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Decomposition-Based Analysis of Queueing Networks Decomposition-Based Analysis of Queueing Networks Ramin Sadre CTIT Ph.D.-thesis Series No. 06-95 Centre for Telematics and Information Technology University of Twente, P.O. Box 217, NL-7500 AE Enschede ISSN 1381-3617 c Ramin Sadre 2006 ° ISBN 90-365-2444-X DECOMPOSITION-BASED ANALYSIS OF QUEUEING NETWORKS DISSERTATION to obtain the degree of doctor at the University of Twente, on the authority of the rector magnificus, prof. dr. W.H.M. Zijm, on account of the decision of the graduation committee, to be publicly defended on Wednesday, January 10th, 2007 at 13.15 by Ramin Sadre born on June 13th, 1974 in Tehran (Iran) This dissertation has been approved by the promotor, prof. dr. ir. B.R. Haverkort. Abstract Model-based numerical analysis is an important branch of the model-based per- formance evaluation. Especially state-oriented formalisms and methods based on Markovian processes, like stochastic Petri nets and Markov chains, have been suc- cessfully adopted because they are mathematically well understood and allow the intuitive modeling of many processes of the real world. However, these methods are sensitive to the well-known phenomenon called state space explosion. One way to handle this problem is the decomposition approach. In this thesis, we present a decomposition framework for the analysis of a fairly general class of open and closed queueing networks. The decomposition is done at queueing station level, i.e., the queueing stations are independently analyzed. Dur- ing the analysis, traffic descriptors are exchanged between the stations, representing the streams of jobs flowing between them. Networks with feedback are analyzed using a fixed-point iteration. Based on the decomposition framework, we have developed an efficient analysis method called FiFiQueues. The method supports open queueing networks with infinite and finite capacity queues. We present the method and discuss its fixed- point behavior, as well as various extensions to the original algorithm. We also show how the method can be applied in the efficient analysis of closed queueing networks. The service processes in FiFiQueues can be arbitrary phase-type renewal pro- cesses. Over the last decade, traffic measurements have shown the presence of prop- erties such as self-similarity and long-range dependency in network traffic. It has been shown that this can be explained by the heavy-tailedness of many of the in- volved distributions. We describe how hyper-exponential distributions, which are a special case of phase-type distributions, can be fitted to heavy-tail distributed measurement data. FiFiQueues’ traffic descriptors are based on the first and second moment of the inter-arrival time and, hence, cannot account for correlations in the traffic streams. To approach this problem, we also introduce MAPs as traffic descriptors. Since a queueing network analysis based on MAP traffic descriptors suffers under the state space explosion problem, we present five different MAP reduction methods in order to decrease the effect of the state space explosion. v Contents 1 Introduction 1 2 Analysis of General Queueing Networks 5 2.1 Thedecompositionapproach. 6 2.2 Application to queueing networks . 10 2.3 Analysis of decomposed networks . 12 2.4 A decomposition framework for queueing networks . 17 2.5 Summaryandconclusions . 18 3 MAPs, PH Renewal Processes and QBD Processes 19 3.1 MarkovianArrivalProcesses(MAPs) . 19 3.2 Phase-type(PH)renewalprocesses . 22 3.3 InfiniteQBDs ............................... 23 3.4 FiniteQBDs................................ 28 3.5 Performanceofthesolutionmethods . 31 3.6 Summaryandconclusions . 33 4 Fitting of long-tailed traffic traces to HEDs 35 4.1 Heavy-tailed distributions and hyper-exponential distributions . 36 4.2 Approximation of HTDs with HEDs . 38 4.3 EM-fittingwithHEDs .......................... 39 4.4 Enhancements of the EM-fitting for traffic traces. 42 4.5 EM-algorithmwithstratification. 44 4.6 Applicationandvalidation . 45 4.7 Summaryandconclusions . 52 5 FiFiQueues 53 5.1 Jackson queueing networks . 53 vii viii CONTENTS 5.2 Whitt’s Queueing Network Analyzer . 57 5.3 FiFiQueues ................................ 61 5.4 Existenceofthefixedpoint . 71 5.5 Traffic descriptors with m moments................... 80 5.6 PH renewal processes as traffic descriptors? . 86 5.7 Summaryandconclusions . 87 6 Performance of FiFiQueues 89 6.1 EvaluationofFiFiQueues . 89 6.2 Performance evaluation of a web server . 94 6.3 FiFiQueues with three-moment descriptors . 102 6.4 Summaryandconclusions . .112 7 Closed Queueing Networks 113 7.1 Existinganalysismethods . .114 7.2 Analysis of closed QNs by decomposition . 116 7.3 ImplementationwithFiFiQueues . 