Stochastic Models in Computer Science A Tutorial

Transcription

1 Stochastic Models in Computer Science A Tutorial Dr. Snehanshu Saha Department of Computer Science PESIT BSC, Bengaluru WCI August 10 to August 13

2 1 Introduction 2 Random Variable 3 Introduction to Probability 4 Probability Distributions - The Motivation 5 Probability Distributions - Explained 6 Applications / Properties 7 Random Graphs 8 Inequalities 9 Queuing Theory 10 Markov Chains

3 Probability Models in Computer Science

4 Applications Machine Learning Randomized Algorithms Computer Graphics Medical Image Analysis Big Data Analytics Speech Recognition Systems Wireless Communication Communication Network

5 Randomness

6 What is a Random Variable?

7 Axioms of Probability Consider a Discrete Random Variable x 1, x 2, x 3,..., x n be the set of possible outcomes. 1 P(x i ) 0 ; i = 1, 2,..., n 2 i P(x i ) = 1

8 Coin Flipping - example A fair coin flipped twice Sample Space: {HH, HT, TH, TT } K : number of heads occurring K P(K )

9 Birth of Statistics - The Battle of Plataea 5th Century BC, Plataea - a city owned by the Persians Defeated by a Greek alliance The wall of Plataea and its height PC: sikyon.com

10 Applications of several Distributions What is a Distribution? A Distribution is a Trend, a Dimension, a Fitting of data or just a Probability!!! Applications of Distributions Apartment Service Management Problem The Last Mile Postman (Poisson) Errors in Astronomical Observations (Gaussian)

11 Gaussian What do we observe about the Gaussian? Symmetry about the y-axis mean forms the axis of the symmetry maximum value is obtained at the mean Gaussian Distribution Given a random variable X, mean µ and standard deviation σ z = X µ σ

12 Binomial Distribution Length of the sequence of event: FIXED Out of each event: Exactly TWO (Success or Failure) Find the probability of obtaining a sequence with k successes and (n-k) failures Binomial Distribution P(K ) = ( n k )θk (1 θ) n k, k = 0, 1, 2,..., n Explain coin flipping problem in light of Binomial Distribution

13 Cumulative Distribution Function Properties of CDF (Continuous Random Variables) F x (x) = P(X x), < x < 1 0 F x (x) 1 2 F x (x 1 ) F x (x 2 ), if x 1 < x 2 3 lim x F x (x) = 1 4 P(a < x b) = F x (b) F x (a) 1 P(x > a) = 1 F x (a) 2 P(x < b) = F x (b) The CDF F x (x) is the area in the histogram up to x

14 Probability Mass Function Properties of PMF (Discrete Random Variables) 1 0 P(K ) 1 2 n i P(K i ) = 1 3 F x (x i ) F x (x i 1 ) = P(x x i ) P(x x i 1 ) = P(x = x i ) = p x (x) A Histogram is the graph of the Probability Mass Function. The total area under the graph = 1

15 Probability Density Function Continuous Random Variable and the PDF a) φ(x w i ) = A i e x a i b i + p(x w i)d x = 1 b) d(x) = p(x w 1) p(x w 2 )

16 Geometric Distribution Consider a sequence of Independent trails, each which is a success with a probability p, 0 < p < 1, or a failure with a probability 1 p. If X represents the trail # of the first success, then X is said to be a Geometric Random Variable having parameter p. Example Let N=# of packets transmitted until first success. P(N = n) = q n 1 (1 q), n = 1, 2, 3,... E (N = n) = n=1 nq n 1 (1 q) = 1 (1 q)

17 Cauchy Distribution Weird Distribution (Ex: DAX, German Stock Exchange) b π PDF f X (x) = ; symmetric about zero b 2 +x 2 Mean E (X ) = + xf X (x)dx = + Variance = + b π b 2 +x 2 dx bx 2 π b 2 +x 2 dx = undefined Variance is infinite

18 Exponential Distribution Properties f X (x) = λe λx, x 0 E (X ) = µ = + λxe λx dx = 1 λ 1 λ rate, As λ, E(X) 0 2 Var(X ) = E (X 2 ) (E (X )) 2 3 Expected time to arrive is inversely proportional to the arrival rate Stochastic equivalent of Laws of Motion

19 Poisson Distribution Properties A sequence of independent trials of a random experiment, the sample space of which has two outcomes, success and failure, is called a Poisson sequence of trials if the probability of success is not constant but varies from one trial to another. f X (x) = e λ. λ K! where λ is the only parameter of Poisson Distribution Limiting Case of Binomial Distribution f x (x) = ( n K )pk q (n K) Binomial Distribution Set p = λ n, λ > 0 p varies from one trial to another n, p 0 P( K successes) = ( n K )pk (1 p) n K e λ. λk K!

