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Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

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Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

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

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

  4. Research Contributions • Definition of Closed QNs with Burstiness (Superset) • Service times are Markovian Arrival Processes (MAPs) • Generalization of PH-Type distributions • MAP Queueing Networks • State-Space Explosion • Transformation: Linear Reduction (LR) of state space • 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

  5. MAP Queueing NetworksModel Definition

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

  7. Markovian Arrival Processes (MAPs) • MAP model both distribution (e.g., high-CV) and burstiness • 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

  8. Example MAP Queueing Network • 3 queues, Population N • Single MAP server with two phases 1-p1-p2 Station 2 p1 M Station 1 M MAP Station 3 FAST p2 SLOW G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

  10. MAP Queueing NetworksDimensionality Reduction

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

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

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

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

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

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

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

  18. MAP Queueing NetworksBound Analysis

  19. Exact Characterization • Necessary conditions of equilibrium (12 equation types) • Example 1: population constraint Conditional Queue-Length 1 Conditional Queue-Length 3 Conditional Queue-Length 2 ? ? ? 1 0 2 1 1 0 cond cond cond Q1 Q3 Q2 = N + + G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

  21. Summary of Linear Reduction • Computational complexity scales linearly with population • Many equations between conditional queue-lengths G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

  22. 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 F(x) G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

  24. Random Validation Methodology • 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 timeR • Error function =worst case relative error G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

  26. 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/ • Supported by NSF grants ITR-0428330 and CNS-0720699 G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

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

  28. References • [HotMetrics] Giuliano Casale, Ningfang Mi, Lucy Cherkasova, Evgenia Smirni: How to Parameterize Models with Bursty Workloads. To be presented at 1st HotMetrics Worshop (6th June 2008), Annapolis, MD, US. • [KPC-Toolbox] Giuliano Casale, Eddy Z. Zhang, Evgenia Smirni. KPC-Toolbox: 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 Characterization. USENIX Annual Technical Conference, General Track 2006: 97-102 G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness

  29. Applicability to Real Workloads 30 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 SIGMETRICS 2008

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