- Performance Analysis:
- Performance Analysis =
the branch of Mathematics
that study
the question:
- How efficient is a system
- Performance Analysis =
the branch of Mathematics
that study
the question:
- The questions studied include:
- How many customers can the system service per time unit (sec)
- How long does the system take to service a customer.
- How long does a customer has to wait on average ???
- And so on.
- Definitions:
- Arrival rate
(Offered load) =
number of customers
that arrives to the
system
per time unit (sec)
- Throughput = number of customers that are served by the system per time unit (sec)
- Arrival rate
(Offered load) =
number of customers
that arrives to the
system
per time unit (sec)
- Computer networks:
- The "customers" in
a computer network are:
- messages !!!
- In computer networking:
- Arrival rate (offered load) = number of messages that arrives per time unit
- Throughput = number of messages that are transmitted per time unit
- The "customers" in
a computer network are:
- Fact:
- In Aloha, a
message may require
multiple transmission attempts
before
the message is
transmitted successfully
(This is due to collisions)
- In Aloha, a
message may require
multiple transmission attempts
before
the message is
transmitted successfully
- Performance parameters defined
for centention base
networks:
- G =
offered load
- G = number of packet transmission attempts per time unit
- S =
throughput
- S = number of successful packet transmission per time unit
- G =
offered load
- Fact about (Mathematical) analysis:
- Often very difficult
- People always make assumptions to make the Mathematical analysis more simple
- Often very difficult
- Simplifying assumptions:
- All packets have the
same length
- Result: the transmission time for every packet is the same duration
- The packet transmission time = 1 time unit
- All packets have the
same length
- Fact:
- In order to study
the performance of
a system with
Mathematical modeling:
we must model the arrival process analytically:
- In order to study
the performance of
a system with
Mathematical modeling:
- Announcement:
- I will only state some
results from
probability theory....
(This material is beyond the scope of a networking course....)
- I will only state some
results from
probability theory....
- Recall:
Arrival rate (offer load)
- Arrival rate (Offered load) = number of customers that arrives to the system per time unit (sec)
- Modeling the
arrival of
messages:
- Modeling the arrival of messages = compute the probability of a certain number of arrivals (for a given arrival rate)
Example:
- Suppose the
average
arrival rate
of customers =
10 customers/sec:
Then:
- What is the probability that there are 0 customer arrival in 1 sec ??
- What is the probability that there are 1 customer arrival in 1 sec ??
- What is the probability that there are 2 customer arrivals in 1 sec ??
- And so on...
Give a formula to compute these probabilities.
Note:
- An average arrival rate
of 10 customer/sec
does not mean:
- There are always
10 customers arrive
every sec.....
Example:
- There are always
10 customers arrive
every sec.....
- There can be heavy and
light arrival periods
Example:
- Therefore:
- The probability that there are 0 customer arrival in 1 sec when the average arrival rate = 10 is not equal to 0 !!!
- The most commonly used
Mathematical model
for customer arrivals is:
- Poisson (arrival) process
Wikipedia page: click here
- The Poisson arrival (probability)
distribution:
- Suppose the (average) arrival rate
of customers =
λ:
Then:
- the probability that there are
k arrivals
in a τ sec period is:
e-λτ (λτ)k Prob[ k arrivals in τ sec ] = ------------- k!
(e = the base of the natural log function (= 0.2718281828....))
- the probability that there are
k arrivals
in a τ sec period is:
- Suppose the (average) arrival rate
of customers =
λ:
- Example:
- Arrival rate λ =
10 (customers/sec)
- Then:
e-10*1 (10*1)0 Prob[ 0 arrivals in 1 sec ] = --------------- 0! = e-10 = 0.00004539992
- Arrival rate λ =
10 (customers/sec)
- What does this result mean for
Aloha:
- The arrival rate (λ) in
the Aloha network =
G messages per
time unit
-
Therefore: the
probability distribution of
the number of arrivals is:
- The arrival rate (λ) in
the Aloha network =
G messages per
time unit