Tamás Bérczes 1 , [email protected], Gábor Guta 2 , [email protected],

Performance Analyzes ofPerformance Analyzes ofa Proxy Cache Server Modela Proxy Cache Server Model

with External Users with External Users using the Probabilistic Model using the Probabilistic Model

Checker PRISMChecker PRISMTamás BérczesTamás Bérczes11, , [email protected],[email protected],Gábor GutaGábor Guta22, , [email protected],[email protected],Gábor KusperGábor Kusper33, , [email protected],[email protected],

Wolfgang SchreinerWolfgang Schreiner22, , [email protected],[email protected],János SztrikJános Sztrik11, , [email protected]@inf.unideb.hu

1.: Faculty of Informatics, University of Debrecen, Hungary, 1.: Faculty of Informatics, University of Debrecen, Hungary, http://www.inf.unideb.huhttp://www.inf.unideb.hu

2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler 2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, http://www.risc.uni-linz.ac.atUniversity, Linz, Austria, http://www.risc.uni-linz.ac.at

3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu

AbstractAbstract• We report our experience with formulating and

analyzing in the probabilistic model checker PRISM a web server performance model with proxy cache server and external users that was previously described in the literature in terms of classical queuing theory. We describe how to model a proxy cache server, a web server, internal and external users in PRISM.

• The main contribution of the paper shows how to model in PRISM an input queue which might receive different type of messages and has to place the answer in different output queues depending on the type of the message.

PRISMPRISM• 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

Why Why PRISMPRISM??ETMCC worst, only small models

MRMC fastest for small models

PRISM hybrid

fast if MTBDD is small

PRISM sparse

fast

VESTA rather slow, no S , statistical errors

YMERslim & very fast, only U ≤t,few statistical errors

The goal of our workThe goal of our work• The two originally distinct areas of the qualitative

analysis (verification) and quantitative analysis (performance modeling) of computing systems have in the last decade started to converge by the arise of stochastic/probabilistic model checking.

• This fact is recognized by both communities. While originally only individual authors hailed this convergence, today various conferences and workshops are intended to make both communities more aware of each others’ achievements.

• One attempt towards this goal is to compare techniques and tools from both communities by concrete application studies.

• The present paper is aimed at exactly this direction.

Case StudyCase Study• We applie PRISM to re-assess a web server

performance models with proxy cache servers with external users that have been previously described and analyzed in the literature:

• T. Bérczes, J. Sztrik: Performance Modeling of Proxy Cache Servers,Journal of Universal Computer Science, 12(9):1139–1153, 2006.

The systemThe system• Using proxy cache server, if any information or file

is requested to be downloaded, first it is checked whether the document exists on the proxy cache server. (We denote the probability of this existence by p). If the document can be found on the PCS then its copy is immediately transfered to the user.

• In the opposite case the request will be sent to the remote Web server. After the requested document arrived to the PCS then the copy of it is delivered to the user.

• We assume that the requests of the PCS users arrive according to a Poisson process with rate lambda , and the external visits at the remote web server form a Poisson process with rate Lambda.

The systemThe 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 two more job-sources:

– Internal users, rate: lambda.– External users, rate: Lambda.

• We have 6 models together.

ParametersParameters• Network Arrival Rate (lambda)• Visit rates for external users (Lambda)• Cache hit rate probability (p)

• Buffer size of the Web server given in requests (K = 100)• Average File Size (F = 5000)• Buffer Size (Bs = 2000) • PCS buffer size (Bxc = Bs)• Initialization Time (Is = 0.004) • PCS initialization time (Ixc = Is)• Static Server Time (Ys = 0:000016)• Static PCS time (Yxc = Ys)• Dynamic Server Rate (Rs = 1310720) • Dynamic PCS rate (Rxc = Rs)• Server Network Bandwidth (Ns = 193000)• Client Network Bandwidth (Nc = 16000)

Programming PRISMProgramming PRISM• Each process contains declarations of its state

variables and 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 and …

• update: performs an update on its state variables.

• label: Transitions in different processes with the same label execute synchronously as a single combined transition whose rate is the product of the rates of the individual transitions.

An Example: An Example: The model of The model of internal internal and external and external usersusers

// generate requests at rate lambdamodule jobs [accept] true -> lambda : true ;endmodule

How to implement a queue?How to implement a queue?• Each node models a queue with a counter,

which contains the number of request in the queue, i.e., we make no distinction between requests.

• Example:module PCS pxwaiting: [0..IP] init 0; …endmodule

How to iHow to implement a queuemplement a queue??• Each node has (generally) two transitions. One (or

more) for receiving requests, one (or more) for serving requests. The first one increases the counter, the second one decreases it.

