Bound Analysis of Closed Queueing Networks with Workload Burstiness Giuliano Casale Ningfang Mi Evgenia Smirni College of William and Mary Department of Computer Science Williamsburg, Virginia ACM SIGMETRICS 2008 Annapolis, June 3, 2008

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

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

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

MAP Queueing Networks Model Definition

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 FAST SLOW Job 1 completion Job 2 completion FAST SLOW Job 1 completion FAST SLOW FAST SLOW FAST SLOW FAST SLOW Job 2 completion

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

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

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)

MAP Queueing Networks Dimensionality Reduction

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 Station 3 empty Station 3 1 job Station 3 2 jobs QueuesStates 3 ~ ~ ~10 12 State Space Explosion Population N=100 Population N=2

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 FAST FAST phase SLOW SLOW phase

Disjoint partitions solved as separate product-form networks Scalability thanks to MVA 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

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

Busy Conditioning G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 15 SIGMETRICS 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!

Idle Conditioning G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 16 SIGMETRICS 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

Linear Reduction (LR) Transformation G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 17 SIGMETRICS 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

MAP Queueing Networks Bound Analysis

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 ? Q3Q3 + + = N Q2Q2 Q1Q1 cond Conditional Queue-Length ? Conditional Queue-Length ? Conditional Queue-Length 1

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

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 ,3023,300 2, ,196,2527,4225, ,526,843,022,54224,93716,044 linearly Computational complexity scales linearly with population Many equations between conditional queue-lengths

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)

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

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

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

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: Supported by NSF grants ITR and CNS G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 26 SIGMETRICS 2008

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 [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): (2007) [Usenix06] Alma Riska, Erik Riedel. Disk Drive Level Workload Characteriz ation. USENIX Annual Technical Conference, General Track 2006: G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 28 SIGMETRICS 2008

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