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.     

Approximate analysis of reader/writer queues

We analyze the performance of queues that serve readers and writers. Readers are served concurrently, while writers require exclusive service. We approximately analyze a first-come-first-serve (FCFS) reader/writer queue, and derive simple formulae for computing waiting times and capacity under the assumption of Poisson arrivals and exponential service. We extend the analysis to handle a one writer queue, and a queue that includes write intention locks. The simple analyses that we present can be used as rules of thumb for designing concurrent systems     

具有优先权的M/G/1重试可修排队系统

在服务台忙的情况下, 到达服务台的顾客以概率 q 进入无限位置的优先队列而以概率 p 进入无限位置的重试轨道 (orbit), 并且按照先到先服务 (FCFS) 规则排队, 假定只有队首的顾客允许重试, 同时考虑服务台可修的因素, 证明了系统稳态解存在的充要条件. 利用补充变量法求得稳态时两个队列与系统的平均队长、顾客等待时间、服务台的各种状态概率以及可靠性指标.     

T-preemptive priority queue and its application to the analysis of an opportunistic spectrum access in cognitive radio networks

We propose a new priority discipline called the T-preemptive priority discipline. Under this discipline, during the service of a customer, at every T time units the server periodically reviews the queue states of each class with different queue-review processing times. If the server finds any customers with higher priorities than the customer being serviced during the queue-review process, then the service of the customer being serviced is preempted and the service for customers with higher priorities is started immediately. We derive the waiting-time distributions of each class in the M/G/1 priority queue with multiple classes of customers under the proposed T-preemptive priority discipline. We also present lower and upper bounds on the offered loads and the mean waiting time of each class, which hold regardless of the arrival processes and service-time distributions of lower-class customers. To demonstrate the utility of the T-preemptive priority queueing model, we take as an example an opportunistic spectrum access in cognitive radio networks, where one primary (licensed) user and multiple (unlicensed) users with distinct priorities can share a communication channel. We analyze the queueing delays of the primary and secondary users in the proposed opportunistic spectrum access model, and present numerical results of the queueing analysis.     

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.     

新颖的离散时间队列系统模型(英文)

带有正负顾客的连续时间单台服务器的队列系统得到了深入研究且已应用于多agent服务系统和计算机网络系统,而带有正负顾客的离散时间Geo/Geo/1队列研究在最近才出现。在拓展离散时间单台服务器Geo/Geo/1队列的基础上,提出了一个具有正负几何到达顾客的离散时间单台服务器GI/M/1队列模型,分析了队列静态长度分布和在RCH与RCE情况下的等待时间长度分布。     

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.     

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。     

An analysis of symmetric polling systems with two priority classes

A cyclic multiqueue system consists of several stations in which messages are enqueued for transmission, and which are served sequentially in cyclic order by a single server. The arrivals at each queue are independent Poisson processes, and the transmission times are generally distributed. Moreover, there is a nonzero switchover time from one station to the next, which is also generally distributed. Messages can be at either of two priority levels: priority 1 (low) or priority 2 (high), and polling occurs either at low priority, in which case both priority 1 and priority 2 messages can be transmitted, or at high priority, in which case only priority 2 messages are transmitted. The service disciplines considered are the exhaustive service discipline and the gated service discipline. In both cases the performance, as measured by the expected delay for high- and for low-priority messages, is evaluated. Part of the analysis is approximate, and simulation results are presented to validate the approximation.     

弹性分组环中的队列长度分析

弹性分组环是城域网发展的重要方向。为了实现基于优先级区分的业务服务质量,弹性分组环采用基于优先级区分的队列以及转发机制。利用M/G/1/K排队模型分析弹性分组环中各类业务缓存中的分组队列长度。与M/G/1排队模型相比,该模型可以获得更准确的结果,实用价值较高。     

Equilibrium joining strategies in the single server queues with negative customers

We consider an M/M/1 queue with negative customers. An arriving negative customer will break the server down and the positive customer being served (if any) is forced to leave the system. Once a breakdown occurs, the server is sent immediately for repair while positive customers are not allowed to join the system during the repair process. When the server is available, positive arrivals decide whether to join or balk the system based on a common reward-cost structure. We consider an observable case that the positive arrivals are informed about the number of customers in the system and an unobservable case without any information. The corresponding Nash equilibrium strategies and the socially optimal joining strategies are explored. We get a socially optimal threshold in the observable case and a mixed joining strategy in the unobservable case. The profit maximization issue is studied, and we derive optimal strategies in two information cases. Finally, numerical examples are provided to show the influence of different parameters on the strategies and social benefit.     

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。     

A discrete-time analysis of the cyclic reservation multiple access protocol

In this paper a performance analysis of the CRMA (Cyclic Reservation Multiple Access) medium access protocol, which is proposed as an access mechanism for high-speed LANs and MANs is presented. An approximate computational method is derived to obtain the distribution functions of performance measures of interest like the medium access delay and the packet transfer time. The analysis is done in discrete-time domain using a decomposition approach for the access delay in conjunction with a G/G/1 queue with control-feedback and a M/G/1 queue with server vacation. In the model the reservation-cancelation backpressure mechanism is also taken into account. Numerical results are obtained to investigate the efficiency of the backpressure scheme and the scaling issues of the interreserve interval under various load conditions and system configurations. Furthermore, results addressing performance aspects like the fairness issues, the jitter of maximum access delay and the system behavior under station-wise saturated conditions are also discussed.     

