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 Union Intersection Complement Properties  Commutation  Associativity  Distribution  De Morgan’s Rule S

5. Axioms and Corollaries Axioms If If A 1, A 2, … are pairwise exclusive Corollaries

6. Conditional Probability Conditional Probability of event A given that event B has occurred If B 1, B 2,…,B n a partition of S, then (Law of Total Probability) S B1B1 B3B3 B2B2 A

7. Bayes’ Rule If B 1, …, B n a partition of S then

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

9. Random Variables

10. 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 x SxSx ζ X(ζ) = x

11. Cumulative Distribution Function Defined as the probability of the event {X≤x} Properties x 2 1 F x (x) ¼ ½ ¾ 10 3 1 x

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

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

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

15. Queuing Theory

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

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

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 inputoutput queue server Delay= queue time +service time

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 M for Markovian (exponential) distribution D for Deterministic distribution G for General (arbitrary) distribution :Where A and B can be omitted if infinite omitted if infinite

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

29. Arrivals follow a Poisson process 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! Readily amenable for analysis Readily amenable for analysis Reasonable for a wide variety of situations Reasonable for a wide variety of situations

30. Model for Interarrivals and Service times  Customers arrive at times t 0 < t 1 <.... - Poisson distributed  The differences between consecutive arrivals are the interarrival times :  n = t n - t n-1   n in Poisson process with mean arrival rate, are exponentially distributed, Pr(  n  t) = 1 - e - t Service times are exponentially distributed, with mean service rate  : Pr(S n  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(T n > t 0 +t | T n > t 0 ) = Pr(T n  t) future events depend only on the present state  This is a Markovian System

32. Exponential Distribution

33. Markov Models Buffer Occupancy n+1 n n-1 n departure 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 P 1 Higher states have decreasing probability Higher utilization causes higher probability of higher states

39. What about P 0 Queue determined by

40. E(n), Average Queue Size

41. Selecting Buffers For large utilization, buffers grow exponentially

42. Throughput Throughput=utilization/service time =  /T s For  =.5 and T s =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

50. Network of M/M/1 Queues

51. M/G/1 Queue Q S 0 S Assume that every customer in the queue pays at rate R when his or her remaining service time is equal to R. Total cost paid by a customer: Expected cost paid by each customer: At a given time t, the customers pay at a rate equal to the sum of the remaining service times of all the customer in the queue. The queue begin first come-first served, this sum is equal to the queueing time of a customer who would enter the queue at time t. The customers pat at rate since each customer pays on the average and customers go through the queue per unit time.