Fuzzy Logic and its Application to Web Caching
By
P.M.Pavan Kiran
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.
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
Classical Set
young = { x P | age(x) 20 } characteristic function:
young(x) = 1 : age(x) 20 0 : age(x) > 20 A=“young”
young(x)
1
{
Fuzzy Set
Classical Logic
Element x belongs to set A or it does not:
(x){0,1}
A=“young”
A(x) 1
Fuzzy Logic
Element x belongs to set A with a certain
degree of membership:
(x)[0,1]
A=“young”
A(x) 1
Types of Membership Functions
x
(x) 1
0 a b c d
Trapezoid: <a,b,c,d>
(x) 1
Singleton: (a,1) and (b,0.5)
(x) 1
Triangular: <a,b,d>
Operators on Fuzzy Sets
Union
x 1
0
AB(x)=min{A(x),B(x)}
A(x) B(x)
x 1
0
AB(x)=max{A(x),B(x)}
A(x) B(x)
Intersection
1
AB(x)=A(x) B(x)
A(x) B(x)
1
AB(x)=min{1,A(x)+B(x)}
A(x) B(x)
Fuzzy Sets & Linguistic Variables
A linguistic variable combines several fuzzy sets.
linguistic variable : temperature
linguistics terms (fuzzy sets) : { cold, warm, hot }
x [C]
(x) 1
0
cold warm hot
60 20
Fuzzy Rules
causal dependencies can be expressed in form of if-then-rules
general form:
if <antecedent> then <consequence>
example:
if temperature is cold and oil is cheap then heating is high
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
Fuzzy Rule Base
Temperature :
cold warm hot
Oil price:
cheap normal expensive
high high medium
high medium low
medium low low
Heating
fuzzy knowledge base Fuzzy Data-Base:
Definition of linguistic input and output variables Definition of fuzzy membership functions
Fuzzy Knowledge Base
Fuzzy Rule-Base:
if temperature is cold and oil price is cheap
x [C]
(x)1 0
cold warm hot 60
20
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.
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.
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.
Figure 2
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. .
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
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
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.
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%).
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
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
The Steam turbine Contrller
The Membership functions
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
Fuzzification and Rule Inferencing
Fuzzification and Rule Inferencing
Defuzzification
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.
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.
Web Caching
Web caching can be implemented at different levels
They are 1.Client
2.Server 3.Network
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.
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.
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.
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.
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.
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.
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.
The Membership functions
The rule base
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.
The performance
The End
Thank You
