Probability Review

1. Probability Review Thinh Nguyen

2. Probability Theory Review • Sample space • Bayes’ Rule • Independence • Expectation • Distributions

3. Sample Space - Events • Sample Point • The outcome of a random experiment • Sample Space S • The set of all possible outcomes • Discrete and Continuous • Events • A set of outcomes, thus a subset of S • Certain, Impossible and Elementary

4. Set Operations S • Union • Intersection • Complement • Properties • Commutation • Associativity • Distribution • De Morgan’s Rule

5. Axioms If If A1, A2, … are pairwise exclusive Corollaries Axioms and Corollaries

6. Conditional Probability • Conditional Probability of event A given that event B has occurred • If B1, B2,…,Bn a partition of S, then (Law of Total Probability) S B1 B2 A B3

7. Bayes’ Rule • If B1, …, Bn a partition of S then

8. Event Independence • Events A and B are independentif • If two events have non-zero probability and are mutually exclusive, then they cannot be independent

9. Random Variables

10. Random Variables • The Notion of a Random Variable • The outcome is not always a number • Assign a numerical value to the outcome of the experiment • Definition • A function X which assigns a real number X(ζ) to each outcome ζ in the sample space of a random experiment S ζ X(ζ) = x x Sx

11. Cumulative Distribution Function • Defined as the probability of the event {X≤x} • Properties Fx(x) 1 x Fx(x) 1 ¾ ½ ¼ 3 x 0 1 2

12. Continuous Probability Density Function Discrete Probability Mass Function Types of Random Variables

13. Probability Density Function • The pdf is computed from • Properties • For discrete r.v. fX(x) fX(x) dx x

14. The expected value or mean of X is Properties The variance of X is The standard deviation of X is Properties Expected Value and Variance

15. Queuing Theory

16. Example • Send a file over the internet packet link (fixed rate) Modem card buffer

17. Delay Models place Computation (Queuing) transmission propagation C B A time

18. Queue Model

19. Practical Example

20. Multiserver queue

21. Multiple Single-server queues

22. Standard Deviation impact

23. Queueing Time

24. Queuing Theory The theoretical study of waiting lines, expressed in mathematical terms input output server queue Delay= queue time +servicetime

25. The Problem Given • One or more servers that render the service • A (possibly infinite) pool of customers • Some description of the arrival and service processes. Describe the dynamics of the system Evaluate its Performance • If there is more than one queue for the server(s), there may also be some policy regarding queue changes for the customers.

26. Common Assumptions • The queue is FCFS (FIFO). • We look at steady state : after the system has started up and things have settled down. State=a vector indicating the total # of customers in each queue at a particular time instant (all the information necessary to completely describe the system)

27. Notation for queuing systems • omitted if infinite :Where A and B can be D for Deterministic distribution M for Markovian (exponential) distribution G for General (arbitrary) distribution

28. The M/M/1 System Poisson Process Exponential server output queue

29. Arrivals follow a Poisson process • Readily amenable for analysis • Reasonable for a wide variety of situations • a(t) = # of arrivals in time interval [0,t] •  = mean arrival rate • t = k ; k = 0,1,…. ; 0 Pr(exactly 1 arrival in [t,t+]) = Pr(no arrivals in [t,t+]) = 1- Pr(more than 1 arrival in [t,t+]) =0 Pr(a(t) = n) = e- t ( t)n/n!

30. Model for Interarrivals and Service times • Customers arrive at times t0 < t1 < .... - Poisson distributed • The differences between consecutive arrivals are the interarrival times :n = tn - t n-1 • n in Poisson process with meanarrival rate, are exponentially distributed, Pr(n t) = 1 - e- t • Service timesare exponentially distributed, with meanservice rate: Pr(Sn s) = 1 - e-s

31. System Features • Service times are independent • service times are independent of the arrivals • Both inter-arrival and service times are memoryless Pr(Tn> t0+t | Tn> t0) = Pr(Tn t) future events depend only on the present state  This is a Markovian System

32. Exponential Distribution

33. Markov Models • n+1 • n • n-1 departure Buffer Occupancy • n arrival

34. Probability of being in state n

35. Steady State Analysis

36. Markov Chains 0 1 ... n-1 n n+1

37. Substituting Utilization

38. Substituting P1 • Higher states have decreasing probability • Higher utilization causes higher probability • of higher states

39. What about P0 Queue determined by

40. E(n), Average Queue Size

41. Selecting Buffers For large utilization, buffers grow exponentially

42. Throughput • Throughput=utilization/service time = /Ts • For =.5 and Ts=1ms • Throughput is 500 packets/sec

43. Intuition on Little’s Law • If a typical customer spends T time units, on the overage, in the system, then the number of customers left behind by that typical customer is equal to

44. Applying Little’s Law

45. Probability of Overflow

46. Buffer with N Packets

47. Example • Given • Arrival rate of 1000 packets/sec • Service rate of 1100 packets/sec • Find • Utilization • Probability of having 4 packets in the queue

48. Example

49. Application to Statistcal Multiplexing • Consider one transmission line with rate R. • Time-division Multiplexing • Divide the capacity of the transmitter into N channels, each with rate R/N. • Statistical Multiplexing • Buffering the packets coming from N streams into a single buffer and transmitting them one at a time. R/N R/N R/N R

50. Network of M/M/1 Queues