Fuzzy Logic and its Application to Web Caching

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Fuzzy Logic and its Application to Web Caching

• By
• P.M.Pavan Kiran

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What is fuzzy logic?

• A type of logic that recognizes more than
  simple true and false values. With fuzzy logic,
  propositions can be represented with degrees of
  truthfulness and falsehood.

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A Simple Example

• The Statement Today is sunny can be
• 100 true if there are no clouds
• 80 true if there are a few clouds
• 50 true if it's hazy and
• 0 true if it rains all day

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Classical Set
young x ? P age(x) ? 20
characteristic function
1 age(x) ? 20 0 age(x) gt 20

myoung(x)
?young(x)
Ayoung
1
0
x years
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Fuzzy Set
Fuzzy Logic Element x belongs to set A with a
certain degree of membership ?(x)?0,1
Classical Logic Element x belongs to set A or it
does not ?(x)?0,1
?A(x)
?A(x)
Ayoung
Ayoung
1
1
0
0
x years
x years
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Types of Membership Functions
Trapezoid lta,b,c,dgt
?(x)
1
0
x
a
b
c
d
Triangular lta,b,dgt
Singleton (a,1) and (b,0.5)
?(x)
?(x)
1
1
0
0
x
a
b
x
a
b
d
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Operators on Fuzzy Sets
Union
Intersection
?A?B(x)min?A(x),?B(x)
?A?B(x)max?A(x),?B(x)
?A(x)
?B(x)
?A(x)
?B(x)
1
1
0
0
x
x
?A?B(x)?A(x) ? ?B(x)
?A?B(x)min1,?A(x)?B(x)
?A(x)
?B(x)
?A(x)
?B(x)
1
1
0
0
x
x
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Fuzzy Sets Linguistic Variables
A linguistic variable combines several fuzzy
sets. linguistic variable temperature linguist
ics terms (fuzzy sets) cold, warm, hot
?(x)
?cold
?warm
?hot
1
0
60
20
x C
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Fuzzy Rules

• causal dependencies can be expressed in form of
  if-then-rules
• general formif ltantecedentgt then ltconsequencegt
• exampleif temperature is cold and oil is
  cheap then heating is high

linguistic values/terms (fuzzy sets)
linguistic variables
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Fuzzy Rule Base
Heating
Temperature cold warm hot
Oil price cheap normal expensive
high high medium high medium low medium low
low
if temperature is cold and oil price is low then
heating is high
if temperature is hot and oil price is normal
then heating is low
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Fuzzy Knowledge Base
fuzzy knowledge base
Fuzzy Data-Base Definition of linguistic input
and output variables Definition of fuzzy
membership functions
?(x)
?cold
?warm
?hot
1
60
20
0
x C
Fuzzy Rule-Base if temperature is cold and oil
price is cheap then heating is high .
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An Example

• In order to illustrate some basic concepts in
  Fuzzy Logic, consider a simplified example of a
  thermostat controlling a heater fan illustrated
  in Figure 1.
• The room temperature detected through a sensor
  is input to a controller which outputs a control
  force to adjust the heater fan speed.

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Conventional Thermostat

• A conventional thermostat works like an on-off
  switch (Figure 2).
• If we set it at 78oF then the heater is
  activated only when the temperature falls below
  75oF .
• When it reaches 81oF the heater is turned off. As
  a result the desired room temperature is either
  too warm or too hot.

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Fuzzy Thermostat

• A fuzzy thermostat works in shades of gray where
  the temperature is treated as a series of
  overlapping ranges.
• For example, 78oF is 60 warm and 20 hot. The
  controller is programmed with simple if-then
  rules that tell the heater fan how fast to run.
• As a result, when the temperature changes the
  fan speed will continuously adjust to keep the
  temperature at the desired level.

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Figure 2
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Designing a Fuzzy Controller

• Our first step in designing such a fuzzy
  controller is to characterize the range of values
  for the input and output variables of the
  controller
• Then we assign labels such as cool for the
  temperature and high for the fan speed, and we
  write a set of simple English-like rules to
  control the system. .

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Design Contd..

