A discrete-time single-server queue with a modified N -policy

We consider a discrete-time single-server queue where the idle server waits for reaching a level N in the queue size to start a batch service of N messages, although the following arrivals during the busy period receive single services. We find the stationary distributions of the queue and system lengths as well as some performance measures. The vacation and busy periods of the system and the number of messages served during a busy period are also analyzed. The stationary distributions of the time spent waiting in the queue and in the system are studied too. Finally, a total expected cost function is developed to determine the optimal operating N-policy at minimum cost.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

Markovian bulk service queue with delayed vacations

An M/M(a, b)/1 queueing system with multiple vacations is studied, in which if the number of customers in the queue is a - 1 either at a service completion epoch or at a vacation completion point, the server will wait for an exponential time in the system which is called the changeover time. During this changeover time if there is an arrival the server will start service immediately, otherwise at the end of the changeover time the server will go for a vacation. The duration of vacation is also exponential. This paper is concerned with the determination of the stationary distribution of the number of customers in the queue and the waiting time distribution of an arriving customer. The expected queue length is also obtained. Sample numerical illustrations are given.     

A new approach to the queue length and waiting time of BMAP/G/1 queues

We apply the factorization principle to derive the generating function of the queue length and the vector Laplace–Stieltjes transform of the waiting time of a BMAP/G/1 queue. The mean performance measures are provided with a computational experience.Scope and purposeThe classical method of obtaining the queue length and waiting time distributions of BMAP/G/1 queues starts with the analysis of the imbedded Markov renewal process at departure epochs. This method is intricate and time consuming when the idle period process is complicated. In this paper, we demonstrate that the factorization property can be applied efficiently and effectively to derive the queue length distributions of BMAP/G/1 queueing systems by avoiding the conventional standard procedures. The approach demonstrated in this paper can be applied to the analysis of many other BMAP/G/1 queueing systems with higher behavioral complexities.     

Analysis of a finite-capacity M/G/1 queue with a resume level

In order to model the buffer pool behavior in a data communication component, an M/G/1/K queue where input is shut down when the queue size (number of messages) attains K until it decreases to a specified level is analyzed. By use of semi-Markov process approach, the queue length distribution at an arbitrary time is found, and the resultant performance measures (utilization, loss probability, and mean response time) are computed.     

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.     

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.     

A framework to determine bounds of maximum loss rate parameter of RED queue for next generation routers

Random early detection (RED) is expected to eliminate global synchronization by random active packet drop. Its packet drop probability is decided by the maximum packet drop probability in its drop function, buffer thresholds, and average queue length. It has been observed that for a large number of connections, a small value of the maximum packet drop probability may not eliminate global synchronization. Furthermore, since RED uses four parameters to regulate its performance, it is necessary to relate its maximum drop probability with those parameters. The objective of this paper is to develop a framework for the bounds of the maximum drop probability of RED, based on TCP channel model and traffic characteristics. The value of the maximum drop probability obtained by our model will make RED queue achieve its targeted goals.     

A maximum entropy priority approximation for a stableG/G/1 queue

Summary The principle of maximum entropy is used under two different sets of mean value constraints to analyse a stableG/G/1 queue withR priority classes under preemptive-resume (PR) and non-preemptive head-of-line (HOL) scheduling disciplines. New one-step recursions for the maximum entropy state probabilities are established and closed form approximations for the marginal queue length distribution per priority class are derived. To expedite the utility of the maximum entropy solutions exact analysis, based on the generalised exponential (GE) distribution, is used to approximate the marginal mean queue length and idle state probability class constraints for both the PR and HOLG/G/1 priority queues. Moreover, these results are used as building blocks in order to provide new approximate formulae for the mean and coefficient of variation of the effective priority service-time and suggest a maximum entropy algorithm for general open queueing networks with priorities in the context of the reduced occupancy approximation (ROA) method. Numerical examples illustrate the accuracy of the proposed maximum entropy approximations in relation to simulations involving different interarrival-time and service-time distributions per class. Comments on the extension of the work to more complex types of queueing systems are included.This work is sponsored in part by the Science and Engineering Research Council (SERC), UK, under grant GR/D/12422 and in part by the Ministry of Higher Education of the Algerian Government     

基于S排队的被动队列管理算法

为了解决网络拥塞问题,基于“弃头”方式建立了一种新的被动队列管理算法（DFSQ）。该算法首先利用S排队推导网络队长的数学表达式,并提出丢包概率和丢包策略。同时,通过仿真实验深入研究了影响网络队长的关键因素,并将DFSQ与随机早期检测算法(RED)、“弃尾”算法进行对比分析,结果表明DFSQ算法性能更优。     

Analysis of total average queue length in multi-hop wireless networks

In upcoming communication environments, multi-hop networking is expected to be pervasive in wireless access and backbone networks because it promises large coverage and increased capacity. Many studies have been devoted to the throughput of multi-hop wireless networks, but the delay performance is not clearly investigated yet. In this Letter, we analyze a multi-hop wireless network to obtain a lower bound of the total average queue length since it is closely related to the delay. We first consider a network with single hop sessions only, and its total average queue length is characterized as a solution of a linear program. Then we present a novel idea of translating a network with multi-hop sessions into a network with single hop sessions only, thereby the linear programming approach can be applied again. Simulation results show that the lower bound is very close to the achievable value of our reference link scheduler. Hence, our bound is tight.     

一种基于速率的主动队列管理机制

提出了一种基于速率的主动队列管理机制。主要思想是根据每流过去的输入速率和输出速率预测下一个采样间隔内的输入速率和输出速率，用上一个采样间隔末的每流队列长度、预计输入、预计输出和期望的队长来决定数据包的丢弃概率，达到稳定队列长度和实现公平性的目的：仿真结果证明在多种网络条件下，算法能够实现稳定队列长度和每流公平性两个目的。     

Performance analysis of a finite-buffer discrete-time queue with bulk arrival,bulk service and vacations

This paper considers a finite-buffer discrete-time Geo^X/G^Y/1/K+B queue with multiple vacations that has a wide range of applications including high-speed digital telecommunication systems and various related areas. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive a set of linear equations to compute the steady-state departure-epoch probabilities based on the embedded Markov chain technique. Next, we present numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as the moments of the number of packets in the queue at three different epochs, the mean delay in the queue of a packet, the loss probability and the probability that server is busy with computational experiences.     

Performance analysis of a threshold-based discrete-time queue using maximum entropy

The provision of guaranteed QoS for various Internet traffic types has become a challenging problem for researchers. New Internet applications, mostly multimedia-based, require differentiated treatments under certain QoS constraints. Due to a rapid increase in these new services, Internet routers are facing serious traffic congestion problems. This paper presents an approximate analytical performance model in a discrete-time queue, based on closed form expressions using queue threshold, to control the congestion caused by the bursty Internet traffic. The methodology of maximum entropy (ME) has been used to characterize closed form expressions for the state and blocking probabilities. A discrete-time GGeo/GGeo/1/{N₁, N₂} censored queue with finite capacity, N₂, external compound Bernoulli traffic process and generalised geometric transmission times under a first come first serve (FCFS) rule and arrival first (AF) buffer management policy has been used for the solution process. To satisfy the low delay along with high throughput, a threshold, N₁, has been incorporated to slow the arrival process from mean arrival rate λ₁ to λ₂ once the instantaneous queue length has been reached, otherwise the source operates normally. This creates an implicit feedback from the queue to the arrival process. The system can be potentially used as a model for congestion control based on the Random Early Detection (RED) mechanism. Typical numerical experiments have been included to show the credibility of ME solution against simulation for various performance measures and to demonstrate the performance evaluation of the proposed analytical model.     

Multi-server retrial queue with second optional service: algorithmic computation and optimisation

We consider an infinite-capacity M/M/c retrial queue with second optional service (SOS) channel. An arriving customer finds a free server would enter the service (namely, the first essential service, denoted by FES) immediately; otherwise, the customer enters into an orbit and starts generating requests for service in an exponentially distributed time interval until he finds a free server and begins receiving service. After the completion of FES, only some of them receive SOS. The retrial system is modelled by a quasi-birth-and-death process and some system performance measures are derived. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of a matrix-analytic approach. A cost model is derived to determine the optimal values of the number of servers and the two different service rates simultaneously at the minimal total expected cost per unit time. Illustrative numerical examples demonstrate the optimisation approach as well as the effect of various parameters on system performance measures.     

