1. Ali Movaghar Winter 2009 Modeling and Analysis of Computer Networks (Delay Models in Data Networks)

2. Delay Components Each link delay consists of four components: The processing delay The queueing delay The transmission delay The propagation delay

3. 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.

4. 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.

5. 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] T i = Time spent in the system by the i-th arriving customer Define We call N t the time average of N(τ) up to time t. N is called the steady-sate time average of N(τ).

6. Little’s Theorem (cont.) Also, define: λ t is called the time average arrival rate over the interval [0, t]. λ is the steady-state arrival rate. Let T t is called the time average the customer delay up to time t. T is the steady-state time average customer delay.

7. Little’s Theorem (cont.) N = λ T

8. Proof of Little’s Theorem

9. 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, N t = λ t T t