CS485 Sylabus

Introduction to Performance Analysis

• Performance Analysis

• Performance Analysis:

• Performance Analysis = the branch of Mathematics that study the question:

 How efficient is a system

• 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.

• Terminology: arrival rate and throughput

• 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)

• 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

• Performance analysis for contention base networks

• Fact:

 In Aloha, a message may require multiple transmission attempts before the message is transmitted successfully (This is due to collisions)

• 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

• Simplifying assumptions....

• Fact about (Mathematical) analysis:

 Often very difficult People always make assumptions to make the Mathematical analysis more simple

• 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

• Mathematical performance analysis

• Fact:

• In order to study the performance of a system with Mathematical modeling:

we must model the arrival process analytically:

• Announcement:

 I will only state some results from probability theory.... (This material is beyond the scope of a networking course....)

• Mathematical modeling of message arrivals

• 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 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 Poisson arrival process

• 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....))

• 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 ```

• Aloha message arrival modeling using Poisson arrival process

• 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: