1 KPC-Toolbox Demonstration Eddy Zheng Zhang, Giuliano Casale, Evgenia Smirni Computer Science Department College of William & Mary

2 What is KPC-Toolbox for?  KPC-Toolbox: MATLAB toolbox Workload Traces  Markovian Arrival Process (MAP)  Why MAP? Very versatile High variabilitytemporal dependence Time Series High variability & temporal dependence in Time Series Easily incorporated into queuing models  Friendly Interface Departure from previous Markovian fitting tools Fit the automatically (no manual tuning)

3 User Interface  Requirement: Matlab installed  Input A trace of inter-event times Or a file that already stores the statistics of the trace  E.g., a file stores the moments, autocorrelations and etc  Help Information Type “ help FunctionName ”,  E.g., “ help map_kpcfit ” Website Keeps Up-To-Date Tool version 

4 A Simple Example of MAP‏  Two state jumps b a c d D1 = D0 = -b-d -a-c Time: a b c d I1 I2 I3 Background Jumps Jumps With Arrivals Arrivals:

5 Challenges  How large is the MAP? MAP(n): determine n?  Which trace descriptors are important? Literature: Moments of interval times, lag-1 autocorrelation long range dependent But, for long range dependent traces? temporal dependence  Need temporal dependence descriptors  MAP Parameterization Construct MAP(n) with matrices D0 and D1 (2n 2 – n entries)

6 Example: Important Trace Statistics 1 2 First, second, third moment and lag-1 autocorrelation accurately fit The queuing prediction ability is not satisfactory! Seagate Web Server Trace Queue Prediction, 80% Utilization Fit With MAP(2)

7 Example: Higher Order Statistics Matter Much Better Results! Queuing Prediction, 80% Utilization ……… Fit with MAP(16) A grid of joint moments and a sequence of autocorrelations fitted, E[X i X i+k X i+k+h ] Seagate Web Server Trace

8  Higher Order Correlations V.S. Moments Correlations capture sequence in the time series Correlations are very important  Summary: first three moments Matching up to the first three moments is sufficient higher order correlations Matching higher order correlations with priority Fitting Guidelines Ref: "KPC-Toolbox: Simple Yet Effective Trace Fitting Using Markovian Arrival Processes", G. Casale, E.Z. Zhang, E. Smirni, to appear in QEST ’ 08

9 Challenge (1): Determine MAP Size  Definition: k ACF coefficient  lag-k ACF coefficient  MAP(n) Property: n Linear Recursive Relationship of n consecutive ACF coeffs  BIC Size Selection: Linear regression model on estimated ACF coeffs BIC value assesses goodness of model size MAP(8) MAP(16) MAP(32)

10 Challenge (2): Trace Descriptor Matching  Kronecker Product Composition (KPC)  KPC Properties: Composition of Statistics Moments are composed from moments of small MAPs  MAP Parameterization by KPC to Match Mean and SCV Exactly Higher order correlations as Close as Possible

11 KPC Tool Overview Trace Extract Statistics Moments ACF Correlations …… Size Selection MAP(2) …… J = log 2 N MAP(2)s MAP(N) Size of MAP N Optimization KPC This work is supported by NSF grants ITR and CNS

12 Thank you!

13  What are higher order correlations? Joint moments of a sequence of inter-arrival times in the time series  Which higher order correlations to fit in KPC? E[X i X i+j X i+j+k ], where i can be arbitrary without loss of generality, and [j,k] chose from a grid of values  E.g., [ ] × [ ] = {[10,10], [10,100], [10,10000], …} Appendix