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
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