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- 1. Probability and Stochastic Processes References: Wolff, Stochastic Modeling and the Theory of Queues , Chapter 1 Altiok, Performance Analysis of Manufacturing Systems , Chapter 2 Chapter 0
- 2. Basic Probability Envision an experiment for which the result is unknown. The collection of all possible outcomes is called the sample space . A set of outcomes, or subset of the sample space, is called an event . A probability space is a three-tuple ( , , Pr) where is a sample space, is a collection of events from the sample space and Pr is a probability law that assigns a number to each event in . For any events A and B, Pr must satsify: Pr( ) = 1 Pr(A) 0 Pr(A C ) = 1 – Pr(A) Pr(A B) = Pr(A) + Pr(B), if A B = . If A and B are events in with Pr(B) 0, the conditional probability of A given B is Chapter 0
- 3. Random Variables Discrete vs. Continuous Cumulative distribution function Density function Probability distribution (mass) function Joint distributions Conditional distributions Functions of random variables Moments of random variables Transforms and generating functions Chapter 0 A random variable is “a number that you don’t know… yet” Sam Savage, Stanford University
- 4. Functions of Random Variables Often we’re interested in some combination of r.v.’s Sum of the first k interarrival times = time of the k th arrival Minimum of service times for parallel servers = time until next departure If X = min( Y , Z ) then therefore, and if Y and Z are independent, If X = max( Y , Z ) then If X = Y + Z , its distribution is the convolution of the distributions of Y and Z. Find it by conditioning. Chapter 0
- 5. Conditioning ( Wolff ) Frequently, the conditional distribution of Y given X is easier to find than the distribution of Y alone. If so, evaluate probabilities about Y using the conditional distribution along with the marginal distribution of X : Example: Draw 2 balls simultaneously from urn containing four balls numbered 1, 2, 3 and 4. X = number on the first ball, Y = number on the second ball, Z = XY . What is Pr( Z > 5)? Key: Maybe easier to evaluate Z if X is known Chapter 0
- 6. Convolution Let X = Y + Z . If Y and Z are independent, Example: Poisson Note: above is cdf. To get density, differentiate: Chapter 0
- 7. Moments of Random Variables Expectation = “average” Variance = “volatility” Standard Deviation Coefficient of Variation Chapter 0
- 8. Linear Functions of Random Variables Covariance Correlation If X and Y are independent then Chapter 0
- 9. Transforms and Generating Functions Moment-generating function Laplace transform (nonneg. r.v.) Generating function ( z – transform) Let N be a nonnegative integer random variable; Chapter 0
- 10. Special Distributions Discrete Bernoulli Binomial Geometric Poisson Continuous Uniform Exponential Gamma Normal Chapter 0
- 11. Bernoulli Distribution “ Single coin flip” p = Pr(success) N = 1 if success, 0 otherwise Chapter 0
- 12. Binomial Distribution “ n independent coin flips” p = Pr(success) N = # of successes Chapter 0
- 13. Geometric Distribution “ independent coin flips” p = Pr(success) N = # of flips until (including) first success Memoryless property: Have flipped k times without success; Chapter 0
- 14. z -Transform for Geometric Distribution Given P n = (1- p ) n-1 p , n = 1, 2, …., find Then, Chapter 0
- 15. Poisson Distribution “ Occurrence of rare events” = average rate of occurrence per period; N = # of events in an arbitrary period Chapter 0
- 16. Uniform Distribution X is equally likely to fall anywhere within interval ( a , b ) Chapter 0 a b
- 17. Exponential Distribution X is nonnegative and it is most likely to fall near 0 Also memoryless; more on this later… Chapter 0
- 18. Gamma Distribution X is nonnegative, by varying parameter b get a variety of shapes When b is an integer, k , this is called the Erlang- k distribution, and Erlang-1 is same as exponential. Chapter 0
- 19. Normal Distribution X follows a “bell-shaped” density function From the central limit theorem, the distribution of the sum of independent and identically distributed random variables approaches a normal distribution as the number of summed random variables goes to infinity. Chapter 0
- 20. m.g.f.’s of Exponential and Erlang If X is exponential and Y is Erlang- k , Fact: The mgf of a sum of independent r.v.’s equals the product of the individual mgf’s. Therefore, the sum of k independent exponential r.v.’s (with the same rate ) follows an Erlang- k distribution. Chapter 0
- 21. Stochastic Processes Poisson process Continuous time Markov chains Chapter 0 A stochastic process is a random variable that changes over time, or a sequence of numbers that you don’t know yet.
- 22. Stochastic Processes Set of random variables, or observations of the same random variable over time: X t may be either discrete-valued or continuous-valued. A counting process is a discrete-valued, continuous-parameter stochastic process that increases by one each time some event occurs. The value of the process at time t is the number of events that have occurred up to (and including) time t . Chapter 0
- 23. Poisson Process Let be a stochastic process where X ( t ) is the number of events (arrivals) up to time t . Assume X (0)=0 and (i) Pr(arrival occurs between t and t + t ) = where o ( t ) is some quantity such that (ii) Pr(more than one arrival between t and t + t ) = o ( t ) (iii) If t < u < v < w , then X ( w ) – X ( v ) is independent of X ( u ) – X ( t ). Let p n ( t ) = P( n arrivals occur during the interval (0, t ). Then … Chapter 0
- 24. Poisson Process and Exponential Dist’n Let T be the time between arrivals. Pr( T > t ) = Pr(there are no arrivals in (0, t ) = p 0 ( t ) = Therefore, that is, the time between arrivals follows an exponential distribution with parameter = the arrival rate. The converse is also true; if interarrival times are exponential, then the number of arrivals up to time t follows a Poisson distribution with mean and variance equal to t. Chapter 0
- 25. When are Poisson arrivals reasonable? The Poisson distribution can be seen as a limit of the binomial distribution, as n , p 0 with constant =np. many potential customers deciding independently about arriving (arrival = “success”), each has small probability of arriving in any particular time interval Conditions given above: probability of arrival in a small interval is approximately proportional to the length of the interval – no bulk arrivals Amount of time since last arrival gives no indication of amount of time until the next arrival (exponential – memoryless) Chapter 0
- 26. More Exponential Distribution Facts Suppose T 1 and T 2 are independent with Then Suppose ( T 1 , T 2 , …, T n ) are independent with Let Y = min( T 1 , T 2 , …, T n ) . Then Suppose ( T 1 , T 2 , …, T k ) are independent with Let W= T 1 + T 2 + … + T k . Then W has an Erlang- k distribution with density function Chapter 0
- 27. Continuous Time Markov Chains A stochastic process with possible values (state space) S = {0, 1, 2, …} is a CTMC if “ The future is independent of the past given the present” Define Then Chapter 0
- 28. CTMC Another Way Each time X ( t ) enters state j , the sojourn time is exponentially distributed with mean 1/ q j When the process leaves state i , it goes to state j i with probability p ij , where Let Then Chapter 0
- 29. CTMC Infinitesimal Generator The time it takes the process to go from state i to state j Then q ij is the rate of transition from state i to state j , The infinitesimal generator is Chapter 0
- 30. Long Run (Steady State) Probabilities Let Under certain conditions these limiting probabilities can be shown to exist and are independent of the starting state; They represent the long run proportions of time that the process spends in each state, Also the steady-state probabilities that the process will be found in each state. Then or, equivalently, Chapter 0
- 31. Phase-Type Distributions Erlang distribution Hyperexponential distribution Coxian (mixture of generalized Erlang) distributions Chapter 0