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Week06-Lecture (20100928)

Week06-Lecture (20100928) - Simulation Modeling and...

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1 Simulation Modeling and Analysis (ORIE 4580/5580/5581) Week 6: Generating Samples of Poisson Processes, Normal Random Variables, and Introduction to Input Modeling (09/28/10 - 09/30/10)
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Announcement and Agenda HW 4 is available on Blackboard and it will be due at 11am on 10/7 For ORIE 5581 Students ONLY: There will be a midterm at 7:30pm on Thursday, September 30 at Rhodes 471. Sample midterm is available on Blackboard (see the “Exam” Folder in Course Documents) Review Session: Tuesday (9/28) at 7:30pm in Rhodes 253 I’ll go over the sample midterm questions Open-book and open-notes (no laptop, iPad, iPhone, iTouch etc) Please bring a calculator! I will be traveling next Thursday (10/7). The lecture will be covered by Fan Zhu. 2
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Motivation: Arrival Process In discrete event simulation (2nd half of the course), we are interested in modeling arrivals of customers to a system Arrival of customers to a bank, ATM, web site, restaurant, etc. Number of telephone calls to 911 centers, etc. Let N(t) denote the number of arrivals during the time interval (0, t] in some system, with N(0) = 0. The stochastic process { N(t) : t 0 } is called an arrival process What does the process { N(t) : t 0 } look like? 3
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Picture, Notation, and Terminology 4 1 T 2 T 3 T 4 T 5 T 0 T 1 A 2 A 3 A 4 A 5 A ) ( t N t N(t) denote the number of arrivals during the time interval (0, t] Observation: the process { N(t) : t 0 } increases by jumps only Let T n be the time of the nth arrival (or jump). Convention: T 0 = 0 Let A n denote the interarrival time between the (n-1) th and the n th arrivals, that is, A n = T n - T n-1
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Definition of the Poisson Process based on Interarrival Time Distribution If the interarrival times A 1 , A 2 , A 3 , .... are independent and identically distributed random variables having an exponential distribution with a parameter λ , then the arrival process is called the Poisson process . λ is called the arrival rate of the Poisson process Example: Suppose the interarrival times for customers arriving at a store are i.i.d. random variables with an exponential distribution with parameter λ = 4 hours -1 . IMPORTANT NOTE: The unit for λ is 1/hour. The expected value of an Exponential( λ ) random variable is 1/ λ On average, the arrivals of two successive customers are separated by 1/ λ = 1/4 hours = 0.25 hours (15 minutes) The rate of customer arrivals is 4 customers per hour. 5
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Important Properties of Poisson Process 6 Pr { N ( t + s ) N ( t ) = k } = e λ s ( λ s ) k k ! k = 0 , 1 , 2 , . . . Number of arrivals during the time period (t, t+s] is independent of the number of arrivals during the interval (0,t] N(t+s) - N(t) is independent of N(t) More generally, if t 1 t 2 t 3 t 4 , then N(t 4 ) - N(t 3 ) is independent of N(t 2 ) - N(t 1 ) For any t and s, the number of arrivals during (t, t+s], N(t+s) - N(t), has a Poisson distribution with parameter λ s The expected # of arrivals during an interval of length s is λ s The probability mass function of N(t+s) - N(t) is given by: IMPORTANT:
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