控制时延的主动队列管理算法

针对现有的主动队列管理(AQM)算法造成的队列时延无法满足VoIP、音视频等流媒体传输需求的问题,提出一种直接控制队列时延的主动队列管理算法--DCQA。该算法使用PID控制器计算路由器缓存的数据包丢弃概率,用其对即将进入缓存排队的数据包做丢包判断并采取相应动作,以控制队列时延在期望值以下。实验仿真了3种网络环境下DCQA的性能,链路利用率分别是99.93%、99.88%和99.95%。并且,队列时延分别有50.45%、51.59%、52.4%被控制在期望值以下,比CoDel算法分别提高了3.6%、40.53%、50.69%。实验结果表明,DCQA在不同的网络环境中都可以获得较高的链路利用率,而且控制队列时延的能力优于CoDel算法,适用于流媒体的传输。     

A single server queue with mixed types of interruptions

Summary The single server M/G/1 queue subject to Poisson interruptions has many useful applications in computer systems modeling. The interruptions are usually characterized by their type of service-preemption discipline. This paper deals with this model in its most general setting, allowing the simultaneous presence of all types of interruptions that may be encountered in real systems. Inspite of the inherent complexity of the analysis, it is possible to derive analytic closed form expressions for interesting performance measures. The results obtained are of theoretical interest as well as of practical significance. In particular, we derive the Laplace Stieltjes transform of the completion time associated with a customer's ervice and obtain the steady-state average number of customers in the system. An application to the modeling of checkpointing and recovery in a transactional system is considered.This work was supported in part by Air Force Office of Scientific Research under grant AFOSR-84-0132, by the Army Research Office under contract DAAG29-84-K0045 and by the National Science Foundation under grant MCS-830200     

A hierarchical task queue organization for shared-memorymultiprocessor systems

There are two basic ways in which waiting ready tasks can be organized: centralized organization or distributed organization. In the centralized organization, a single central task queue is maintained. In the latter case, private ready queue of tasks. Ideally, a central ready queue global to all processors is desired over the distributed organization because the centralized organization provides perfect load sharing. However, the centralized organization is not suitable for large parallel systems because the central task queue could become a system bottleneck. The distributed organization, on the other hand, creates the load imbalance problem, which results in performance deterioration, while techniques have been proposed to reduce the ill-effects of task queue contention in the centralized organization and load imbalance in the distributed organization these techniques introduce problems of their own. We propose the use of a hierarchical task queue organization to incorporate the best features of these two organizations. Our study into the performance of this hierarchical organization shows that a properly designed hierarchical organization provides performance very close to that of the centralized organization while eliminating the ready queue contention problem, we also provide an analysis that identifies and provides guidance for designing the hierarchical task queue organization that avoids ready queue access contention. A brief discussion of task scheduling policies is also included     

Transient behaviour of finite capacity tandem queues with blocking

The transient output process from tandem qucueing systems with no wailing positions is analysed. Our system is subject to blocking; that is, a unit cannot move to the second queue until it completes service at the first queue and the second queue is not busy. Expressions arc derived for the expected output time of the nth unit when the system is initially empty and these expressions are used to derive the throughput of these systems in equilibrium. For the systems studied, placing a faster server ahead of a slower server maximizes the throughput.     

Reducing run queue contention in shared memory multiprocessors

The performance of parallel processing systems, especially large systems, is sensitive to various types of overhead and contention. Performance consequences may be serious when contention occurs for hardware resources such as memory or the interconnection network. Contention can also occur for software resources such as critical data structures maintained by either system or application software. A run queue is one such critical data structure that can affect overall system performance. There are two basic types of run queues, centralized and distributed. Both present performance problems. There are also several techniques to mitigate their drawbacks, but none is completely satisfactory. Instead, the author proposes a different run queue organization, a hierarchical organization that inherits the best features of the centralized and the distributed queue organizations while avoiding their pitfalls. Thus, the hierarchical organization is suitable for building large-scale multiprocessor systems