20 Poisson Distribution contd. Note Events where probability of success is small and the number of trials is large such that n p = λ is of moderate magnitude, can be modeled by P(K ) e λ. λk K! Poisson is the limiting case of Binomial Distribution Reproductive property of the Poisson Distribution

21 Memoryless Property P(X s + t X s) = P(X t) where X is the amount of time of waiting from a given point, say 0 Conditional Probability that the time to receive a call after s + t th amount of time = Prob (You received the call after the t th amount of time) Not dependent on the fact that you have already waited for an s th amount of time!!! No Memory of s Memoryless

22 Applications / Properties Jensen s Inequality For any convex function g(x ), and any random variable X; E [g(x )] g(e [X ]) Convex Function 0 α 1, any x 0 < x 1 g(αx 0 + (1 α)x 1 ) α.g(x 0 ) + (1 α).g(x 1 )... g is Convex Can we apply this property repeatedly? g( n i=0 α i x i n i=0 α i g(x i )... as long as x i < x i+1, ; i = 0, 1,...

23 Why Random Graphs? Telephone Internet Social Network Power Grid WSN MANET

24 Random Graph Undirected Graph: G (V, E ) where V : Set of Vertices and E : Set of Edges Path: A sequences of edges where {(i 0, i 1 ), (i 1, i 2 ),,..., (i k 1, i k ), (i k, j)} are distinct edges, is called a path from vertex i to vertex j

25 Preliminaries Clique Clique in an undirected graph is a subset of its vertices such that every two vertices in the subset are connected by an edge. Sub graphs in Social Networks denotes a set of people who know each other Maximum Clique Prob( clique of size K G(n, p)) ( n K )p(k 2 ) Independent Set Vertex Cover: Minimum number of vertices to cover all edges

26 Random Graphs Description Let G = (V, E );where V = {1, 2,..., n}and E = {(i, X (i))} n i=1 X (i) are independent Random Variables P{X (i) = j} = P j, n j=1 P j = 1 where (i, X (i)) an edge that starts from vertex i; i = 1, 2,..., n i.e. from each vertex, another vertex is randomly chosen and joined by an edge, according to the probabilities P j Probability Space Define a new Probability Space G(n, p) Ω = Ω E Sample Space is a string of length ( n 2 ) x : {E } {0, 1}= whether an ith edge belongs to the edge set Ω

27 Random Graphs - Problems Map Coloring Problem Influence Modeling Problem Bio-informatics Application Party Problem Community Clustering Problem

28 Ramsey Number With a reasonably large number of vertices n, you will always find a complete graph of r vertices; if not; an independent set of r vertices Ramsey Number= R(r) denotes how big the graph needs to be in order to get a clique or an independent set of r vertices R(1) = 1 ;R(2) = 2 R(3) = 6 i.e. with 6 vertices, you will find a complete graph of 3 vertices OR an independent set of 3 vertices What is R(r) = K? How small should the Ramsey number be, K min in R(r) = K? Bounds of Ramsey 2 r 2 < R(r) 2 2r 3 As long as R(r) = 2 r 2, it is impossible to find a graph which has either K clique or K independent set

29 Inequalities Markov s Inequality Gives an upper bound for the percent of distribution that is above a particular value What is the probability that the value of a R.V. X is far from its expectation? Definition If X is a non-negative R.V., then for any c > 0, then P{X c} E[X] c Example The average height of a kid is 4 ft. What is the probability of finding a kid whose height is greater than 6 ft? How about greater than 7 ft?

