Transcript 1. dia - EKTF.hu

Analyzing a Proxy Cache Server
Performance Model with the
Probabilistic Model Checker PRISM
Tamás Bérczes1,
Gábor Guta2,
Gábor Kusper3,
Wolfgang Schreiner2,
János Sztrik1,
[email protected],
[email protected],
[email protected],
[email protected],
[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
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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.
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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.
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The System
Performance Models of a Firm’s Proxy Cache Server
Client
sends back the file
in a loop (F > Bxc)
requests a file with rate lamba
and with average file size F.
Proxy Cache Server(PCS)
Case A:
With probability p the request
can be answered by the PCS.
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The System
Performance 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)
requests the file from
a remote web server
with rate (1-p)lambda
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Web Server
sends back the file
in a loop (F > Bs)
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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
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Original Model
Original Equation for Response
Time
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Original Response Time Diagram
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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
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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 Markov chain (CTMC)
• The model can be analyzed by queries
– in the language of Continuous Stochastic Logic
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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
distributed) rate.
with
a
certain
(exponentially
• update: performs an update on its state variables:
vname’ = vname-1
• label: Transitions in different
processes with the same
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label execute synchronously as a single combined
Example
// generate requests at rate lambda
// [label] guard -> rate : update;
module jobs
[accept] true -> lambda : true;
endmodule
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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
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How to Implement a Queue?
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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
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[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
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[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
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[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 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);
[panswer] (pxwaiting > 0) ->
(1/Ixc)*p
: (pxc’ = pxc-1);
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endmodule
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[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
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A Question!
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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
Solution B
module Server
module ServerInputQueue
RC[0..size] init 0;
RC[0..size] init 0;
RC<size->lambda: RC’=RC+1;
RC<size -> lambda: RC’=RC+1;
[pcs] RC>0-> 1/Is*1/(Ys+Bs/Rs):
[serve] RC>0 -> 1/Is: RC’ =RC-1;
RC ’ =RC-1; endmodule
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;
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endmodule
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 ]
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Response Time:
Original Numerical Results and
Results Computed by PRISM
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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.
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Original Model
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Corrected Model
Original and Corrected Equations
for Response Time
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Response Time:
Numerical Results for Corrected
Equation and
Results Computed by PRISM
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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
about the adequacy of the model.
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Thank you for your attention!
Tamás Bérczes1,
Gábor Guta2,
Gábor Kusper3,
Wolfgang Schreiner2,
János Sztrik1,
[email protected],
[email protected],
[email protected],
[email protected],
[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
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