Queueing network models of packet switching networks part 2: Networks with population size constraints

An overview of the application of product-form queueing networks to the performance analysis of store-and-forward packet communication networks is presented. Queueing network models with closed chains and other forms of population size constraints are considered. The modeling and analysis of networks with window flow controlled virtual channels are described first. Models for the performance analysis of strategies for nodal buffer management and permit-oriented network congestion control are then presented.     

Better Capacity Planning with Queueing Network Models

Capacity planning is the key to orderly growth, and performance modeling is the key to orderly capacity planning. Queueing network models have emerged as the preferred performance modeling technology for many capacity planning applications. This article familiarizes data center operations managers with the capabilities of queueing network models and assists performance analysts in using this planning tool by developing and modifying a hypothetical system model using a typical software package for queueing network modeling.     

Better Capacity Planning with Queueing Network Models

Capacity planning is the key to orderly growth, and performance modeling is the key to orderly capacity planning. Queueing network models have emerged as the preferred performance modeling technology for many capacity planning applications. This article familiarizes data center operations managers with the capabilities of queueing network models and assists performance analysts in using this planning tool by developing and modifying a hypothetical system model using a typical software package for queueing network modeling.     

Performance Analysis of a Client–Server System Using Queueing Networks: A Case Study

Queueing networks have been widely used to evaluation performance of mainframe computer systems. In contrast, few results have been reported for modern open systems, so it was not clear whether queueing networks are useful for modern systems or not. We think this situation has partly been due to lack of handy evaluation tools. This paper presents tow tools that we developed to evaluate open system performance. On is a measuring tool that is capable of accurately obtaining the service times of system resources requested by an application transaction. The other is an estimating tool which calculates various performance measures based on queueing network models. This paper also describes a case study in which the performance of a medium-sized UNIX client–server system (up to 24 clients) is estimated using the tools and these estimates are then compared with experimental results. The estimates closely agree with the measured results and are accurate enough for practical applications. Based on this, we conclude that queueing network models are also useful for modern systems.     

Integrated Performance Models for Distributed Processing in Computer Communication Networks

This paper deals with the analysis of large-scale closed queueing network (QN) models which are used for the performance analysis of computer communication networks (CCN's). The computer systems are interconnected by a wide-area network. Users accessing local/remote computers are affected by the contention (queueing delays) at the computer systems and the communication subnet. The computational cost of analyzing such models increases exponentially with the number of user classes (chains), even when the QN is tractable (product-form). In fact, the submodels of the integrated model are generally not product-form, e.g., due to blocking at computer systems (multiprogramming level constraints) and in the communication subnet (window flow control constraints). Two approximate solution methods are proposed in this paper to analyze the integrated QN model. Both methods use decomposition and iterative techniques to exploit the structure of the QN model such that computational cost is proportional to the number of chains. The accuracy of the solution methods is validated against each other and simulation. The model is used to study the effect that channel capacity assignments, window sizes for congestion control, and routing have on system performance.     

Cycle time distribution of cyclic networks with blocking

An algorithm is provided to compute the cycle time distribution for cyclic closed exponential queueing networks with N finite capacity nodes and blocking. The blocking phenomenon is due to the nodes with finite capacity queues which can be used to represent systems with limited resources. Moreover, a recursive algorithm is provided to evaluate the cycle time moments, and a closed form solution, both in terms of distribution and moments of the cycle time, is provided for the model under special constraints.     

基于VOQ交换结构的仲裁算法分析

信头阻塞（HOL）限制了采用FIFO输入队列交换机的吞吐率，而使用虚输出队列（VOQ）技术可以完全消除HOL阻塞，给出了VOQ交换机模型，提出了对VOQ仲裁算法的分类方法和评价指标，分析了基于VOQ交换结构的MSM和MWM近似算法，并对其性能进行了分析比较。     

MRE hierarchical decomposition of general queueing network models

Summary The principle of Minimum Relative Entropy (MRE), given fully decomposable subset and aggregate mean queue length, utilisation and flow-balance constraints, is used in conjunction with asymptotic connections to infinite capacity queues, to derive new analytic approximations for the conditional and marginal state probabilities of single class general closed queueing network models (QNMs) in the context of a multilevel variable aggregation scheme. The concept of subparallelism is applied to preserve the flow conservation and a universal MRE hierarchical decomposition algorithm is proposed for the approximate analysis of arbitrary closed queueing networks with single server queues and general service-times. Heuristic criteria towards an optimal coupling of network's units at each level of aggregation are suggested. As an illustration, the MRE algorithm is implemented iteratively by using the Generalised Exponential (GE) distributional model to approximate the service and asymptotic flow processes in the network. This algorithm captures the exact solution of separable queueing networks, while for general queueing networks it compares favourably against exact solutions and known approximations.This work is sponsored by the Science and Engineering Research Council (SERC), UK, under grant GR/F29271     

On the nonconcavity of throughput in certain closed queueing networks

Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance.We show that for closed queueing networks of m arbitrarily connected single server queues with n customers, throughput, as a function of a scaled, constrained workload, is not concave. In fact, the function appears to be strictly quasiconcave. There is a constraint on the total workload that must be allocated among the servers in the network. However, for closed networks of two single server queues, we prove that our scaled throughput is concave when there are two customers in the network and strictly quasi-concave when there are more than two customers. The mathematical properties of both the scaled throughput and reciprocal throughput are demonstrated graphically for closed networks of two and three single server queues.     

Approximate analysis of product-form type queueing networks with blocking and deadlock

A numerical procedure for analyzing exactly closed exponential queueing networks with finite queues is presented first. Due to the finiteness of these queues, blocking and deadlock may occur. Deadlocks are assumed to be detected and resolved instantaneously. The numerical procedure is then incorporated in an approximation algorithm for analyzing closed exponential queueing networks of the product-form type, in which some of the queues are finite. These finite queues are assumed to be linked together to form a single subnetwork. The approximation algorithm is based on a variant of Norton's theorem. Comparisons between the approximate results and exact numerical results were carried out and the relative error was observed to be small.     

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.     

Analysis of finite buffered multistage combining networks

Analyzing the performance of finite buffered multistage networks has been considered a difficult task because of dynamic blocking effects due to finite sized buffers. With the multistage networks enhanced with combining capability, in which multiple requests directed to a shared location combine together to form a single request to be forwarded, the analysis becomes even more difficult due to the interactions between combining probability and queuing delay. Performance bounds for combining networks are known under two extreme assumptions: infinite combining queues and saturated finite combining queues. We analyze multistage combining networks with the consideration of blocking due to finite combining queues. Our analysis provides iterative solutions for combining probability, blocking probability, and queuing delay     

基于分层排队网络模型的分布式系统性能评价

关于并发或分布式系统的性台匕评价是一个广泛研究的课题,提供有效的数学理论工具、直观的模型描述方法和有效的模型分析方法,是系统性能评价所面临的关键问题。传统的性能模型——排队网络模型已很难分析这样复杂的系统,分层排队网络（Layered Queueing Net,LQN）模型是排队网络模型的扩展,可以用来分析相互依赖任务间的冲突。介绍了分层排队网络模型的原理及研究现状,并以DBMS为例,建立了DBMS的分层排队网络模型。     

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.     

Nonstationary analysis of circuit-switched communication networks

Circuit-switched communication networks have been analyzed extensively in the stationary case, i.e. where the arrival and/or service rates are independent of time. In this paper, we study a circuit-switched network where the rates of external arrivals at the network are time-dependent functions. The circuit-switched network is modelled as a nonstationary queueing network with population constraints, which is analyzed approximately in order to obtain the blocking probability functions. Using this method we model two circuit-switched networks, namely, a traffic-groomed tandem optical network and a single-orbit LEO satellite network.     

Queueing network software for systems modelling

Several requirements are placed on queueing models of computer systems. These include credibility, accuracy, timeliness and cost. Modelling software can have critical impact on all of these requirements. We survey the characteristics of major pieces of queueing software. Based on this survey we synthesize a set of design objectives for queueing software. Finally, we discuss our own queueing network software, the Research Queueing Package (RESQ), in light of these objectives.     

Synchronous versus asynchronous operation of a packet switch with combined input and output queueing

A single-stage non-blocking N × N packet switch with combined input and output queueing is considere. The limited queueing at the output ports partially resolves output port contention. Overflow at the output queues is prevented by employment of a backpressure mechanism and additional queueing at the input ports. This paper investigates the performance of the switch under two different modes of operation: asynchronous and synchronous or slotted. For the purpose of comparison a switch model is developed. Assuming Poisson packet arrivals, several performance measures are obtained analytically. These include the distribution of the delay through the switch, the input queue length distribution, packet losses at the inputs in the case of finite input queues, and the maximum switch throughput. The results obtained demonstrate a slight performance advantage of asynchronous over synchronous operation. However, the maximum switch throughput is the same for both modes of operation.     

Selecting Capacities in Computer Communication Systems

This paper considers the selection of capacities, in two classes of open queueing network models of computer communication systems: 1) local-balanced queueing networks with multiple classes of customers and 2) the Reiser-Kobayashi diffusion approximation model. The problem of selecting optimal bandwidths for communication lines and switches and computing machinery is difficult due to 1) the economy of scale exhibited by components (i.e., the bandwidth per dollar increases with the total cost of the component) and 2) the discrete nature of computer/communication components (for instance it is possible to lease communication lines with 2400-Bd or 4800-Bd band-widths but not one with a 2401.3-Bd bandwidth). This paper develops aun algorithm to select servers taking both of these characteristics of server costs into account. The contribution of this paper is to discuss 1) the optimization of more general network models with 2) the consideration of more realistic tariffs and 3) more constraints on network behavior than were previously analyzed.     

Performance evaluation of multicast communication in packet-switched networks

In this paper we describe the theoretical background and practical application of QNA-MC (queueing network analyser supporting multicast), a tool for the analytical evaluation of multicast protocols. QNA-MC is based on the QNA method, which (approximately) analyses open networks of GI|G|m queues. In contrast to standard QNA, QNA-MC allows for the specification and evaluation of multicast routes. As in real multicast communication, packets leaving a particular node can be copied and deterministically routed to several other nodes. In order to analyse such queueing networks, QNA-MC converts the multicast routes to a suitable input for standard QNA. From the results delivered by QNA, QNA-MC then derives several performance measures for multicast streams in the network. A validation of QNA-MC, via a comparison to simulation results, shows that QNA-MC yields very good results. Finally, we give a detailed application example by evaluating different multicast routing algorithms for a realistic video conferencing scenario in the European MBONE.     

Simulation of queueing network with time varying arrival rates

Queueing models are important in the analysis of many aspects of system or human behaviour. In such widely divergent fields as administrative processes, health services, traffic control, production control, computer communication networks, time sharing, and architecture, queueing analysis is indisposable for better understanding and ultimately synthesis of the system.However straight mathematical analysis is difficult, as the difference equations describing a queueing network are non-linear and of infinite dimensional. Due to this difficulty, time varying cases are seldomly included in the study of analytic solutions of queueing systems. In existing literature, the mean arrival rates are assumed independent of time. This assumption is of dubious validity as the arrival rate in actual operations varies with a twenty four hours period. While one can assume further that at every instant in time, the equilibrium condition for constant arrival rate is reached for the current arrival rate, it has never been verified that such is the case.An alternative approach is to reduce the queueing problem to finite dimensional and to simulate it on computer as is proposed in the present paper. The basic assumptions are (1) time-varying random arrival rate with Poisson statistics and (2) exponentially distributed service time. An algorithm and flow diagram for doing this is given. With the simulation techniques developed in this paper, we can study the dynamics of varying arrival rate. Simulation of a simple network shows that as the arrival rate approaches capacity, the divergence between simulated results and previous analytical results for the equilibrium condition become increasingly large. A conclusion is therefore that except for systems with very light loading, analytical results based on constant arrival rate are not accurate. Simulation is necessary if one is to obtain a true picture of the system.