Bound Analysis of Closed Queueing Networks with Workload Burstiness Giuliano Casale Ningfang Mi...

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


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



Model Definition

Markovian Arrival Processes (MAPs)

Hyper-exponential: samples independent of past history

Two-phase MAP with burstiness (high-CV+correlations)


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


0.50.5

0.50.5

FASTFAST

SLOWSLOW


FASTFAST

SLOWSLOW

FASTFAST

SLOWSLOW

FASTFAST

SLOWSLOW

FASTFAST

SLOWSLOW


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]


3 queues, Population N

Single MAP server with two phases

Example MAP Queueing Network


MM

MMStation 1Station 1

Station 2Station 2

Station 3Station 3

pp11

pp22

MAPMAP

1-p1-p11-p-p22

FASTFAST

SLOWSLOW

Roadmap Bound Derivation


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)


Dimensionality Reduction

State Space Dimensionality


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)


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


Partition 1FASTFAST phase

Partition 1SLOWSLOW phase

Decomposition performance

Decomposition unable to approximate MAP QN performance


Busy Conditioning


200200

110110

020020

011011

101101 002002 200200

110110

020020

011011

101101 002002

Overlapping States = Not Lumping/Decomposition

Station 3 busyFASTFAST phase


Station 3 busySLOWSLOW phase




More information available to partitions: assume a station is busybusy

No longer a product-form network: we lose scalability!

Idle Conditioning


200200

110110

020020

011011

101101 002002 200200

110110

020020

011011

101101 002002

Station 2 idleFASTFAST phase



Station 1 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


011011

101101 002002200200

110110

020020

200200

110110

020020

011011

101101 002002


00 11

Conditional Queue-Length Station 1

?

00 11


?

11 22


?

Population N=2

Loss of information to reduce dimension

Number of states scales well with model size


Bound Analysis

Necessary conditionsNecessary conditions of equilibrium (12 equation types)

Example 1:Example 1: population constraint

Exact Characterization


11 22?

Q3+ + = NQ2Q1condcondcond

Conditional Queue-Length 3

00 11?


00 11?


Example 2: Example 2: Flow Balance Assumption (FBA)

Marginal balanceMarginal balance: fine grain probabilistic version of FBA

XIN(k) = XOUT (k)

Exact Characterization


XIN XOUT XIN=XOUT

MAPMAPXIN(k)

k jobs

XOUT (k)

XIN(k), XOUT(k) function of conditional queue-lengths

Summary of Linear Reduction


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


F(x)

LR Bounds Example


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 time R

Error function = worst case relative error

LR Bounds: Worst Case Error


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


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


Applicability to Real Workloads

3 queues, 16-phases MAP fitting the Bellcore-Aug89 trace


Bound Analysis of Closed Queueing Networks with Workload Burstiness Giuliano Casale Ningfang Mi...

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