Week 5

Question 1

Typical Discrete-Time Output Performance Measures

Answer 1

Di = delay of the ith customer in queue

Average delay: Đ(N) = ΣDi/N
Maximum delay: D*(N) = maxDi
Proportion of customers that have waited more than 3 minutes: P3(N) = Σ I(3,∞)(Di)/N
-> where I(3,∞)(Di) = 1 if Di > 3, 0 else

Question 2

Typical Continuous-Time Output Performance Measures

Answer 2

Time-average length of queue: Q̄(T) = ∫Q(t)dt)/T (integral from 0 to T)
Maximum length of queue: Q*(T) = max Q(t)
Proportion of time that there was more than 2 people in queue: P2(T) = ∫I(2,∞)(Q(t))dt/T

Server utilization: U(T) = ∫B(t)dt/T, where B(t) is busy function = 1 if server busy at time t

Holding cost in inventory: H(T) = h∫S(t)dt/T, where S(t) = # of stocks at time t

Question 3

Terminating vs. Non-Terminating Simulation

Answer 3

Terminating simulation: there is a natural event that specifies the end of each replication
Non-terminating simulation: there is no natural event that specifies the end of a replication

Question 4

Steady-State Parameter

Answer 4

A performance measure that is a characteristic of the steady-state distribution

• Theoretically, it does not depend on initial conditions
• Practically, we must ensure that run is long enough so that the impact of initial-condition is minimized

Not all non-terminating systems are steady-state: there could be a periodic “cycle” in the long run, giving rise to steady-state cycle parameters

Question 5

When is steady-state relevant?

Answer 5

24 hours/day, manufacturing companies

Question 6

Recursive Computation of Mean and Variance

Answer 6

1. Initialize Xbar 0 = 0, S1^2 = 0
2. Update mean: Xbar k+1 = Xbar k + (X k+1 - Xbar k)/(k+1)
3. Update variance: Sk+1^2 = (1 - 1/k)Sk^2 + (k+1)(Xbar k+1 - Xbar k)^2

Question 7

Mean Square Error (MSE)

Answer 7

E[ (Xbar - θ)^2 ] = σ^2/n

We can stop generating data when σ/sqrt(n) small
LOLN: at least 100 draws

Question 8

Empirical Distribution Function

Answer 8

X1, …, Xn with CDF F
Consider a realization x1, …, xn of these r.v.s

Fe(x) = 1/n Σ I{xi ≤ x}, I{xi ≤ x} = 1 if xi ≤ x

Question 9

Bootstrapping Purpose

Answer 9

To calculate the MSE of the empirical distribution

Impossible to compute analytically when n is large

Question 10

Bootstrapping

Answer 10

• For r = 1, …, R
• Draw x1^r, … xn^r from Fe, that is drawn from the data:
  
  • Let s be a draw from U[0,1]
  • Set j = floor(ns)
  • Return xj
• Compute Mr = [θ^*(x1^r, … xn^r) - θ(Fe)]^2
• Estimate of MSE(Fe) and, therefore of MSE(F):
  1/R Σ Mr

-Typical value for R: 100

Question 11

Bootstrapping Understanding

Answer 11

Take θ(Fe) from empirical distribution
Find θ^ for each sample
Find MSE for each sample: [θ^ - θ(Fe)]^2
Take average of MSEs to find bootstrap approximation

Question 12

Confidence Interval Formula

Answer 12

Xbar (n) ± t(n-1, 1-α/2)sqrt(S^2(n)/n)

Question 13

Idea of Specified Precision

Answer 13

Make the half-width of the CI small enough

δ(α, n) = t(n-1, 1-α/2)sqrt(S^2(n)/n)

Question 14

Absolute Precision

Answer 14

Specify β > 0, want n big enough s.t. δ(α, n) < β

Requires at least some knowledge of context to set meaningful β

Question 15

Relative Precision

Answer 15

Card not working - delta/xbar < gamma; don’t need to know much to set meaningful

Question 16

Sequential Approach

Answer 16

Idea: Increase n until δ(α, n) is small enough

Algorithm:

1. Choose a confidence level and an acceptable value β for CI length
2. Generate at least 100 data values
3. Continue to generate additional data, stopping when δ(α, n*) < β
4. The estimate is given by Xbar(n) = 1/n ΣXi

Question 17

Two-Stage Approach

Answer 17

Idea: One quick way to estimate n* is to use fixed estimates of S(n) based on a small # of replications
1. Use a pilot sample (small number of trial runs e.g. n1 replications) to get estimates of sample variance
2. Substitute these in the formulas for the required sample sizes
n* = min{i≥n1 : t(i-1, 1-α/2)*sqrt(Sn1^2/i) < β)

Question 18

Using Normal Distribution instead of t-student

Answer 18

n* = Sn1^2* (Z(α/2)/β)^2

Question 19

Replication/Deletion Approaches Disadvantages

Answer 19

No completely reliable method to identify an appropriate warm-up period
Too long -> wasteful of data
Too short -> point-estimator bias, which can have serious consequences

Question 20

Reason to Use Bootstrap

Answer 20

Have a small but significant sample of an unknown distribution and want to construct e.g. confidence intervals