1 / 31

Analyzing a Proxy Cache Server Performance Model with the Probabilistic Model Checker PRISM

Analyzing a Proxy Cache Server Performance Model with the Probabilistic Model Checker PRISM. Tamás Bérczes 1 , tberczes@inf.unideb.hu, Gábor Guta 2 , Gabor.Guta@risc.uni-linz.ac.at, Gábor Kusper 3 , gkusper@aries.ektf.hu, Wolfgang Schreiner 2 , Wolfgang.Schreiner@risc.uni-linz.ac.at,

mira
Download Presentation

Analyzing a Proxy Cache Server Performance Model with the Probabilistic Model Checker PRISM

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Analyzing a Proxy Cache Server Performance Modelwith the Probabilistic Model Checker PRISM Tamás Bérczes1, tberczes@inf.unideb.hu, Gábor Guta2, Gabor.Guta@risc.uni-linz.ac.at, Gábor Kusper3, gkusper@aries.ektf.hu, Wolfgang Schreiner2,Wolfgang.Schreiner@risc.uni-linz.ac.at, János Sztrik1, jsztrik@inf.unideb.hu 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 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. requests a file with rate lamba and with average file size F. The SystemPerformance Models of a Firm’s Proxy Cache Server sends back the file in a loop (F > Bxc) Client Proxy Cache Server(PCS) Case A: With probability p the request can be answered by the PCS. WWV 2009

  5. requests the file from a remote web server with rate (1-p)lambda The SystemPerformance Models of a Firm’s Proxy Cache Server Case B: With probability 1-p the request must be forwarded to a remote web server. Client Proxy Cache Server(PCS) sends back the file in a loop (F > Bs) Web Server 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. Original Model 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: • Discrete-time Markov chain (DTMC) • Markov decision processes (MDP) • Continuous-time Markovchain (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 • Mathematical model: • Continuous-time Markovchain (CTMC) • The model can be analyzed by queries • in the language of Continuous Stochastic Logic WWV 2009

  12. Programming PRISM • Each process contains declarations of • state variables: vname: [minv..maxv] INIT initv; • state transitions of form: [label] guard -> rate : update; • guard: A transition is enabled to execute if its guard condition evaluates to true; • rate: it executes with a certain (exponentially distributed) rate. • update: performs an update on its state variables: vname’ = vname-1 • label: Transitions in different processes with the same label execute synchronously as a single combined transition. WWV 2009

  13. Example // generate requests at rate lambda // [label] guard -> rate : update; module jobs [accept] true -> lambda : true; endmodule WWV 2009

  14. The System in PRISM • 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; [accept] pxc < IP -> 1 :(pxc’ = pxc+1); [sforward] (pxc > 0) -> (1/Ixc)*(1-p) : (pxc’ = pxc-1); [panswer] (pxc > 0) -> (1/Ixc)*p : (pxc’ = pxc-1); endmodule WWV 2009

  18. [label] guard -> rate : update; • 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); [panswer] (pxc > 0) -> (1/Ixc)*p : (pxc’ = pxc-1); endmodule WWV 2009

  19. [label] guard -> rate : update; • 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 iftis a time, then1/tis a rate. • Example, where Ixc isthe PCS initializationtime: module PCS … [sforward] (pxc > 0) -> (1/Ixc)*(1-p) : (pxc’ = pxc-1); [panswer] (pxwaiting > 0) -> (1/Ixc)*p : (pxc’ = pxc-1); endmodule WWV 2009

  20. [label] guard -> rate : update; • 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. module PCS … [sforward] (pxc > 0) -> (1/Ixc)*(1-p) : (pxc’ = pxc-1); endmodule module server_input_queue … [sforward] (sic <IA)->1: (sic’=sic+1); … endmodule WWV 2009

  21. A Question! WWV 2009

  22. Implement this server!Which is the right solution? • Users send request with rate lambda. • The server initialization time is Is, the buffer size is Bs, static server time is Ys, and the dynamic server rate is Rs. Solution A module Server RC[0..size] init 0; RC<size->lambda: RC’=RC+1; [pcs] RC>0-> 1/Is*1/(Ys+Bs/Rs): RC ’ =RC-1; endmodule Solution B module ServerInputQueue RC[0..size] init 0; RC<size -> lambda: RC’=RC+1; [serve] RC>0 -> 1/Is: RC’ =RC-1; endmodule module ServerOutputQueue AC[0..size] init 0; [serve] AC<size -> 1: AC’=AC+1; [pcs] AC>0 -> 1/(Ys+Bs/Rs): AC’=AC-1; endmodule WWV 2009

  23. Who to Compute the Expected Response Time? • Program: module PCS pxc: [0..IP] init 0; … endmodule • Reward: rewards "time" true : (pxc+…)/lambda; endrewards • CSL query (R: expected value, S: steady-state): R{"time"}=? [ S ] WWV 2009

  24. Response Time:Original Numerical Results andResults Computed by PRISM WWV 2009

  25. Errors inPerformance 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. Original Model WWV 2009

  27. Corrected Model 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 queuingnetworks by • representing every network node as a module • withexplicit qualitative and quantitative descriptions • Thus, it forces us to be much more precise about the system model • which mayfirst look like a nuisance, • but shows its advantage when we want to argue about theadequacy of the model. WWV 2009

  31. Thank you for your attention! Tamás Bérczes1, tberczes@inf.unideb.hu, Gábor Guta2, Gabor.Guta@risc.uni-linz.ac.at, Gábor Kusper3, gkusper@aries.ektf.hu, Wolfgang Schreiner2, Wolfgang.Schreiner@risc.uni-linz.ac.at, János Sztrik1, jsztrik@inf.unideb.hu 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 WWV 2009

More Related