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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 MaryDepartment of Computer Science
Williamsburg, Virginia
ACM SIGMETRICS 2008
Annapolis, June 3, 2008
Integrate in queueing networks service time burstiness Long peaks (“burstsbursts”) 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 2SIGMETRICS 2008
Classes of Closed Queueing Networks
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 3SIGMETRICS 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) Service times are Markovian Arrival Processes Markovian Arrival Processes (MAPs)
Generalization of PH-Type distributions
MAP Queueing Networks
2. State-Space Explosion Transformation: Linear ReductionLinear 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 4SIGMETRICS 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 6SIGMETRICS 2008
0.50.5
0.50.5
FASTFAST
SLOWSLOW
Job 1 completionJob 1 completion
0.50.5
0.50.5
0.50.5
0.50.5
Job 2 completionJob 2 completion
0.50.5
0.50.5
FASTFAST
SLOWSLOW
Job 1 completionJob 1 completion
FASTFAST
SLOWSLOW
FASTFAST
SLOWSLOW
FASTFAST
SLOWSLOW
FASTFAST
SLOWSLOW
Job 2 completionJob 2 completion
1
1
2 3
23
Markovian Arrival Processes (MAPs)
MAP model both distribution (e.g., high-CV) and burstiness
Generalization of the method of phasesmethod 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 7SIGMETRICS 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 8SIGMETRICS 2008
MM
MMStation 1Station 1
Station 2Station 2
Station 3Station 3
pp11
pp22
MAPMAP
1-p1-p11-p-p22
FASTFAST
SLOWSLOW
Roadmap Bound Derivation
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 9SIGMETRICS 2008
Case Study
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200Job Population
Thro
ughp
ut LR Upper
ExactThroughputLR Lower
Dimensionality ReductionDimensionality Reduction Bound AnalysisBound 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 11SIGMETRICS 2008
JOB DistributionJOB Distribution
= Job Completions
202000
111100
020200
010111
101011 000022
Station 3 empty
Station 3 1 job
Station 3 2 jobs Queues States
3 ~104
5 ~106
10 ~1012
State Space ExplosionState Space Explosion
Population N=100Population N=2
MAP QN State Space (Markov chain)
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 12SIGMETRICS 2008
Station 3Station 3MAPMAP
FASTFAST
SLOWSLOW
200200
110110
020020
011011
101101 002 002002
011011
020020
110110
101101 200200
FASTFAST phase SLOWSLOW phase
Disjoint partitions solved as separate product-form networks
Scalability thanks to MVA
200200
110110
020020
011011
101101 002002 200200
110110
020020
011011
101101 002002
Decomposition-Aggregation
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 13SIGMETRICS 2008
Partition 1FASTFAST phase
Partition 1SLOWSLOW 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 14SIGMETRICS 2008
Busy Conditioning
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 15SIGMETRICS 2008
200200
110110
020020
011011
101101 002002 200200
110110
020020
011011
101101 002002
Overlapping States = Not Lumping/Decomposition
Station 3 busyFASTFAST phase
Station 1 busyFASTFAST phase
Station 3 busySLOWSLOW phase
Station 1 busySLOWSLOW phase
Station 2 busySLOWSLOW phase
Station 2 busyFASTFAST phase
More information available to partitions: assume a station is busybusy
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 16SIGMETRICS 2008
200200
110110
020020
011011
101101 002002 200200
110110
020020
011011
101101 002002
Station 2 idleFASTFAST phase
Station 3 idleFASTFAST phase
Station 1 idleFASTFAST phase
Station 1 idleSLOWSLOW phase
Station 2 idleSLOWSLOW phase
Station 3 idleSLOWSLOW phase
How do we restore scalability? How do we use the new information?
Alternatively assume a certain station is idleidle
Linear Reduction (LR) Transformation
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 17SIGMETRICS 2008
011011
101101 002002200200
110110
020020
200200
110110
020020
011011
101101 002002
Station 3 busyFASTFAST phase
00 11
Conditional Queue-Length Station 1
?
00 11
Conditional Queue-Length Station 2
?
11 22
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 conditionsNecessary 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 19SIGMETRICS 2008
11 22?
Q3+ + = NQ2Q1condcondcond
Conditional Queue-Length 3
00 11?
Conditional Queue-Length 2
00 11?
Conditional Queue-Length 1
Example 2: Example 2: Flow Balance Assumption (FBA)
Marginal balanceMarginal balance: fine grain probabilistic version of FBA
XIN(k) = XOUT (k)
Exact Characterization
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 20SIGMETRICS 2008
XIN XOUT XIN=XOUT
MAPMAPXIN(k)
k jobs
XOUT (k)
XIN(k), XOUT(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 21SIGMETRICS 2008
Queues Jobs Num states LR states LR eqs
3 100 10,302 3,300 2,709
5 100 9,196,252 7,422 5,379
10 100 8,526,843,022,542 24,937 16,044
Computational complexity scales linearlylinearly 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 22SIGMETRICS 2008
F(x)
LR Bounds Example
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 23SIGMETRICS 2008
Random Validation Methodology
G.Casale, N.Mi, E.Smirni. Bound Analysis of Closed Queueing Networks with Workload Burstiness 24SIGMETRICS 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 25SIGMETRICS 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: 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 26SIGMETRICS 2008
http://www.cs.wm.edu/MAPQN/
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 28SIGMETRICS 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 30SIGMETRICS 2008