Analysis of GI/M/s/c queues using uniformisation

We combine uniformisation, a powerful numerical technique for the analysis of continuous time Markov chains, with the Markov chain embedding technique to analyze GI/M/s/c queues. The main steps of the proposed approach are the computation of

• (1)the mixed-Poisson probabilities associated to the number of arrival epochs in the uniformising Poisson process between consecutive customer arrivals to the system; and
• (2)the conditional embedded uniformised transition probabilities of the number of customers in the queueing system immediately before customer arrivals to the system.

To show the performance of the approach, we analyze queues with Pareto interarrival times using a stable recursion for the associated mixed-Poisson probabilities whose computation time is linear in the number of computed coefficients. The results for queues with Pareto interarrival times are compared with those obtained for queues with other interarrival time distributions, including exponential, Erlang, uniform and deterministic interarrival times. The obtained results show that much higher loss probabilities and mean waiting times in queue may be obtained for queues with Pareto interarrival times than for queues with the other mentioned interarrival time distributions, specially for small traffic intensities.     

An enhanced algorithm to solve multiserver retrial queueing systems with impatient customers

The homogenization of the state space for solving retrial queues refers to an approach, where the performance of the M/M/c retrial queue with impatient customers and c servers is approximated with a retrial queue with a maximum retrial rate restricted beyond a given number of users in the orbit. As a consequence, the stationary distribution can be obtained by the matrix-geometric method, which requires the computation of the rate matrix. In this paper, we revisit an approach based on the homogenization of the state space. We provide the exact expression for the conditional mean number of customers based on the computation of the rate matrix R with the time complexity of O(c). We develop simplified equations for the memory-efficient implementation of the computation of the performance measures. We construct an efficient algorithm for the stationary distribution with the determination of a threshold that allows the computation of performance measures with a specific accuracy.     

Throughput in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We first define the average rate as observed by users and by the queue. In an M/M/1 processor-sharing queue, we give closed-form expressions for these average rates, and prove a strict ordering amongst them. We prove that the queue service rate (in bps) is an increasing function of the minimum required average transmission rate, and give a closed-form expression for the marginal cost associated with such a performance requirement. We then consider the effect of using connection access control by modeling an M/M/1/K processor-sharing queue. We give closed-form expressions for average transmission rates, and discuss the relationship between the queue service rate (in bps), the queue limit, the average rate, and the blocking probability     

Alternating priority queues with non-zero switch rule

There are two queues and a single server. The server serves the two queues alternately according to a non-zero switch rule. That is, the server continues to serve without interruption in queue i until k_i customers are served or until the queue there becomes empty, whichever happens first, i = 1, 2. Customers arrive at the two queues according to two independent Poisson processes. The service times of the customers in each queue have a general distribution. In this paper, distributions of busy periods and waiting times are studied and pointed out the difficulties to obtain an explicit analytic expression for the mean waiting time. The non-zero switch rule is compared with the zero switch rule (k₁ = k₂ = ∞), in terms of the mean waiting times. Since an analytic comparison of waiting times is found to be difficult, a numerical comparison is done by means of simulation. The simulation results show also, how the waiting times are sensitive to the switch parameters k₁ and k₂.     

大时滞网络中的拥塞控制算法

主动队列管理(AQM)通过网络中间节点有目的的分组丢弃实现了较低的排队延时和较高的有效吞吐量,是近年来TCP端到端拥塞控制的一个研究热点.已有的大多数AQM算法在设计过程中都没有充分考虑到大时滞对算法性能的影响.首先通过仿真试验证实了已有的几种典型算法控制的队列在大时滞网络中无一例外地出现了剧烈的振荡,导致瓶颈链路利用率下降和延时抖动加剧.为此,在进行了适当模型拟合处理的基础上,应用控制理论中的内模补偿原理设计了鲁棒的延时补偿主动队列管理(delay compensation-active queue management,简称DC-AQM)算法,克服了大时滞给队列稳定性造成的不利影响.仿真实验结果表明,新算法在大时滞小期望队列长度的网络配置中表现出的综合性能明显优于已有的算法,链路利用率是其他算法的3～4倍.     

Violation Probability in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We focus here upon performance requirements in the form of an upper bound on the probability of failing to achieve a specified minimum transmission rate or a specified minimum average rate. For an M/G/l processor-sharing queue, we give a closed-form expression for this violation probability. We derive closed-form expressions for the marginal service rate with respect to the violation probability and to the minimum transmission rate, and characterize when each is binding. We then consider the effect of using connection access control by modeling an M/G/l/K processor-sharing queue, and discuss the relationship between queue service rate, queue limit, violation probability, and blocking probability. Finally, we consider a two-class discriminatory processor-sharing queue, and discuss what combinations of class weighting and service rate can be used to achieve specified minimum rate violation probabilities for both classes.     

Revisiting priority queues for image analysis

Many algorithms in image analysis require a priority queue, a data structure that holds pointers to pixels in the image, and which allows efficiently finding the pixel in the queue with the highest priority. However, very few articles describing such image analysis algorithms specify which implementation of the priority queue was used. Many assessments of priority queues can be found in the literature, but mostly in the context of numerical simulation rather than image analysis. Furthermore, due to the ever-changing characteristics of computing hardware, performance evaluated empirically 10 years ago is no longer relevant. In this paper I revisit priority queues as used in image analysis routines, evaluate their performance in a very general setting, and come to a very different conclusion than other authors: implicit heaps are the most efficient priority queues. At the same time, I propose a simple modification of the hierarchical queue (or bucket queue) that is more efficient than the implicit heap for extremely large queues.     

The behaviour of a single queue in a general queueing network

Summary We develop a method based on diffusion approximations in order to compute, under general conditions, the queue length distribution of a single queue in a network of queues. Several applications of this approach to computer network performance analysis and to time-sharing systems are presented. The accuracy of model predictions are evaluated by comparison with known exact results in particular cases, with simulation experiments and with the approximation method of Kobayashi and Reiser.     

Fuzzy-Based Dynamic Distributed Queue Scheduling for Packet Switched Networks

Addressing the problem of queue scheduling for the packet-switched system is a vital aspect of congestion control. In this paper, the fuzzy logic based decision method is adopted for queue scheduling in order to enforce some level of control for traffic of different quality of service requirements using predetermined values. The fuzzy scheduler proposed in this paper takes into account the dynamic nature of the Internet traffic with respect to its time-varying packet arrival process that affects the network states and performance. Three queues are defined, viz low, medium and high priority queues. The choice of prioritizing packets influences how queues are served. The fuzzy scheduler not only utilizes queue priority in the queue scheduling scheme, but also considers packet drop susceptibility and queue limit. Through simulation it is shown that the fuzzy scheduler is more appropriate for the dynamic nature of Internet traffic in a packet-switched system as compared with some existing queue scheduling methods. Results show that the scheduling strategy of the proposed fuzzy scheduler reduces packet drop, provides good link utilization and minimizes queue delay as compared with the priority queuing (PQ), first-in-first-out (FIFO), and weighted fair queuing (WFQ).     

Dynamic routing of real-time jobs among parallel EDF queues: A performance study

This paper introduces an analytical method for approximating the performance of a firm real-time system consisting of a number of parallel infinite-capacity single-server queues. The service discipline for the individual queues is earliest-deadline-first (EDF). Real-time jobs with exponentially distributed relative deadlines arrive according to a Poisson process. Jobs either all have deadlines until the beginning of service or deadlines until the end of service. Upon arrival, a job joins a queue according to a state-dependent stationary policy, where the state of the system is the number of jobs in each queue. Migration among the queues is not allowed. An important performance measure to consider is the overall loss probability of the system. The system is approximated by a Markovian model in the long run. The resulting model can then be solved analytically using standard Markovian solution techniques. Comparing numerical and simulation results for at least three different stationary policies, we find that the existing errors are relatively small.     

Smart priority queue algorithms for self-optimizing event storage

Low run-time overhead, self-adapting storage policies for priority queues called smart priority queue (SPQ) techniques are developed and evaluated. The proposed SPQ policies employ a low-complexity linear queue for near-head activities and a rapid-indexing variable bin-width calendar queue for distant events. The SPQ configuration is determined by monitoring queue access behavior using cost-scoring factors and then applying heuristics to adjust the organization of the underlying data structures. To illustrate and evaluate the method, an SPQ-based scheduler for discrete event simulation has been implemented and was used to assess the resulting efficiency, components of access time, and queue usage distributions of the existing and proposed algorithms. Results indicate that optimizing storage to the spatial distribution of queue access can decrease HOLD operation cost between 25% and 250% over existing algorithms such as calendar queues.     

Sticky States in Banyan Network Queues and Their Application to Analysis

A finite-buffered banyan network analysis technique designed to model networks at high traffic loads is presented. The technique specially models two states of the network queues: queue empty and queue congested (roughly, zero or one slots free). A congested queue tends to stay congested because packets bound for the queue accumulate in the previous stage. The expected duration of this state is computed using a probabilistic model of a switching module feeding the congested queue. A technique for finding a lower arrival rate to an empty queue is also described. The queues themselves are modeled using independent Markov chains with an additional congested state added. The new analysis technique is novel in its modeling the higher arrival rate to a congested queue and a lower arrival rate to an empty queue. Comparison of queue state distributions obtained with the analysis and simulation shows that an important feature of congestion is modeled.     

MultiCL: Enabling automatic scheduling for task-parallel workloads in OpenCL

The OpenCL specification tightly binds a command queue to a specific device. For best performance, the user has to find the ideal queue-device mapping at command queue creation time, an effort that requires a thorough understanding of the underlying device architectures and kernels in the program. In this paper, we propose to add scheduling attributes to the OpenCL context and command queue objects that can be leveraged by an intelligent runtime scheduler to automatically perform ideal queuedevice mapping. Our proposed extensions enable the average OpenCL programmer to focus on the algorithm design rather than scheduling and to automatically gain performance without sacrificing programmability. As an example, we design and implement an OpenCL runtime for task-parallel workloads, called MultiCL, which efficiently schedules command queues across devices.Our case studies include the SNU benchmark suite and a real-world seismology simulation. To benefit from our runtime optimizations, users have to apply our proposed scheduler extensions to only four source lines of code, on average, in existing OpenCL applications. We evaluate both single-node and multinode experiments and also compare with SOCL, our closest related work. We show that MultiCL maps command queues to the optimal device set in most cases with negligible runtime overhead.     

The influence of service-time variability in a closed network of queues

This paper describes the effect of service-time variability on the standard performance measures of a closed network of single-server queues with the first-come first-served discipline and one job class. Several service-time variability principles are proposed to serve as rough practical guidelines. The most interesting one states that the mean queue length at a bottleneck queue typically decreases when the variability of the service time at that queue is increased. The principles are supported here by numerical examples and theorems in special cases. The principles are also applied to test approximation procedures.     

Proof of stability conditions for token passing rings by Lyapunovfunctions

A token passing ring can be described as a system of M queues with one server that rotates around the queues sequentially. Georgiadis-Szpankowski (1992) considered rings where the token (server) performs x ∇ l_j services on queue j, where x is the size of queue j upon arrival of the token, and l_j is a fixed limit of service for queue j. The token then spends some random time switching to the next queue. For j=1, ..., M, arrivals to queue j are Poisson with rate λ_j, and service times have mean s _j and are independent of the arrival and switchover processes. The purpose of this paper is to give an alternate and simpler proof of the stability conditions given by Georgiadis-Szpankowski using Lyapunov functions. An additional assumption is made about the second moments of the service and switchover times being finite     

