Ali MovagharWinter 2009

Modeling and Analysis of Computer Networks

(Delay Models in Data Networks)

Delay Components

Each link delay consists of four components:

• The processing delay

• The queueing delay

• The transmission delay

• The propagation delay

Queueing Models

In the context of a data network,

• Customers represent packets assigned to a communication link for transmission.

• Service time corresponds to the packet transmission time and is equal to L/C, where L is the packet length in bits and C is the link transmission capacity in bits/sec.

Quantities of Interest

• The average number of customers in the system.

• The average delay per customer.

These quantities will be estimated in terms of known information such as:

• The customer arrival rate.

• The customer service rate.

Little’s Theorem

Let

N(t) = Number of customers in the system at time t

α(t) = Number of customers who arrive in the interval [0, t]

β(t) = Number of customers who depart in the interval [0, t]

Ti = Time spent in the system by the i-th arriving customer

Define

• We call Nt the time average of N(τ) up to time t. N is called the steady-sate time average of N(τ).

0

1( )

t

tN N dt

lim tt

N N

Little’s Theorem (cont.)

Also, define:

• λt is called the time average arrival rate over the interval [0, t]. λ is the steady-state arrival rate.

Let

• Tt is called the time average the customer delay up to time t. T is the steady-state time average customer delay.

( )t

t

t

lim tt

( )

0

( )

t

iit

TT

t

lim t

t

T T

Little’s Theorem (cont.)

N = λ T

Proof of Little’s Theorem

Proof of Little’s Theorem (cont.)

• The shaded area between α(τ) – β(τ) can be expressed as

• And if t is any time for which the system is empty [N(t)=0], the shaded area is also equal to

• Dividing both expressions above with t, we obtain

• or equivalently, Nt = λt Tt

0( )

tN d

( )

1

t

ii

T

( )( )

1

01

1 1 ( )( )

( )

ttt ii

ii

TtN d T

t t t t