Experiments with improved approximate mean value analysis algorithms

Approximate mean value analysis (MVA) is a popular technique for analyzing queueing networks because of the efficiency and accuracy that it affords. In this paper, we present a new software package, called the improved approximate mean value analysis library (IAMVAL), which can be easily integrated into existing commercial and research queueing network analysis packages. The IAMVAL packages include two new approximate MVA algorithms, the queue line (QL) algorithm and the fraction line (FL) algorithm, for analyzing multiple class separable queueing networks. The QL algorithm is always more accurate than, and yet has approximately the same computational efficiency as, the Bard–Schweitzer proportional estimation (PE) algorithm, which is currently the most widely used approximate MVA algorithm. The FL algorithm has the same computational cost and, in noncongested separable queueing networks where queue lengths are quite small, yields more accurate solutions than both the QL and PE algorithms.     

Blending randomness in closed queueing network models

Random environments are stochastic models used to describe events occurring in the environment a system operates in. The goal is to describe events that affect performance and reliability such as breakdowns, repairs, or temporary degradations of resource capacities due to exogenous factors. Despite having been studied for decades, models that include both random environments and queueing networks remain difficult to analyse. To cope with this problem, we introduce the blending algorithm, a novel approximation for closed queueing network models in random environments. The algorithm seeks to obtain the stationary solution of the model by iteratively evaluating the dynamics of the system in between state changes of the environment. To make the approach scalable, the computation relies on a fluid approximation of the queueing network model. A validation study on 1800 models shows that blending can save a significant amount of time compared to simulation, with an average accuracy that grows with the number of servers in each station. We also give an interpretation of this technique in terms of Laplace transforms and use this approach to determine convergence properties.     

An analysis of an approximation algorithm for queueing networks

Queueing network models have proved to be useful aids in computer system performance prediction. Numerous approximation algorithms have been devised to allow queueing networks to be analyzed with a reduced computational expense. In particular, for separable queueing networks several popular iterative approximation algorithms are available.

One disadvantage of these algorithms has been the lack of results concerning their behavior, such as whether or not iteration convergence is guaranteed. This paper presents an analysis of an approximate MVA algorithm proposed by Schweitzer (1979). It is proved that the equations defining the algorithm have a unique solution when there is only a single customer class, and iteration initializations that yield monotonic convergence to this solution are exhibited. It is also proved that the solution is pessimistic relative to the exact queueing network solution.     

The tree MVA algorithm

A new algorithm to solve product form queueing networks, especially those with large numbers of centers and chains, is presented. This algorithm is a Tree version of Mean Value Analysis (MVA). Tree MVA is analogous to the Tree version of Convolution developed by Lam and Lien. Like Tree Convolution, Tree MVA allows exact solution of large networks which are intractable with previous sequential algorithms. As with the sequential versions of Convolution and MVA, Tree MVA has better numerical properties than Tree Convolution. Further, Tree MVA avoids the computational complexity of sequential MVA in networks with several queue dependent centers. Thus, we consider Tree MVA to be the best algorithm for general product form networks.     

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

This paper is concerned with reliable multistation series queueing networks. Items arrive at the first station according to a Poisson distribution and an operation is performed on each item by a server at each station. Every station is allowed to have more than one server with the same characteristics. The processing times at each station are exponentially distributed. Buffers of nonidentical finite capacities are allowed between successive stations. The structure of the transition matrices of these specific type of queueing networks is examined and a recursive algorithm is developed for generating them. The transition matrices are block-structured and very sparse. By applying the proposed algorithm the transition matrix of a K-station network can be created for any K. This process allows one to obtain the exact solution of the large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures can be calculated.Scope and purposeThe exact analysis of queueing networks with multiple servers at each workstation and finite capacities of the intermediate queues is extremely difficult as for even the case of exponential operation (service or processing) times the Markovian chain that models the system consists of a huge number of states which grows exponentially with the number of stations, the number of servers at each station and the queue capacity of each intermediate queue of the resulting system. The scope and purpose of the present paper is to analyze and provide a recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks. By applying the proposed algorithm one may create the transition matrix of a K-station queueing network for any K. This process allows one to obtain the exact solution of the resulting large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures of the system can be obtained.     

The distribution of network states during residence times in product form queueing networks

Product form queueing networks are so named because their equilibrium state probabilities have a simple product form. This simple form has lead to computationally efficient techniques for obtaining solutions of these networks. Recent work on the probability of network states at those instants when a customer completes service at a service center (departure instants) or arrives at a service center (arrival instants) has revealed a similar product form. The arrival instant result provides a simple, intuitive explanation of the Mean Value Analysis solution technique for product form networks with First-Come-First-Served service centers.In this paper we derive the distribution of network states seen by a particular customer while resident at a particular service center. This distribution too has a relatively simple product form. We use this information to explain in an intuitive way the MVA solution technique for a more general class of network, those containing load independent Processor Sharing and last-Come-First-Served-Preemptive-Resume service centers. It is hoped that, just as the intuitive explanation of the response time formula for FCFS centers has led to approximation techniques for non-separable FCFS centers, this new information may provide approximation techniques for non-separable centers with other scheduling disciplines such as preemptive and non-preemptive priority.     

An approximate MVA algorithm for exponential, class-dependent multiple servers

Mean value analysis (MVA) is an efficient algorithm for determining the mean sojourn time, the mean queue length, and the throughput in a closed multiclass queueing network. It provides exact results for the class of product-form networks. Often different classes have different service requirements in FCFS queues, but such networks are not of product form. There are several possibilities to compute performance measure for such nodes and networks. In this paper we present an approximation formula for multiple-server FCFS queues with class-dependent service times as a Norton flow equivalent product node, where the departure rate of any class depends on the number of customers of all classes in the queue. We will use this approximation in the sojourn time formula of some exact and approximate MVA algorithms.     

Two schedulers to provide delay proportion and reduce queueing delay simultaneously

The proportional delay differentiation model provides controllable and predictable delay differentiation, that is, the packet delay proportion between two classes of services is consistent on any measured timescale. Previous studies have focused on improving the accuracy of the achieved delay proportion between classes, and have not considered reducing the packet queueing delay, since these proposed scheduling algorithms are independent of the packet service time, such that the mean queueing delay is invariant, as specified by the conservation law. This paper proposes maximum WTP (MWTP) and variance WTP (VWTP) schedulers, modified from the waiting-time priority (WTP) algorithm which is an excellent scheduler for performing proportional delay differentiation. All of the proposed schedulers account for the packet transmission time. Simulation results indicate that when the link utilization is moderate, the two schedulers not only yield more accurate delay proportions than the WTP scheduler, regardless of whether the timescale is long or short, but also reduce the mean queueing delay. The effects of load distribution, packet size, and coefficient of variation (CoV) of packet sizes, on the performance of all schedulers are also investigated. Our proposed schedulers always outperform WTP.     

Exact analysis of performance models by the Method of Moments

Performance evaluation of distributed systems and service-oriented architectures is often based on stochastic models, such as closed queueing networks which are commonly solved by the Mean Value Analysis (MVA) algorithm. However, the MVA is unable to solve models with hundreds or thousands of users accessing services of multiple classes, a configuration that is often useful to predict the performance of real-world applications. This paper introduces the Method of Moments (MoM), the first exact algorithm for solving closed queueing networks with large population sizes.Compared to the MVA algorithm, which is based on a recursive evaluation of mean queue-lengths, MoM defines a recursion on higher-order moments of queue-lengths that is solved at each step by a linear system of equations. This approach dramatically decreases the costs of an exact analysis compared to the MVA approach. We prove that MoM requires log-quadratic time and log-linear space in the total population size, whereas MVA complexity expressions grow combinatorially as the product of class populations. This extends the feasibility of exact methods to a much larger family of multiclass performance models than those that can be solved by the MVA algorithm.     

Mean-value analysis and convolution method for queue-dependent servers in closed queueing networks

In this paper, we formulate a recursive relation for the marginal probabilities of a closed network with K customers in terms of the same network with K ? 1 customers. Mean-value analysis (MVA) is an application of this relation, together with Little's formula. It is shown that the convolution method, too, can be based on the same recursive result. This leads to a new convolution algorithm called normalized convolution algorithm (NCA), which like MVA works entirely with probabilities and throughputs rather than with quantities such as normalization contacts. NCA avoids a difficult problem, the occurrence of floating-point over-flows in the original convolution algorithm.We shall also solve a numerical stability problem found in MVA. Finally, we show how MVA and the convolution algorithm can be combined in the same problem to yield a hybrid method retaining the best properties of both methods.     

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.     

基于多点传送的DAP-NAD-CJ改进算法的研究

对用于多点传送的DAP-NAD-CJ(周期性跳跃的确定性适应优先网络访问延迟)算法做了改进。新的算法指定在参差不齐的无线信道空位和网络基站之间进行多点传送,以便多点传送的来源信号能尽可能快地进入通道。模拟试验表明,修改后的算法在单一的优先权和多重优先权的网络上都优于DAP-NAD-CJ算法。     

A generalized method of moments for closed queueing networks

We introduce a new solution technique for closed product-form queueing networks that generalizes the Method of Moments (MoM), a recently proposed exact algorithm that is several orders of magnitude faster and memory efficient than the established Mean Value Analysis (MVA) algorithm. Compared to MVA, MoM recursively computes higher-order moments of queue lengths instead of mean values, an approach that remarkably reduces the computational costs of exact solutions, especially on models with large numbers of jobs.In this paper, we show that the MoM recursion can be generalized to include multiple recursive branches that evaluate models with different numbers of queues, a solution approach inspired by the Convolution algorithm. Combining the approaches of MoM and Convolution simplifies the evaluation of normalizing constants and leads to large computational savings with respect to the recursive structure originally proposed for MoM.     

A performance model of compressionless routing in k-ary n-cube networks

Several analytical models of fully adaptive routing in wormhole-routed networks have recently been reported in the literature. All these models, however, have been discussed for routing algorithms with deadlock avoidance. Recent studies have revealed that deadlocks are quite rare in the network, especially when enough routing freedom is provided. Thus, the hardware resources, e.g. virtual channels, dedicated for deadlock avoidance are not utilised most of the time. This consideration has motivated researchers to introduce fully adaptive routing algorithms with deadlock-recovery. This paper proposes a new analytical model to predict message latency in k-ary n-cubes with compressionless routing, a fully adaptive algorithm that uses deadlock-recovery. The proposed model uses results from queueing systems with impatient customers to capture the effects of the timeout mechanism used in this routing algorithm to deal with message deadlock. The validity of the model is demonstrated by comparing results predicted by the analytical model against those obtained through simulation experiments.     

A diffusion approximate solution to the g/g/k queueing system

In the opening sections of this paper we present a diffusion approximate solution to a G/G/k queueing station under steady state condtions. The approximation technique is computationally simple and practical in that it requires knowledge of the means and variances of the interarrival and service time distributions, but not their distributional forms. Several numeric examples attesting to the good quality of the approximation are also included. We then discuss extending the basic approximation technique to account first for a feed back option and then to networks of multiserver queueing stations.     

无线传感器网络中一种抗无线局域网干扰的信道分配机制

无线传感器网络(WSN)易受到与其共享信道的无线局域网(Wifi)干扰,造成通信可靠性及吞吐量下降.当具有不同优先级的多个WSN受到Wifi干扰时,如何按优先级分配信道,并兼顾整体通信可靠性及吞吐量是一个重要问题.针对该问题,作者提出了一种抗Wifi干扰的信道分配机制EasiCAP(Channel Allocation for wireless sensor networks with Priority).该机制利用基于干扰强度和活跃比率的干扰特征模型(External Interference Model,EIM)度量WSN中各信道的Wifi干扰;同时,采用以接收端为中心的模型(Internal Interference Model,IIM)度量各WSN之间的干扰.然后,各WSN根据EIM和IIM度量的结果,采用局部化贪婪信道分配算法独立、实时地选择信道,通过保持信道、切换信道及抢占信道操作实现按优先级分配信道,并尽可能降低所有网络所受干扰之和.实际测量和仿真结果表明,EasiCAP可为各WSN提供与其优先级相对应的通信可靠性和吞吐量;而且该机制下的平均通信可靠性及吞吐量也比现有方法高.此外,EasiCAP未带来过大的开销.     

PSO-based algorithm for home care worker scheduling in the UK

This paper presents the novel application of a collaborative population-based meta-heuristic technique called Particle Swarm Optimization (PSO) to the scheduling of home care workers. The technique is applied to a genuine situation arising in the UK, where the provision of community care service is a responsibility of the local authorities. Within this provision, optimization routes for each care worker are determined in order to minimize the distance traveled providing that the capacity and service time window constraints are not violated. The objectives of this paper are twofold; first to exploit a systematic approach to improve the existing schedule of home care workers, second to develop the methodology to enable the continuous PSO algorithm to be efficiently applied to this type of problem and all classes of similar problems. For this problem, a particle is defined as a multi-dimensional point in space which represents the corresponding care activities and assignment priority. The Heuristic Assignment scheme is specially designed to transform the continuous PSO algorithm to the discrete job schedule. The Earliest Start Time Priority with Minimum Distance Assignment (ESTPMDA) technique is developed for generating an initial solution which guides the search direction of the particle. Local improvement procedures (LIP), i.e. insertion and swap, are embedded in the PSO algorithm in order to further improve the solution quality. The proposed methodology is implemented, tested, and compared with existing solutions on a variety of real problem instances.     

Performance analysis of a virtual circuit protocol with local control

In many current packet switching networks, packets are transmitted sequentially by means of so-called virtual circuits. In this paper, a queueing network model for a virtual circuit with local buffer size constraints and blocking at the intermediate nodes is derived and analysed. The analysis is based on the decomposition of the fixed virtual route into its individual links. Each hop is modelled as a closed BCMP queueing network. The introduction of the steady-state blocking probability p as the characterizing parameter of the interface between two consecutive links permits the recursive computation of important design parameters, such as throughput, utilization, and mean transmission time. A modified rejection probability, introduced to describe the behaviour at the interface more accurately, supports the previous results.     

A hybrid method for performance analysis of G/G/m queueing networks

Open queueing networks are useful for the performance analysis of numerous real systems. Since exact results exist only for a limited class of networks, decomposition methods have been extensively used for approximate analysis of general networks. This procedure is based on several approximation steps. Successive approximations made in this approach can lead to a considerable error in the output. In particular, there are no general accurate formulas for computing the mean waiting time and the inter-departure variance in general multiple-server queues. This causes the results from decomposition methods when applied to G/G/m queueing networks to be very approximative and to significantly deviate from actual performance values. We suggest substituting some approximate formulae by low-cost simulation estimates in order to obtain more accurate results when benefiting from the speed of an analytical method. Numerical experiments are presented to show that the proposed approach provides improved performance.