<aside> 📖 Queues with Random Arrival Processes

• Usually, arrival processes are complicated, so we often model them using random processes.
• The study of queues with random arrival processes is called Queueing Theory.
• Queues with random arrival processes have some interesting properties. </aside>

Property #1 “Burstiness increases delay”

Time evolution of a queue

• A departure with dotted line is called “shadow departure”, it was a departure opportunity where we could have sent a packet, but the queue was empty.

Property #2 “Determinism minimizes delay”

i.e. random arrivals wait longer on average than simple periodic arrivals.

Property #3 “Little’s Result”

$$ L = \lambda d $$

Where:

• $L$ is the average number of customers in the system
• $\lambda$ is the arrival rate, in customers per second
• $d$ is the average time that a customer waits in the system

Result holds so long as no customers are lost/dropped

<aside> 📖 Why Poisson process?

• It is the models aggregation of many independent random events
• It makes the math easy

But be warned that:

1. Network traffic is very bursty
2. Packet arrivals are not Poisson
3. But it models quite well the arrival of new flows (such as web requests and new flow arrivals) </aside>

M/M/1 queue