30 Application of Markov s Inequality in Delay Estimation Table: Avg. Delay per Job Request Figure: Average Delay Comparison

31 Application of Markov s Inequality in Delay Estimation contd. Table: Total Delay per Job Request Figure: Total Delay Comparison

32 Inequalities contd. Chebyshev s Inequality Let X be a R.V. (not necessarily non-negative), c > 0, then P( X E [X ] c) Var(X) c 2 Boole s Inequality P( n i=1 A i ) n i=1 P(A i ), where {A i } n i=1 set of events

33 Inequalities contd. Chernoff Bound φ(t) = E [e tx ]; X R.V. then for any c > 0 P{X c} e tc φ(t); t > 0 P{X c} e tc φ(t); t < 0 Jensen s Inequality of Expectations If f is a convex function E [f (x)] f (E [X ]) ; provided that the expectation exists

34 Queuing Theory - An Introduction λ: Arrival rate µ: Departure rate p n (t): State Transition Probabilities, i.e. at time t, there are n customers in the system p 1 (t + t) = λ t p 0 (t) + p 1 (t) λp 1 (t) µ t p 1 (t) + µ t p 2 (t) + 0( t)

35 Queuing Theory contd. Based on Memoryless property as t 0: p 0 (t + t) = (1 λ t)p 0 (t) + µ tp 1 (t) + 0( t) p 1 (t + t) = λ tp n 1 (t) + (1 (λ µ)) tp n (t) + µ tp n+1 (t) + 0( t)

36 Key terminologies Memoryless Property No need to remember when the last customer arrived Push the limit t 0, so that there is practically little or no difference between t and t + t Arrivals/Departures can t be tracked between time intervals Memoryless Steady State Probability p n=limt P{X(t)=n}, n=0,1,2,... Limiting or long-run probability that there will be exactly n customers in the system If p 0 = 0.3, then, in the long run, the system is empty of customers 30% of the time.

37 Key terminologies contd. Service Distribution (Ex: FCFS) Arrival one at a time, Depart one at a time Customers are serviced in the order they arrived Traffic Intensity Exponential inter-arrival times with mean = 1 λ times with mean = 1 µ ρ = λ µ < 1 with ρ < 1 and service Rate Equality State 0: rate at which process leaves = λp 0 and rate at which the process arrives = µp 1 Balance Equations (Job Flow) i.e. λp 0 = µp 1

38 Key terminologies contd. Limiting Behavior of the System Steady State Probabilities can be computed as: 1 0 = λp 0 + µp = λp n 1 (λ + µ)p n + µp n+1 3 p n also satisfies n=0 p n = 1 Solutions to the System Prob( n jobs in the system): p n = (1 ρ)ρ n ; n = 0, 1, 2,... Expected # jobs in the system = L s ρ 1 ρ Expected # of jobs in the queue = L q = L s ρ Expected waiting time in the system W s = L s λ Expected waiting time in the queue W q = L q λ

39 Types of Queues M/M/1 Queue M: Memoryless arrival time (Poisson) M: Memoryless service time (Exponential) 1: One Server M/G/1 Queue M: Memoryless arrival time (Poisson) G: General service time 1: One Server

40 Markov Chains - An Introduction What is a Markov Chain? Mathematical model of a random phenomenon evolving with time such that the past affects the future only through the present Example P{X n+1 = j X n = i, X n 1 = i 1,..., X 0 = i 0 } = P i,j for all states i 0, i 1,..., i n 1, i, j and for all n 0 For a Markov Chain, the conditional distribution of any future state X n+1, given the past states X 0, X 1,..., X n 1 and the present state X n, is independent of the past states and depends only on the present state X n

41 Examples Mouse in a Cage Bank Account Simple random walk (Drunkard s Walk) Simple Random Walk (Drunkard s Walk) in a city Actuarial Chains

42 Chapman-Kolmogorov s Equations Defines n step transition probabilities P n ij to be the probability that a process in state i will be in state j after n additional transitions. That is: P n ij = P{X n+k = j X k = i}, n 0, i, j 0 P 1 ij = P ij n step transition probabilities p n+m ij = k=0 p n ik.pm kj for all n, m 0, all i, j

43 Thank you THANK YOU! Acknowledge Sara Punagin, M.Tech student of PESIT Bangalore South Campus, for help in preparing the slides.