• Example:module PCS pxwaiting: [0..IP] init 0; [accept] pxwaiting < IP -> 1 :

(pxwaiting’ = pxwaiting+1); [sforward] (pxwaiting > 0) & (1-p > 0) ->

(1/Ixc)*(1-p) : (pxwaiting’ = pxwaiting-1); [panswer] (pxwaiting > 0) & (p > 0) ->

(1/Ixc)*p : (pxwaiting’ = pxwaiting-1);endmodule

How to iHow to implement a queuemplement a queue??• 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] (pxwaiting > 0) & (1-p > 0) ->

(1/Ixc)*(1-p) : (pxwaiting’ = pxwaiting-1); [panswer] (pxwaiting > 0) & (p > 0) ->

(1/Ixc)*p : (pxwaiting’ = pxwaiting-1);endmodule

How to iHow to implement a queuemplement a queue??• 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] (pxwaiting > 0) & (1-p > 0) ->

(1/Ixc)*(1-p) : (pxwaiting’ = pxwaiting-1);endmodulemodule S_I … [sforward] waiting<IA->1: (waiting’=waiting+1); …endmodule

Who to compute the expected Who to compute the expected response time of PCS?response time of PCS?

• Program:module PCS pxwaiting: [0..IP] init 0; …endmodule• Reward: rewards "time" true : (pxwaiting)/lambda; endrewards• CSL query (R: expected value, S: steady-state): R{"time"}=? [ S ]

Expected response time of Expected response time of the systemthe system

• p = 0.25• Lambda =

100• computed

(by PRISM) results are the same as the numerical ones up to the 4-5th digit

Implement input queue with Implement input queue with two output queues, The goal:two output queues, The goal:

• Implement an input queue which might receive different type of messages and has to place the answer in different output queues depending on the type of the message.

SolutionSolution• If we know the incoming rates of the two

types of messages, then we can compute the probability that a message is of type A or B.

• Let the two rates lambda2 and Lambda. If lambda3 = lambda2 + Lambda, then: – lambda2/lamdba3 is the probability that a

message belongs to type A,– 1-(lambda2/lamdba3) is the probability that a

message belongs to type B.

Program of the solutionProgram of the solution• lambda2/lamdba3: message is of type A• 1-(lambda2/lamdba3): message is of type Bmodule S_R … // forwarding of block to internet queue [sanswer] (irwaiting > 0) & (q > 0) -> 1/(Ys+Bs/Rs)*q *(lambda2/lambda3): (irwaiting’ = irwaiting-1) ; // forwarding of block to external users [extanswer] (irwaiting > 0) & (q > 0) -> 1/(Ys+Bs/Rs)*q *(1-(lambda2/lambda3)): (irwaiting’ = irwaiting-1) ;endmodule

Implement this server! Implement this server! Which is the right solution?Which is the right solution?

• Internal users send request with rate alfa. External ones with rate beta. Answers to internal users are sent to a PCS.

• The server initialization time is Is, the buffer size is Bs, static server time is Ys, and the dynamic server rate is Rs.

module Server RC[0..size] init 0; [inter] RC<size->alfa: RC’=RC+1; [exter] RC<size->beta:RC’=RC+1; [pcs] RC>0-> 1/Is*1/(Ys+Bs/Rs)* (alfa / (alfa+beta)): RC ’ =RC-1; [out] RC>0-> 1/Is*1/(Ys+Bs/Rs)* (1-alfa/(alfa+beta)):RC ’=RC-1;endmodule

module ServerInputQueue RC[0..size] init 0; [inter] RC<size->alfa: RC’=RC+1; [exter] RC<size->beta:RC’=RC+1; [serve] RC>0 -> 1/Is: RC’ =RC-1;endmodulemodule ServerOutputQueue AC[0..size] init 0; [serve] AC<size -> 1: AC’=AC+1; [pcs] AC>0 -> 1/(Ys+Bs/Rs)* (alfa/(alfa+beta)): AC’=AC-1; [out] AC>0 -> 1/(Ys+Bs/Rs)* (1-alfa/(alfa+beta)): AC’=AC-1;endmodule

ConclusionConclusion• The PRISM modeling language can be

quite conveniently used to describe queueing networks by representing every network node as an automaton (“module”) with explicit (qualitative and quantitative) descriptions of the interactions between automata.

• This 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 about the adequacy of the model.

Thank you for your Thank you for your attention!attention!

Tamás BérczesTamás Bérczes11, , [email protected],[email protected],Gábor GutaGábor Guta22, , [email protected],[email protected],Gábor KusperGábor Kusper33, , [email protected],[email protected],

Wolfgang SchreinerWolfgang Schreiner22, , [email protected],[email protected],János SztrikJános Sztrik11, , [email protected]@inf.unideb.hu

1.: Faculty of Informatics, University of Debrecen, Hungary, 1.: Faculty of Informatics, University of Debrecen, Hungary, http://www.inf.unideb.huhttp://www.inf.unideb.hu

2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler 2.: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, http://www.risc.uni-linz.ac.atUniversity, Linz, Austria, http://www.risc.uni-linz.ac.at

3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu3.: Esterházy Károly College, Eger, Hungary, http://www.ektf.hu

Documents

Tamás Bérczes 1 , [email protected], Gábor Guta 2 , [email protected],