117 7.4 Validation .................................122 7.5 Summaryandconclusions . .130 8 Tool Support 133 8.1 TheFiFiQueuesnetworkdesigner . .133 8.2 Integrating FiFiQueues in the M¨obius framework . 135 8.3 Steady-state simulation of queueing networks . 136 8.4 Summaryandconclusions . .144 9 MAP-Based Traffic Descriptors 145 9.1 MAPsastrafficdescriptorsinQNs . .146 9.2 Reduction by removing equivalent states . 150 9.3 Finite output process approximation for MAP MAP 1 queues . 152 | | 9.4 Approximation for MAP MAP 1( K)queues ..............154 | | | 9.5 Approximation of processes with positive, exponentially decreasing autocovariance ..............................155 9.6 FittingPHrenewalprocessestoMAPs . .157 9.7 Summaryandconclusions . .158 10 Performance of MAP-Based Traffic Descriptors 159 CONTENTS ix 10.1 Reduction by removing equivalent states . 159 10.2 Approximations for MAP MAP 1queues ................162 | | 10.3 Approximations for MAP MAP 1 K queues. .170 | | | 10.4 Approximation of processes with positive, exponentially decreasing autocovariance ..............................173 10.5 Summaryandconclusions . .178 11 Conclusion 181 A The Power Spectrum of a MAP 183 B The Simulator 185 B.1 Introduction................................185 B.2 Overview..................................186 B.3 Example..................................191 C Publications by the Author 195 Bibliography 197 Index 207 x CONTENTS Chapter 1 Introduction From the beginning of modern telecommunication systems, their performance eval- uation has been of particular interest. The complexity and costs of the involved technologies and the rapidly increasing demand made it necessary to carefully plan communication networks from a technological and an economical point of view. As a consequence, the question regarding the optimal amount of resources to fulfill given or future demands arose. Basically, performance evaluation of a system should give an answer to this question. Of course, the evaluation has to follow the same economical considerations as the evaluated object. Hence, the most direct approach, the measurement, is usu- ally not an option since it requires an installed and running system. Model-based approaches try to circumvent the problem by applying the evaluation methods to a virtual version of the object of interest, the model. Although such models can have any arbitrary complexity to reflect every detail of the real thing, their practical usefulness is limited by the available methods and tools to construct and evalu- ate them. Simulation approaches accept a wide range of model classes but often result in long computation time if information related to rare events is needed. Ap- propriate actions have to be taken in order to deal with such situations [7, 112]. For the model-based numerical analysis, especially methods based on Markovian processes have been successfully adopted. Analysis methods like discrete-time, resp. continuous-time Markov chains (DTMCs, resp. CTMCs) extend the classical deter- ministic state machines by the concept of state probabilities. This state-oriented approach allows the intuitive modeling of many processes of the real world, espe- cially if the design behind those processes also is state-based (which is, for example, the case for network protocols [89]). However, these methods are sensitive to the well-known phenomenon called state space explosion. For any realistic system, a direct naive representation of the system behavior by a model leads to a state space that is far too large for any analytical treatment. One way to handle this problem is to avoid the state-oriented approach, for example by flow-oriented analysis [3] or simulation [53]. This will, however, 1 2 1 Introduction not be the topic of this thesis. Instead, we will follow a decomposition approach which aims to reduce the number of states in a model. The decomposition approach achieves this by dividing a large model into smaller submodels which can then be independently analyzed. A fixed-point iteration scheme is used to combine the solu- tions of the submodels in order to compute an approximate solution for the original, overall model. The decomposition and the iteration-based solution approach were successfully applied to various model classes in the past. However, most attempts bore monolithic tool implementations that could not be easily extended or combined with other formalisms or solution methods. As a reaction, tools like SMART [22] have been developed that offer multiple modeling formalisms and permit the easy integration of new solution methods. The M¨obius framework and tool [23, 26] can be considered as the next step of this development; it provides an infrastructure for formalisms and solvers with minimal assumptions about their functionality, the supported model classes, and how submodels are combined. But the reduced com- putational complexity is not
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