1. Stochastic Processes1 CPSC 601.43: Stochastic Processes Instructor: Anirban Mahanti Email: mahanti@cpsc.ucalgary.ca Reference Book “Computer Systems Performance Evaluation and Prediction” by P. Fortier and H. Michel, Digital Press, 2004.

2. Stochastic Processes2 Outline r Definitions m Discrete, continuous, independent, stationary r Bernoulli Process r Poisson Process r Birth Death Process r Markov Process (next time)

3. Stochastic Processes3 r Definition: A family of random variables, denoted X(t), where one value of the random variable X exists for each value of t. r Example T = {heads, tails} <- the index set x = {0, 1} <- the state space X(heads) = 0; X(tails) = 1;

4. Stochastic Processes4 Stochastic Processes (2) Examples r Number of commands, N(t), received by a time-sharing computer system during some time interval (0, t) -> discrete state, continuous index r Number of heads returned, N(n), by tossing a fair coin n times? Stochastic Processes discrete continuous

5. Stochastic Processes5 Stochastic Processes (3) Independent increments x(t 2 ) - x(t 1 ) != x(t 4 ) – x(t 3 ) E.g., number of phone calls, N(t), handled by a call center between noon and 3pm on a weekday. Stationary Increments x(t+h) – x(s+h) == x(t) – x(s) t1t1 t3t3 t2t2 t4t4 s t s+h t+h

6. Stochastic Processes6 Bernoulli Process Let X i ( i>= 0) be an independent and identically distributed Bernoulli random variable, such that: X i = 1, with probability p, and, X i = 0, with probability (1-p) Let S n = X 1 + X 2 + … + X n (i.e., counting number of successes in n trials) S n is a Bernoulli process. Why?

7. Stochastic Processes7 Poisson Process

8. Stochastic Processes8 Merging Poisson Streams λ1λ1 λ2λ2 λ=λ1+λ2λ=λ1+λ2

9. Stochastic Processes9 Dividing Poisson Streams papa pbpb λ λpaλpa λpbλpb

10. Stochastic Processes10 More on Poisson Process r Number of occurrences in intervals of equal length are identically distributed. r Poisson process is “memory less”, i.e., past history does not aid in predicting future events. r Probability of k arrivals in an interval of length t (k is an integer >= 0) follows the Poisson Density Function

11. Stochastic Processes11 Birth Death (BD) Processes A continuous parameter, discrete state space stochastic process {X(t), t >= 0} E(n), n = 0, 1, 2, 3 … describe the state X(t) = n means that X(t) is in state E(n) at time t

12. Stochastic Processes12 Properties of BD Processes r State changes only in increments of +- 1 r E n >= 0 r If the system is in state E n at time t, the probability of a transition to state E n+1 during interval (t,t+h) is λ n h + o(h), and to state E n-1 is µ n h + o(h), where λ n = birth rate µ n = death rate r Probability of more than one transition during an interval of length h is o(h)

13. Stochastic Processes13 Time Dependent Solutions for BD

14. Stochastic Processes14 Equilibrium Solution for BD r Rate entering = Rate leaving λ 0 p 0 = µ n p 1 and p 0 + p 1 = 1 So, now you can solve! λ0λ0 µ1µ1 1 0