Asymptotic bounds of throughput in series-parallel queueing networks

We propose a piece-wise linear upper bound on the throughput rate from a network of series-parallel queues where arrivals occur through a single infinite queue. This bound is tight and is observed to be extremely accurate in forecasting the actual throughput rate. We also describe the monotonicity of throughput as a function of the arrival rate and specify a condition under which the upper bound may be computed. We approximate analytically the throughput measured as a function of the arrival rate for two tandem exponential queues, where the first queue has an infinite buffer while the second queue has a finite buffer. We extend this analysis to elementary split and merge queueing networks. We demonstrate the generality and robustness of this asymptotic property, for larger series-parallel networks with general service times and specify the set up of a single simulation experiment which can be used to retrieve the throughput for any arrival rate, as well as other networks performance measures.     

Stable Scheduling Policies for Maximizing Throughput in Generalized Constrained Queueing Systems

We consider a class of queueing networks referred to as "generalized constrained queueing networks" which form the basis of several different communication networks and information systems. These networks consist of a collection of queues such that only certain sets of queues can be concurrently served. Whenever a queue is served, the system receives a certain reward. Different rewards are obtained for serving different queues, and furthermore, the reward obtained for serving a queue depends on the set of concurrently served queues. We demonstrate that the dependence of the rewards on the schedules alter fundamental relations between performance metrics like throughput and stability. Specifically, maximizing the throughput is no longer equivalent to maximizing the stability region; we therefore need to maximize one subject to certain constraints on the other. Since stability is critical for bounding packet delays and buffer overflow, we focus on maximizing the throughput subject to stabilizing the system. We design provably optimal scheduling strategies that attain this goal by scheduling the queues for service based on the queue lengths and the rewards provided by different selections. The proposed scheduling strategies are however computationally complex. We subsequently develop techniques to reduce the complexity and yet attain the same throughput and stability region. We demonstrate that our framework is general enough to accommodate random rewards and random scheduling constraints.     

Approximating queue lengths inM(t)/G/1 queue using the maximum entropy principle

Using the discrete time approach a model is developed for obtaining the expected queue length of theM(t)/G/1 queue. This type of queue occurs in different forms in transportation and traffic systems and in communications and manufacturing systems. In order to cut down the very high computational efforts required to evaluate the performance measures in such queues by exact methods, the Maximum Entropy Principle is used to approximate the expected queue length which is one of the most commonly used performance measures. A procedure is then developed for reducing the error encountered when this approximation is adopted. The results from this paper will encourage the practitioners to use the appropriate time-varying queueing models when the need arises instead of resorting to very poor approximations.     

Optimally efficient estimation of the statistics of rare events inqueueing networks

Because of their rarity, the estimation of the statistics of buffer overflows in networks of queues by direct simulation is very costly. An asymptotically optimal (as the overflow recurrence time becomes large) scheme has been proposed by others, using importance sampling. Two aspects of this scheme are addressed. First, in the existing approach, a numerical minimization is required to generate the simulation network. An equivalent analytic minimization is described. A simple procedure for constructing the optimal simulation network is included. Second, it is shown that the average behaviour of the simulation system is the same as the average behavior of the original network in the period leading up to a buffer overflow. For a sufficiently large buffer size, the optimal simulation system depends only on the statistics of the service rate of one queue (that of the least serviced buffer) and the arrival process, assuming that no two service rates are actually equal, and does not depend in any way on the statistics of the service rates of buffers other than the one dominating the overflow statics     

缓冲交叉开关交换结构性能分析

分析了一种缓冲交叉开关交换结构在突发流量到达下的性能.通过建立分析模型,给出了每个输入端口拥有单个或多个输入队列的缓冲交叉开关结构的饱和吞吐.结果显示,对于单输入队列结构而言,随着突发平均长度的增加,饱和吞吐迅速从1下降,并收敛于0.5.随着每个输入端口输入队列数目的增加,饱和吞吐率逐渐接近1.仿真实验验证了分析模型的准确性.该结果可以用于指导基于缓冲交叉开关的路由交换设备的优化设计.