Policy function scheduling

Scheduling disciplines have traditionally been specified in terms of a queue structure and algorithms for routing jobs within this structure. Alternatively, a discipline may be formally defined by a policy function, a function of job and system parameters. A policy function scheduler is a parameterized scheduler that — when supplied with a specific policy function — behaves like the specified discipline. The formal definition allows performance measures of a discipline (e.g., the response function) to be expressed in terms of the defining policy function. We review the principles of formal definitions, summarize previous queueing-theoretical results concerning response functions of policy function schedulers, and extend them to multiple preemptive job classes with processor-sharing subclasses. For a large variety of disciplines and job classes, we also express the policy functions in terms of the resulting response functions. Given a desired realizable performance goal, this relation serves to determine the discipline that achieves it. Policy function schedulers with their explicit relation between policy and response functions, which we plot for several different job characteristics, thus offer increased precision in controlling the performance of a computer system.     

Sojourn times in polling systems with various service disciplines

We consider a polling system of N queues Q₁,…,Q_N, cyclically visited by a single server. Customers arrive at these queues according to independent Poisson processes, requiring generally distributed service times. When the server visits Q_i, i=1,…,N, it serves a number of customers according to a certain polling discipline. This discipline is assumed to belong to the class of branching-type disciplines, which includes the gated and exhaustive disciplines. The special feature of our study is that, within each queue, we do not restrict ourselves to service in order of arrival (FCFS); we are interested in the effect of different service disciplines, like Last-Come–First-Served, Processor Sharing, Random Order of Service, and Shortest Job First, on the sojourn time distribution of a typical customer that arrives to the system during steady-state. After a discussion of the joint distribution of the numbers of customers at each queue at visit epochs of the server to a particular queue, we determine the Laplace–Stieltjes transform of the cycle-time distribution, viz., the time between two successive visits of the server to, say, Q₁. This yields the transform of the joint distribution of past and residual cycle time, w.r.t. the arrival of a tagged customer at Q₁. Subsequently concentrating on the case of gated service at Q₁, we use that cycle-time result to determine the (Laplace–Stieltjes transform of the) sojourn-time distribution at Q₁, for each of the scheduling disciplines mentioned above.Next to locally gated polling disciplines, we also consider the globally gated discipline. Again, we consider various non-FCFS service disciplines at the queues, and we determine the (Laplace–Stieltjes transform of the) sojourn-time distribution at an arbitrary queue.     

基于离散时间排队的ARQ性能分析

基于自动请求重传(ARQ)协议的工作原理,提出基于离散时间带有启动机制的Geom/G/1排队模型。使用嵌入马尔可夫链方法推导出排队系统的稳态队长、等待时间、忙期和忙循环等性能指标的解析表达式,给出ARQ协议中数据帧的平均响应时间、信道利用率、系统吞吐量等性能指标的解析表达式。利用仿真工具Matlab进行计算机仿真,数值例子证明了性能指标解析表达式的正确性。     

Performance comparison of some discrete service time systems

This paper analyzes and compares the slotted time operation and the non-slotted time operation of a singleserver system with deterministic service time. These two operations are commonly used to model discrete service time systems in computer or digital communications. But due to the similarity in their operations and performance, the two models may be mixed up with each other. This paper examines by means of queueing analysis the performance of a few classes of systems that differ in the service order of their customers. The means and variances of the queue lengths, waiting times and interdeparture times of FCFS systems using the slotted time operations are first obtained from their respective LST and generating function equations. These are then used for comparison with those in non-slotted time systems. The results show that although the slotted time operation in the FCFS systems can be approximated by the non-slotted time operations under heavy traffic condition, the performances under other regions and service disciplines (e.g., the LCFS and the priority systems) may deviate significantly. They must be properly adjusted if one wishes to use the simpler equations of the non-slotted time operations to approximate the slotted time operations. The comparison graphs provided in this paper supply adjustments guidelines for the careful designers.     

Analysis of a finite buffer queue with different scheduling and push-out schemes

A queueing system M₁, M₂/G₁, G₂/1/N with different scheduling and push-out scheme is analyzed in this paper. This work is motivated by the study of the performance of an output link of ATM switches with traffic of two classes with different priorities. However, the queueing model developed in this paper is more general than that of the output link of ATM switches with two-class priority traffic. General service time distributions are allowed for classes 1 and 2 and a general service discipline function, ₁(i, j), is introduced where ₁(i, j) is the probability that a class 1 packet will be served, given that there are i class 1 and j class 2 packets waiting for service. An exact solution is obtained for the loss probabilities for classes 1 and 2, the queue length distribution and the mean waiting time for class 1. The queue length distribution and the mean waiting time for class 2 are calculated approximately. It is shown that the approximation is an upper bound and the error due to the approximation is very small when the loss probability of class 2 is small (e.g., less than 0.01).     

基于时延的动态优先级调度算法

队列管理是提高网络QoS的一种有效方法。在基于时延的调度算法（BDS）基础上将时间片与优先级相结合,提出了一种基于时延的动态优先级调度算法（DDPQS）。为了实现该算法,针对进入缓冲区的每个子队列设置一个计数器,以调整的计数器值为基准来动态的改变队列的优先级,从而达到队列调度的效果;又从研究该算法的过程中,发现其局限性,即计数器值对时间片过于敏感的问题,于是进一步采用设置阈值进行区分的方法来优化。优化前后的仿真结果表明,时延和吞吐率性能具有明显改善。     

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.