Little’s Law

Regardless of the distribution of arriving items and how the system chooses to process them, it always holds that L = λW if there exists a steady state!

example

为什么发明指数分布

exponential distribution

minimum property

if X,Y independent exponential with parameters r1 and r2, then min{X,Y} exponential with parameter r1+r2.

poisson process

Superposition property

Thinning-out property

discrete-time Bernoulli process

continuous-time Poisson process

the inter-arrival times are independent and exponentially distributed with mean 1/r.

Processor sharing

Flow-level behavior of TCP data transfers is often be modeled by PS model
key performance metric
mean sojourn time

Jackson Network

In queueing networks,the output of one queue may be the input of another.
In stready-state,the output process if M/M/1,M/M/k queue follows a Poisson process, because no traffic is lost in such queues,the arrival rate must be equal to the departure rate.

product form network

Product form networks are easier to analyze
Jackson showed that any arbitrary open network of m server queues with exponentially distributed service times has a product form