Page 116 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes Little's Theorem

We start our analysis of queueing systems by understanding Little's Theorem. Little's theorem
states that:
The average number of customers (N) can be determined from the following equation:

N = T
Here lambda is the average customer arrival rate and T is the average service time for a customer.
Proof of this theorem can be obtained from any standard textbook on queueing theory. Here we
will focus on an intuitive understanding of the result. Consider the example of a restaurant
where the customer arrival rate (lambda) doubles but the customers still spend the same amount
of time in the restaurant (T). This will double the number of customers in the restaurant (N). By
the same logic if the customer arrival rate remains the same but the customers service time
doubles, this will also double the total number of customers in the restaurant.
Queueing System Classification

With Little's Theorem, we have developed some basic understanding of a queueing system. To
further our understanding we will have to dig deeper into characteristics of a queueing system
that impact its performance. For example, queueing requirements of a restaurant will depend
upon factors like:
1. How do customers arrive in the restaurant? Are customer arrivals more during lunch and
dinner time (a regular restaurant)? Or is the customer traffic more uniformly distributed
(a cafe)?

2. How much time do customers spend in the restaurant? Do customers typically leave the
restaurant in a fixed amount of time? Does the customer service time vary with the type of
customer?

3. How many tables does the restaurant have for servicing customers?
The above three points correspond to the most important characteristics of a queueing system.
They are explained below:

Arrival Process  The probability density distribution that determines the customer
arrivals in the system.
 In a messaging system, this refers to the message arrival probability
distribution.
Service Process  The probability density distribution that determines the customer
service times in the system.
 In a messaging system, this refers to the message transmission time
distribution. Since message transmission is directly proportional to
the length of the message, this parameter indirectly refers to the
message length distribution.
Number of Servers  Number of servers available to service the customers.
 In a messaging system, this refers to the number of links between the
source and destination nodes.

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