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Bound Analysis of Closed Queueing Networks with Workload Burstiness Giuliano Casale Ningfang Mi Evgenia Smirni {casale,ningfang,esmirni}@cs.wm.edu College of William and Mary Department of Computer Science Williamsburg, Virginia ACM SIGMETRICS 2008 Annapolis, June 3, 2008

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Integrate in queueing networks service time burstiness bursts Long peaks (bursts) of consecutively large requests Real workloads often characterized by burstiness Seagate (disks, [Usenix06,Perf07] ),HPLabs (multi-tier, [HotMetrics] ) Workload Burstiness G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 2 SIGMETRICS 2008

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Classes of Closed Queueing Networks G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 3 SIGMETRICS 2008 Product-Form Networks High Service Variability Networks (GI) Queueing Networks with Burstiness (G) BCMP assumptions Exact Solution: MVA General Independent Service/FCFS Approximations: AMVA, Decomposition No prior formalization Can analyze also GI/Product-Form Burstiness: High-variability and correlation of service times

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Research Contributions 1.Definition of Closed QNs with Burstiness (Superset) Markovian Arrival Processes Service times are Markovian Arrival Processes (MAPs) Generalization of PH-Type distributions MAP Queueing Networks 2.State-Space Explosion Linear Reduction Transformation: Linear Reduction (LR) of state space 3.Linear Reduction Bounds LR of state space + Linear Programming Mean error 2% on random models G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 4 SIGMETRICS 2008

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MAP Queueing Networks Model Definition

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Markovian Arrival Processes (MAPs) Hyper-exponential: samples independent of past history Two-phase MAP with burstiness (high-CV+correlations) G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 6 SIGMETRICS 2008 0.5 0.5 FAST SLOW Job 1 completion 0.5 0.5 0.5 0.5 Job 2 completion 0.5 0.5 FAST SLOW Job 1 completion FAST SLOW FAST SLOW FAST SLOW FAST SLOW Job 2 completion 1 1 2 3 2 3

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Markovian Arrival Processes (MAPs) MAP model both distribution (e.g., high-CV) and burstiness method of phases Generalization of the method of phases Building block: exponential distribution Easy to integrate in Markov chains and queueing models Tools and fitting algorithms KPC-Toolbox: automatic fitting from traces [Demo, QEST08] G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 7 SIGMETRICS 2008

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3 queues, Population N Single MAP server with two phases Example MAP Queueing Network G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 8 SIGMETRICS 2008 M M Station 1 Station 2 Station 3 p1p1p1p1 p2p2p2p2 MAP 1-p 1 -p 2 FAST SLOW

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Roadmap Bound Derivation G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 9 SIGMETRICS 2008 Dimensionality Reduction Bound Analysis Conditioning Transformation Characterization Bounding (Linear Programming)

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MAP Queueing Networks Dimensionality Reduction

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State Space Dimensionality G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 11 SIGMETRICS 2008 JOB Distribution = Job Completions 200 110 020 011 101 002 Station 3 empty Station 3 1 job Station 3 2 jobs QueuesStates 3 ~10 4 5 ~10 6 10 ~10 12 State Space Explosion Population N=100 Population N=2

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MAP QN State Space (Markov chain) G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 12 SIGMETRICS 2008 Station 3 MAP FAST SLOW 200 110 020 011 101 002 002 011 020 110 101 200 FAST FAST phase SLOW SLOW phase

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Disjoint partitions solved as separate product-form networks Scalability thanks to MVA 200 110 020 011 101 002 200 110 020 011 101 002 Decomposition-Aggregation G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 13 SIGMETRICS 2008 Partition 1 FAST FAST phase Partition 1 SLOW SLOW phase

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Decomposition performance Decomposition unable to approximate MAP QN performance G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 14 SIGMETRICS 2008

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Busy Conditioning G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 15 SIGMETRICS 2008 200 110 020 011 101 002 200 110 020 011 101 002 Overlapping States = Not Lumping/Decomposition Station 3 busy FAST FAST phase Station 1 busy FAST FAST phase Station 3 busy SLOW SLOW phase Station 1 busy SLOW SLOW phase Station 2 busy SLOW SLOW phase Station 2 busy FAST FAST phase busy More information available to partitions: assume a station is busy No longer a product-form network: we lose scalability!

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Idle Conditioning G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 16 SIGMETRICS 2008 200 110 020 011 101 002 200 110 020 011 101 002 Station 2 idle FAST FAST phase Station 3 idle FAST FAST phase Station 1 idle FAST FAST phase Station 1 idle SLOW SLOW phase Station 2 idle SLOW SLOW phase Station 3 idle SLOW SLOW phase How do we restore scalability? How do we use the new information? idle Alternatively assume a certain station is idle

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Linear Reduction (LR) Transformation G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 17 SIGMETRICS 2008 011 101 002 200 110 020 200 110 020 011 101 002 Station 3 busy FAST FAST phase 0 1 Conditional Queue-Length Station 1 ? 0 1 Conditional Queue-Length Station 2 ? 1 2 Conditional Queue-Length Station 3 ? Population N=2 Loss of information to reduce dimension Number of states scales well with model size

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MAP Queueing Networks Bound Analysis

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Necessary conditions Necessary conditions of equilibrium (12 equation types) Example 1: Example 1: population constraint Exact Characterization G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 19 SIGMETRICS 2008 1 2 ? Q3Q3 + + = N Q2Q2 Q1Q1 cond Conditional Queue-Length 3 0 1 ? Conditional Queue-Length 2 0 1 ? Conditional Queue-Length 1

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Example 2: Example 2: Flow Balance Assumption (FBA) Marginal balance Marginal balance: fine grain probabilistic version of FBA X IN (k) = X OUT (k) Exact Characterization G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 20 SIGMETRICS 2008 X IN X OUT X IN =X OUT MAP X IN (k) k jobs X OUT (k) X IN (k), X OUT (k) function of conditional queue-lengths

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Summary of Linear Reduction G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 21 SIGMETRICS 2008 QueuesJobsNum statesLR statesLR eqs 3 10010,3023,300 2,709 51009,196,2527,4225,379 101008,526,843,022,54224,93716,044 linearly Computational complexity scales linearly with population Many equations between conditional queue-lengths

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Linear Reduction (LR) bounds Intelligent guess of conditional queue-length probabilities Best guess searched by linear programming Objective function Utilizations Throughput ( Response Time) Mean queue-lengths Linear programming analysis Unknowns: marginal subspace probabilities Constraints: exact characterization LR lower bounds: solve min F(x) subject to constraints LR upper bounds: solve max F(x) subject to constraints G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 22 SIGMETRICS 2008 F(x)

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LR Bounds Example G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 23 SIGMETRICS 2008

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Random Validation Methodology G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 24 SIGMETRICS 2008 Validation on 10,000 random queueing networks Arbitrary routing, three queues Random two-phase MAP distribution and burstiness LR bounds compared to exact for populations 1000 jobs Reference metric: response time R Error function = worst case relative error

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LR Bounds: Worst Case Error G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 25 SIGMETRICS 2008

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Conclusion Major extension of closed QNs to workload burstiness Linear Reduction state-space transformation LR Bounds Future work delay servers/load-dependent MAP service (we have it ) mean-value analysis version (no state space, we almost have it) open queueing networks (not yet) Online resources: http://www.cs.wm.edu/MAPQN/http://www.cs.wm.edu/MAPQN/ Supported by NSF grants ITR-0428330 and CNS-0720699 G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 26 SIGMETRICS 2008

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http://www.cs.wm.edu/MAPQN/

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References [HotMetrics] Giuliano Casale, Ningfang Mi, Lucy Cherkasova, Evgenia Smir ni: How to Parameterize Models with Bursty Workloads. To be presented at 1 st HotMetrics Worshop (6 th June 2008), Annapolis, MD, US. [KPC-Toolbox] Giuliano Casale, Eddy Z. Zhang, Evgenia Smirni. KPC-Toolb ox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes. To be presented at QEST 2008 Conference, St.Malo, France, Sep 2008. [Performance07] Ningfang Mi, Qi Zhang, Alma Riska, Evgenia Smirni, Erik Riedel. Performance impacts of autocorrelated flows in multi-tiered systems. Perform. Eval. 64(9-12): 1082-1101 (2007) [Usenix06] Alma Riska, Erik Riedel. Disk Drive Level Workload Characteriz ation. USENIX Annual Technical Conference, General Track 2006: 97-102 G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 28 SIGMETRICS 2008

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Applicability to Real Workloads 3 queues, 16-phases MAP fitting the Bellcore-Aug89 trace G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 30 SIGMETRICS 2008

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