• Inside the controller all temperature regulating
  actions will be based on how the current room
  temperature falls into these ranges and the rules
  describing the system behavior. The controller's
  output will vary continuously to adjust the fan
  speed

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The Rule Base

• The temperature controller described above can be
  defined in four simple rules
• IF temperature IS cold THEN fan_speed IS high
• IF temperature IS cool THEN fan_speed IS medium
• IF temperature IS warm THEN fan_speed IS low
• IF temperature IS hot THEN fan_speed IS zero

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The Process

• A fuzzy controller works similar to a
  conventional system it accepts an input value,
  performs some calculations, and generates an
  output value. This process is called the Fuzzy
  Inference Process and works in three steps
  illustrated in Figure 3
• (a) Fuzzification where a crisp input is
  translated into a fuzzy value,
• (b) Rule Evaluation, where the fuzzy output truth
  values are computed, and
• (c) Defuzzification where the fuzzy output is
  translated to a crisp value.

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Fuzzification

• During the fuzzification step the crisp
  temperature value of 78oF is input and translated
  into fuzzy truth values.
• For this example, 78oF is fuzzified into warm
  with truth value 0.6 (or 60) and hot with truth
  value 0.2 (or 20).

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Rule Evaluation

• For 78oF only the last two of the four rules will
  fire.
• IF temperature IS warm THEN fan_speed IS low
• with truth value 0.6
• IF temperature IS hot THEN fan_speed IS zero with
  truth value 0.2

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Defuzzification

• During the defuzzification step the 60 low and
  20 zero labels are combined using a calculation
  method called the Center of Gravity (COG) in
  order to produce the crisp output value of 13.5
  RPM for the fan speed

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The Steam turbine Contrller
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The Membership functions
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The Rule Base

• rule 1 IF temperature IS cool AND pressure IS
  weak, THEN throttle is P3.
• rule 2 IF temperature IS cool AND pressure IS
  low, THEN throttle is P2.
• rule 3 IF temperature IS cool AND pressure IS
  ok, THEN throttle is Z.
• rule 4 IF temperature IS cool AND pressure IS
  strong, THEN throttle is N2.

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Fuzzification and Rule Inferencing
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Fuzzification and Rule Inferencing
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Defuzzification
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Why Web Caching?

• It is widely recognized that slow web sites are
  primary source of user dissatisfaction.
• Web Caching is a mechanism widely employed to
  reduce the latency to retrieve web pages.

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What is web Caching?

• The idea of web caching is to store popular web
  objects closer to the user who requests them
  such that they can be retrieved faster. Caching
  also has the effect of reducing the load on the
  web servers and traffic over network.

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Web Caching

• Web caching can be implemented at different
  levels
• They are
• 1.Client
• 2.Server
• 3.Network

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Web Caching

• The web server and web browser are responsible
  for caching at server and client side
  respectively. Proxy servers are used for caching
  at the network level.
• A proxy server acts as an intermediary between
  clients and web servers.

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Proxy server

• Many organizations use proxy servers in front of
  their LANs to save network bandwidth and speed up
  web page requests serving them locally.
• Upon receiving requests from multiple clients the
  proxy server checks from its cache whether the
  page is already present.

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Proxy Server

• In Case of a cache miss the proxy server forwards
  the request to the web server. Once the page is
  returned by the server, the proxy sends it back
  to the client and stores a copy in the local
  cache for further requests.
• If the cache is full one or more pages have to be
  evicted from the cache.

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The Replacement Policies

• The efficiency and performance of proxy caches
  depend on their design and management.
  Replacement policies play a key role for
  effectiveness of caching.
• The goal of these policies is to make best use
  of the available resources by dynamically
  selecting the pages to cached or evictes.

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The fuzzy Database for web caching

• 1. Identification of the input output variables.
• 2. Definition of the membership functions.
• 3. Construction of the rule base.

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The Variables

• The proper choice of process state input variable
  is essential to the characterization of the
  operation of a fuzzy system.
• Three variables as input are chosen, they being
• Size, Frequency of access, Access recency, i.e
  time elapsed since the last access.

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The Variables

• The output variable is probability of
  replacement.
• The membership functions of each of these
  variables is plotted from the analysis of various
  proxy servers and their workloads.

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The Membership functions
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The rule base
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The process

• Measurement of the values of the input data from
  the server.
• Fuzzification
• Inference from fuzzy rules using Max-Min
  Inference.
• Defuzzification using the COG method.

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The performance
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The End

• Thank You