CS302 Topics: * More on Simulation and Priority Queues * Random Number Generation

Transcription

1 CS302 Topics: * More on Simulation and Priority Queues * Random Number Generation Thursday, Oct. 5, 2006 Greetings from Salvador, Brazil!! (i.e., from my recent trip to present a plenary lecture on multi-robot systems) First IFAC Workshop on Multivehicle Systems (MVS'06) October 2-3, 2006 Centro de Convenções da Bahia Salvador - Bahia - Brazil

2 Announcements Lab 4 (Algorithm Analysis) due next Wed., Oct. 11 Lab 5 (Stock Reports) now available; due Wed., Oct. 25 Don t procrastinate!! You may delay, but time will not. -- Benjamin Franklin Midterm: Tuesday, Oct. 1 Open book, open notes, no electronic devices [Review during Tuesday s class (Oct. 10)]

3 Recall: Discrete Event Simulations Here, we re interested in procesess and systems that change at discrete instants. We re interested in discrete events, which occur at an instant of time. Because some systems are complex, we can t model them and derive an analytical solution. Thus, we can simulate them to answer interesting questions.

4 Recall: The Banking Simulation We have a bank: Customers arrive and wait in a line until one of k tellers is available Customer arrival governed by probability distribution Service time governed by a probability distribution We want to know: How long (on average) does a customer have to wait? How long (on average) is the line? What is the optimal number of tellers? Enough so that customers don t have to wait too long Not so many that we waste our money paying tellers

5 Recall: The general process for discrete event simulation loop State : all variables that describe a system Initialize (State) Time 0 Generic Simulation Program: Initialize (futureeventset) While (futureeventset not empty) do: Event next event in futureeventset (i.e., deletemin) Time Time (Event) State function of old state and event Update(futureEventSet)

6 Event Generation Tricky part of simulation: Choosing how to generate random events Example: Say we want transaction times to average 10 minutes. Here are 2 sequences that average 10: But, these have very different characteristics we need to define the distribution we re interested in

7 Event Generation (con t.) The distribution defines the characterization of the data in ways other than just the mean (average) Here are some common distributions: Uniform Gaussian/Normal Exponential

8 Distributions: Uniform Distribution: Defines characterization of random numbers in more specific ways than just an average value Mean m Uniform distribution: For our purposes, we ll have random numbers 0 r 2m, with every value between 0 and 2m equally likely f( x) = 1 ( ) for b a a x b 0 otherwise

9 Uniform Distribution in Banking Example Service time: represented by uniform distribution

10 Uniform distributions in C++ (pseudo-random number generators) srand48(i): takes integer i and uses it as a seed to the random number generator (just do this once) drand48(): returns a double uniformly distributed in range [0,1) drand is a random number generator that follows a uniform distribution with a mean of 0.5 What do you do if you want a mean of m? Multiply the result of drand48 by 2m.

11 Note on how pseudo-random number generators work drand48() generates a sequence of 48-bit integers X i according to following: X = n 1 ( ax + n c) mod m, for n + 0 a = 0x5DEECE66D c = 0xB srand48(seedval) sets the high order 32-bits of X i to the argument seedval Low order 16-bits are set to arbitrary value 0x330E Note: these numbers aren t really random. Sequence repeats Numbers are predictable

12 Example of generating random numbers from uniform distribution /* generate 20 random numbers using a uniform distribution with a mean of mean */ srand48(3); // provide an initial "seed" to random generator for (i = 0; i < 20; i++) { d = drand48()*mean*2; printf("%d \t %lf \n", i, d); }

13 Distributions: Exponential Exponential distribution λ = = 1 1 mean 120 f( x) = e λx Exponential distribution plotted on log axis In our bank simulation, customer arrivals are modeled using an exponential distribution Here, mean = 120

14 How to generate non-uniform random numbers in C++? Unfortunately, there is no C++ standard library implementation of a general random number generator (at least yet) drand() just generates a uniform distribution But there are lots of probability distributions: Poisson e f( k, λ) = k! λ k λ Exponential f ( x) = e λx Gaussian 1 f( x, µσ, ) = e σ 2π ( x µ ) 2 2σ 2

15 How do we implement an arbitrary probability distribution in C++? General idea: Come up with a mapping from a uniform distribution to our arbitrary distribution Bigger chunks of uniform distribution map to numbers that are more likely to occur in our arbitrary distribution arbitrary distribution uniform distribution

16 Use histogram to represent arbitrary distribution Histogram: bins values along x axis Rather than use the continuous representation, we use a discrete representation (with the bins)

17 Bin width of histogram is important Here s an example using a distribution describing the eruption durations of the Old Faithful geyser in Yellowstone Nat l. Park:

18 How to create histogram file? Evaluate function describing distribution at discrete values (representing center of bins) Output bin-number and function value to a file Example: exponential function with mean of 120 λ = = 1 1 mean 120 f( x) = e λx So that we can just deal with integers, we ll multiply by and round gx ( ) = f( x) output file (e.g., expon_120)

19 How do we generate random numbers matching arbitrary distribution? General idea: Come up with a mapping from a uniform distribution to our arbitrary distribution Bigger chunks of uniform distribution map to numbers that are more likely to occur in our arbitrary distribution arbitrary distribution uniform distribution

20 We ll store this mapping in a red-black tree Create red-black tree total 0 Do following for each line of histogram file Get x and y value total total + y insert (key,value) = (total, x) into redblack tree arbitrary distribution uniform distribution

21 Example Distribution that models our system (e.g., arrival time): Resulting histogram file:

22 Example (con t.) Create red-black Tree total 0 Do following for each line of histogram file Get x and y value total total + y insert (key,value) = (total, x) into redblacktree Histogram file: x y Resulting (key, value) pairs:

23 Example (con t.) (key, value) pairs: Resulting Red-black tree:

24 Example (con t.) When you want a random # according to original distribution: Generate random number from uniform distribution between 0 and total (i.e., uniform distribution with mean total/2) Find node in red-black Tree with next greater value Return value of this node (this will be your rand. # from orig. distribution)

25 Example (con t.) Random numbers from uniform distribution [0,113): Generated random numbers from original distribution (using rbtree lookup):

26 Example (con t.) Generated random numbers from original distribution (using red-black tree lookup): Notice how generated numbers match distribution (as more and more numbers are generated, it will match the original distribution more closely)

27 Summarizing histogram random number generator Step 1 Generate histogram file: Evaluate function describing distribution at discrete values (representing center of bins) Output bin-number and function value to a file Step 2 Create red-black tree: total 0 Do following for each line of histogram file Get x and y value total total + y insert (key,value) = (total, x) into red-black tree Step 3 Generate random numbers: Generate random number from uniform distribution between 0 and total (i.e., uniform distribution with mean total/2) Find node in red-black tree with next greater value Return value of this node (this will be your random number from the original distribution)

28 Another example Here is original distribution: What are values in histogram file? x y

29 Another example, con t. Histogram file: What are resulting (key, value) pairs? What do we do next? Put into a red-black tree

30 Another example, con t What next? Generate random number from uniform distribution in what range? [0,121) Lookup next greater value in red-black tree Value = new random number 10 9

31 Another example, con t. Random numbers from uniform distribution [0,121): Generated random numbers from original distribution (using red-black Tree lookup):

32 Another example, con t. Compare to original distribution: Generated random numbers from original distribution (using red-black Tree lookup): Again, notice how generated numbers match distribution (as more and more numbers are generated, it will match the original distribution more closely)

33 Keep in mind how this histogram generator works General idea: Come up with a mapping from a uniform distribution to our arbitrary distribution Bigger chunks of uniform distribution map to numbers that are more likely to occur in our arbitrary distribution arbitrary distribution uniform distribution