WWV 2009 1 Analyzing a Proxy Cache Server Performance Model with the Probabilistic Model Checker PRISM Tamás Bérczes 1 , [email protected], Gábor Guta 2 , [email protected]linz.ac.at, Gábor Kusper 3 , [email protected], Wolfgang Schreiner 2 , [email protected]linz.ac.at, János Sztrik 1 , [email protected]1.: Faculty of Informatics, University of Debrecen, Hungary, http://www.inf.unideb.hu 2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, http://www.risc.uni-linz.ac.at 3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu
31
Embed
WWV 20091 Analyzing a Proxy Cache Server Performance Model with the Probabilistic Model Checker PRISM Tamás Bérczes 1, [email protected], Gábor Guta.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
WWV 2009 1
Analyzing a Proxy Cache Server Performance Model with the
1.: Faculty of Informatics, University of Debrecen, Hungary, http://www.inf.unideb.hu2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria,
http://www.risc.uni-linz.ac.at3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu
WWV 2009 2
Motivation
• The two originally distinct areas– qualitative analysis (verification)– quantitative analysis (performance modeling)
• have in the last decade started to converge by the arise of– stochastic/probabilistic model checking.
• One attempt towards this goal is to compare techniques and tools from both communities by concrete application studies.
• The presented paper is aimed at exactly this direction.
WWV 2009 3
Case Study
• We apply PRISM to re-assess some web server performance models with proxy cache servers that have been previously described and analyzed in the literature:– L. P. Slothouber. A Model of Web Server
Performance. Proceedings of the 5th International World Wide Web Conference, 1996.
– I. Bose and H. K. Cheng. Performance Models of a Firm’s Proxy Cache Server. Decision Support Systems, 29:47–57, 2000.
WWV 2009 4
The System Performance Models of a Firm’s Proxy Cache Server
Client
Proxy Cache Server(PCS)
Case A:With probability p the request can be answered by the PCS.
requests a file with rate lambaand with average file size F.
sends back the filein a loop (F > Bxc)
WWV 2009 5
The System Performance Models of a Firm’s Proxy Cache Server
Proxy Cache Server(PCS)
Case B:With probability 1-p the request must be forwardedto a remote web server.
Web Server
requests the file froma remote web serverwith rate (1-p)lambda
sends back the filein a loop (F > Bs)
Client
WWV 2009 6
Derived Constants
• l1 = lambda1 = p * lambda
• l2 = lambda2 = (1-p) lambda
• Let q be the probability, that the Web Server can send the requested file at once.
• q = min{1, Bs / F)
• l2’ = lambda2prime = lambda2 / q
WWV 2009 7
Orig
inal
Mod
el
WWV 2009 8
Original Equation for Response Time
WWV 2009 9
Original Response Time Diagram
WWV 2009 10
PRISM
• A Probabilistic Model Checker, developed at University of Oxford
• Supports 3 models:
1. Discrete-time Markov chain (DTMC)
2. Markov decision processes (MDP)
3. Continuous-time Markov chain (CTMC); we use this one
WWV 2009 11
PRISM
• In PRISM one gives the model by a– Finite state transition system
• qualitative aspects of the system
– Associate rates to the individual state transitions• quantitative aspects of the system
• This network consists of four queues: – one models the Proxy Cache Server– two model the Web server, input/output– one models the loop to download the requested file.
• We have a job-source:– Users generates jobs, rate: lambda.
• We have 5 models together:– module jobs– module PCS– module server_input_queue– module server_output_queue– module client_queue
WWV 2009 15
How to Implement a Queue?
WWV 2009 16
State Variables
• Each model has a counter, which contains the number of request in the represented queue.
• Note: We make no distinction between requests.• Example:
module PCS pxc: [0..IP] init 0; …endmodule
WWV 2009 17
[label] guard -> rate : update;
• Each module has (generally) two state transitions.• One transition (or more) for receiving requests.• One transition (or more) for serving requests.• The first type increases the counter.• The second one decreases it.module PCS pxc: [0..IP] init 0;
• Each module has (generally) two transitions.• One transition (or more) for receiving requests.• Guard: there is place in the queue.• One transition (or more) for serving requests.• Guard: there is at least one request in the queue.module PCS pxc: [0..IP] init 0; [accept] (pxc < IP) -> 1 :(pxc’ = pxc+1); [sforward] (pxc > 0) -> (1/Ixc)*(1-p) : (pxc’ = pxc-1);
• The rate of the server transactions has generally this shape:• 1/t * p, where
– t is the time for processing a request and– P is the probability of the branch for which the transaction
corresponds.• Note that if t is a time, then 1/t is a rate.• Example, where Ixc is the PCS initialization time:module PCS … [sforward] (pxc > 0) -> (1/Ixc)*(1-p) : (pxc’ = pxc-1);
• If two queues, say A and B, are connected, then the server transaction of A and the receiver transaction of B have to be synchronous, i.e., they have to have the same label.
• The rate of the receiver transactions are always 1, because product of rates rarely makes sense.
Response Time:Original Numerical Results andResults Computed by PRISM
WWV 2009 25
Errors in Performance Models of a Firm’s Proxy Cache Server.
• Client Network Bandwidth and Server Network Bandwidth are modeled as queues.
• “branching” should not start after the Server Network, but before.
• One queue is missing to simulate the looping process of sending and receiving files by the client.
WWV 2009 26
Orig
inal
Mod
el
WWV 2009 27
Cor
rect
ed M
odel
WWV 2009 28
Original and Corrected Equations for Response Time
WWV 2009 29
Response Time:Numerical Results for Corrected
Equation andResults Computed by PRISM
WWV 2009 30
Conclusion
• The PRISM modeling language can describe queuing networks by – representing every network node as a module– with explicit qualitative and quantitative descriptions
• Thus, it forces us to be much more precise about the system model– which may first look like a nuisance,– but shows its advantage when we want to argue
1.: Faculty of Informatics, University of Debrecen, Hungary, http://www.inf.unideb.hu2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria,
http://www.risc.uni-linz.ac.at3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu