1. CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Stochastic processes Bernoulli and Poisson processes (Sec. 6.1,6.3.,6.4)

2. Introduction  Example: Count the number of cars in a service station, each time a car departs:  In between, two departures, some cars may arrive:  Family of random variables:

3. Introduction (contd..)  State space of the process:  Parameter index:

4. Classification of processes  Discrete vs. continuous state-space:  Discrete vs. continuous parameter space: :  Four types of processes:

5. Discrete-state, discrete-parameter process  Example: Number of cars in a service station, at the departure of each car.

6. Discrete-state, continuous-parameter process  Example: Number of cars in a service station at time t.

7. Continuous-state, discrete-parameter process  Example: Average waiting time for service, at the departure of each car.

8. Continuous-state, continuous-parameter process  Example: Total service time of all the cars in the system, at time t.

9. Bernoulli process  Sequence or a family of Bernoulli random variables:  Type:  Parameters:

10. Bernoulli process (contd..)  Random variable Yn – Number of successes in n trials:  Random variable Ti – Number of trials until the first success:

11. Poisson process  Count the number of event arrivals in an interval:  Successive occurrence of events:

12. Poisson process (contd..)  Superposition of Poisson processes:

13. Poisson process (contd..)  Decomposition of a Poisson process: