Arena Variables and Function
DISC(0.3, 1, 0.8, 2, 1.0, 3) |
DISC generates random discrete values based on cumulative probabilities; pair each probability with its corresponding value. |
TNOW |
Current simulated time |
NR(Res) |
Res Servers currently in service |
NQ(Queue) |
Number of customers in Queue |
Mod.NumberOut |
Customers who have left the mode |
Arena Set Types
Resource, Counter, entity type, entity picture |
Arena Key Modules
Assign |
Give new value to an attribute |
Decide |
Route customers probabilistically or conditonally |
Separate / Clone |
Split one customer into two or more clones |
Route |
Move entities station to station (advanced transfer) |
Enter / Leave |
Usually paired together for station management |
Seize - Delay - Release |
equals single Process module |
Queue block |
Cannot connect with a process module |
Example of finding X for Pois
Universal truths
−ln(U) |
∼Exponential(1) |
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lnu |
messy cdf
See F(X) |
Replace with Uniform(0,1) |
Multiply by 3 |
U(0,3) |
Subtract 1 |
U(−1,2) |
Mean |
(-1+2)/2 = 1/2 |
What is the mean of the random variable 3F(X)−1?
Inverse Transformation
Given U, find Z |
invNorm(U, 0, 1) |
Given Z, find U |
normCdf(-9999, Z, 0, 1) |
Exponential(λ) |
X = -ln(1-U)/λ |
Uniform(a,b) |
X = a + (b-a)·U |
Weibull(a,b) |
X = a * (-ln(U))1/b |
Triangular |
If U<0.5: √(2U); If U≥0.5: 2-√(2(1-U)) |
Bernoulli(p) |
If U < 1-p → 0; Else → 1 |
Poisson(λ) |
Build CDF, match U |
Discrete Unif(1,n) |
[n·U] |
Erlang(k,λ) |
-(1/λ)ln(∏Ui) |
Geometric |
ln(1-U)/ln(1-p) |
For discrete: Find smallest x where F(x) ≥ U
For continuous: Use inverse CDF formulas
Box-Muller generates TWO Normal(0,1) values from TWO Uniform(0,1) values
XOR
XOR |
is only true if different |
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Expected Value, Variance, ...
Discrete E[X] |
SUM[x * f(x)] |
Continous E[X] |
SUM[x * f(x) dx] |
Variance of X |
E[X2] - (E[X])2 |
Standard Deviation of X |
SQRT[Var(X)] |
Expected Value Joint pdf
Joint p.d.f.: f(x,y) = 2xy²; Domain: 0 < x < 1, 0 < y < 1 |
Find: E[2X-1] |
E[2X - 1] = ∫₀¹ ∫₀¹ (2x - 1) · 2xy² dy dx |
Random Number Generators
Bad generators |
Midsquare number generator, Random number tables, von Neumann's mid-square method, Fibonacci generator, Additive congruential generator, RANDU |
Good generators |
Linear Congruential Generators (modern cycle length > 2191; Mersenne Twister (219937) |
Randu |
65539Xi mod(231) |
Desert island |
16807Xi mod(231-1) mod(2147483647) |
Desert island technique |
Z = [SUM(Ui)-n/2] / [SQRT(n * 1/12] |
Inverse Transform Method Key Problems
If X ~ Normal(0,1), what’s the distribution of Φ(X)? |
Unif(0,1) |
If U∼Uniform(0,1) and Φ(x) is the CDF of the standard normal, what is the distribution of 2Φ −1 (U)+3? |
Φ−1(U) turns a Uniform(0,1) into a Normal(0,1). Multiply by 2 → scales the standard deviation by 2 = Normal(0,4). Add 3 → shifts the mean to 3. = Normal(3,4) |
-3ℓn(U²V²) where U, V ~ i.i.d. Unif(0,1) |
= -6ℓn(U) - 6ℓn(V) ~ Exp(1/6) + Exp(1/6) ~ Erlang₂(1/6) |
Joint Probability Mass Function
E[XY] |
Summe von x * y * f(xy) |
Joint p.m.f.
Are independent if |
X and Y are independent if and only if pX,Y(x,y) = pX(x) ⋅ pY(y) |
For example |
P(X=1, Y=0) = P(Y=0) * P(X=1) |
E[XY] |
Example: (1)(0)(0.2) + (1)(1)(0.0.0) + ... |
Cov(X,Y) |
E[XY] - E[Y]E[X] |
Variance X + Y |
Var(X) + Var(Y) + 2Cov(X,Y) |
Variance X - Y |
Var(X) + Var(Y) - 2Cov(X,Y) |
Theorem |
Cov(X,Y) = 0 if X, Y indepedent. Converse not true. |
Correlation |
p = Cov(X,Y) / SQRT(Var(X) * Var(Y)) |
pdf, cdf
pdf -> cdf |
integrate with x, 0 as limit |
cdf F-1(U) |
solve(F(x)=U, x) |
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Complete Distribution Reference Table
Box Muller Method
Z1 |
√(-2·ln(U₁)) · cos(2π·U₂) |
Z2 |
√(-2·ln(U₁)) · sin(2π·U₂) |
Z1/Z2 |
cot(2π·U₂) ~Cauchy |
Z2/Z1 |
tan(2π·U₂) ~Cauchy |
Radian-Modus einschalten!
Chi-Square Distribution
If Z₁, Z₂, Z₃ are i.i.d. Nor(0,1), find c such that P(Z₁² + Z₂² + Z₃² < c) = 0.99
Calculator: chiSqInv(0.99, 3) → 11.34
X₁, X₂, X₃, X₄ are i.i.d. Nor(0,1). Find Pr(X₁ + X₂ - X₃ - X₄ > 10)
Calculator normCdf(10,unendlich,0,n/2) |
Find distribution of U1 and U2
Find distribution of -4(U₁ + U₂) - 2 |
U₁ + U₂ ~ Tria(0, 1, 2) |
Apply the transformation -4(U₁ + U₂) = 4(Tria(0, 1, 2) ) |
The minimum becomes: -4(2) = -8; The mode becomes: -4(1) = -4; The maximum becomes: -4(0) = 0 |
4·Tria(0, 1, 2) = Tria(-8, -4, 0) |
Subtract 2 |
Tria(-10, -6, -2) |
Acceptance-Rejection
Goal |
Generate random samples from a hard-to-sample distribution (f(x)). |
Idea |
Sample from an easier distribution (h(x)), then accept or reject each sample based on how well it fits f(x). |
Example |
If a random variable X has the beta distribution, then its p.d.f. is of the form f(x) = Γ(α+β) Γ(α)Γ(β) xα−1(1−x)β−1, 0 <x<1, for parameters α and β >0, and where Γ(·) is the gamma function. How might you generate such a random variate? Pick the best answer. |
Goodness of fit
Find critical value |
Inverse X2; Area = 1-alpha; df = n-1; invx2(0.9,3) |
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If critical value is bigger than accept H0 |
Hypothesis Testing Errors
Type I Error (False positive) |
α = P(Reject H₀ | H₀ is true) , Fire alarm when no fire |
Type II Error (False negative) |
β = P(Fail to reject H₀ | H₀ is false); No fire alarm when there IS a fire |
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