的优越性.     

多信道SW-ARQ系统基于带休假的M/G/1排队模型时延性能分析*

在考虑分组长度、信道传输质量的影响下,运用带休假的M/G/1排队模型分析SW-ARQ系统的时延性能指标。分别获得了单信道SW-ARQ系统和多信道SW-ARQ系统分组平均等待时延和系统平均时延解析表达式,并将两种系统的时延性能指标分别进行分析比较。分析结果表明：当分组长度较大时,采用多信道SW-ARQ可以获得较小的系统平均时延,提高系统的传输可靠性。最后通过数值模拟比较分析了单信道SW-ARQ系统和多信道SW-ARQ系统下分组长度、传输正确率以及到达率对系统平均时延的影响。     

The interdeparture time distribution for each class in the ∑Mi/Gi/1 queue with set-up times and repeated server vacations

This paper studies the interdeparture time distribution of one class of customers who arrive at a single server queue where customers of several classes are served and where the server takes a vacation whenever the system becomes empty or is empty when the server returns from a vacation. Furthermore, the first customer in the busy period is allowed to have an exceptional service time (set-up time), depending on the class to which this customer belongs. Batches of customers of each class arrive according to independent Poisson processes and compete with each other on a FIFO basis. All customers who belong to the same class are served according to a common generally distributed service time. Service times, batch sizes and the arrival process are all assumed to be mutually independent. Successive vacation times of the server form independent and identically distributed sequences with a general distribution.For this queueing model we obtain the Laplace transform of the interdeparture time distribution for each class of customers whose batch size is geometrically distributed. No explicit assumptions of the batch size distributions of the other classes of customers are necessary to obtain the results.The paper ends by showing how the mathematical results can be used to evaluate a protocol that controls access to a shared medium of an ATM passive optical network. The numerical results presented in the last section of this paper show that the bundle spacing principle that is used by the permit distribution algorithm of this protocol introduces high delays and in many cases also more variable interdeparture times for the ATM cells of individual connections. An alternative algorithm is proposed that does not cope with these performance short comings and at the same time conserves the good properties of the protocol.     

Analysis of waiting time of M/G/1/K system with vacations under random scheduling and LCFS

This paper considers an M/G/1/K queueing system with multiple vacations under random scheduling and LCFS disciplines. As for M/G/1/K vacation models, Lee obtained the results of the joint distribution of the number of messages in the system and the remaining service or vacation time for a message. Using these expressions, we derive LSTs of the waiting time distribution under the two service disciplines. We show the calculation method for the first and the second moments of the waiting time under two service disciplines. Furthermore, we illustrate some numerical results of the mean waiting times and the coefficient of variations of the waiting time under FCFS, random scheduling and LCFS.     

基于PME重尾分布服务时间的M/G/1模型排队性能研究

通信网络的业务源普遍存在着白相似性（或称为长相关性）,传统的假定业务到达间隔服从负指数分布的Poisson模型或其改进形式已不再适用．但在利用M／G／1模型对白相似业务源进行排队分析时,由于重尾分布服务时间的LST变换无闭合形式,进行排队性能分析非常困难．该文通过引入一类混合指数分布证明此类分布服从Pareto重尾分布,并得到相应的LST变换闭合形式及服务时间渐进级数,同时将形状参数y=3／2时的服务时间及其LST变换推广到更一般的情形,从而较为有效地解决了重尾分布的信源排队等待时间分析问题．     

The Geometry of M/D/1 Queues and Large Deviation

Since an M/D/1 queue is represented by a Markov chain, we can consider the set of all the M/D/1 queues as a subset of Markov chains. A geometric structure is induced from the geometric structure of the set of Markov chains, which forms an exponential family. In this paper, we show that in the large deviation of the tail probability of the queue length of an M/D/1, the rate function and a twisted Markov chain, etc., are represented in terms of the geometry. Moreover, in the importance sampling (IS) simulation for the M/D/1 queue, we elucidate the geometric relation between the underlying distribution and a simulation distribution, and evaluate the variance of an IS estimate by geometric quantities.     

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.