Problems with parameterization (example:keeper usage): average duration: 1.27, min: 0.106, max: 6.46 possible outcome for keeper and crane queues? Assignment 4

For validation, simulate long enough.

Half width determination by Arena: statistical analysis of samples. Arena help file: Half Width (Runtime Confidence Intervals—Within a Replication) Some sections contain a column called "Half Width". This statistic is included to help you determine the reliability of the results from your replication. Three results are possible in the "Half Width" category: Insufficient: The formula used to calculate half width requires the samples to be normally distributed. That assumption may be violated if there is a small number (fewer than 320) of samples. If that is the case, Arena will return the message "Insufficient" for that variable’s half width, indicating there is insufficient data to accurately calculate the half width. Running the simulation for a longer period of time should correct this. Correllated: The formula used to calculate half width requires the samples to be independently distributed. Data that is correlated (the value of one observation strongly influences the value of the next observation) results in an invalid confidence interval calculation. If data is determined to be correlated, the message "Correllated" is returned for that variable’s half width. Running the simulation for a longer period of time should correct this. A value: If a value is returned in the Half Width category, this value may be interpreted by saying "in 95% of repeated trials, the sample mean would be reported as within the interval sample mean ± half width". Half widths

Arena has to be trusted w.r.t. confidence values. Computations "inside Arena" not clear! - determine average and variance  ^2 - test for insufficient/correlated - half width = C.  for some constant C. Half width calculation Homework: Appendices B,C,D of lecture notes. Concepts from probability theory and statistics. ?

Increase confidence in simulation. Divide simulation run into subruns. Add initial run (move away from initial state). Randomization Random generator is in fact deterministic! Replaying same model gives same result. Still, it has all characteristics of "true" random generator (no "fairness"). DCT case: 15% increase BF trucks should diminish keeper queues. Short simulation: keeper queues might get longer! (see half width)

Example computation: page 30 of lecture notes. n = 30 subruns, each with sample of measure x (occupation rate). Replications Make sure that replications/subruns are independent (e.g. no queue length dependencies). When done right, division into subruns allows to compute confidence intervals.

Sample average/stddev: x = , s = how do you compute s ?  - confidence interval:

Computed confidence interval should match Arena's half width. Arena half width should (for large n) approximate cf. Chapter 6 of lecture notes Confidence interval matching This example: 1.96( / 5.51)= So with 95% probability, occ.rate in [0.936,0.945]

 - confidence interval: Consequences: - You can be 99.99% confident, but not 100%. - Four times longer simulation halves confidence int. - high variance = low confidence Function z : normal distribution surface (table lookup).

Comparisons Many simulation studies (e.g. DCT example) are about relative shortage of resources, leading to queues. Compare possible solutions through simulation. Simulation yields to following reports: Number waitingAverageHalf Width res1.Queue Sol1: Number waitingAverageHalf Width res1.Queue Sol2: Sol1 better?

Longer simulation needed to get following result: Number waitingAverageHalf Width res1.Queue Number waitingAverageHalf Width res1.Queue Number waitingAverageHalf Width res1.Queue Number waitingAverageHalf Width res1.Queue

Variance and confidence Large half widths caused by high subrun variance, require very long simulations for acceptable confidence. For instance, compare First samples: less variance, higher confidence, shorter simulation run

Variance reduction 1 Large half widths caused by high subrun variance, require very long simulations for acceptable confidence. Various techniques to reduce subrun variance, e.g. antithetic random numbers (see lecture notes) Subrun result depends on sequence of random numbers Next subrun: use antithetic sequence: Frequent arrivals, long processing times in a subrun compensated by infrequent arrivals and short times in next subrun.

Variance reduction 2 Second approach: sharing of e.g. arrival patterns. Less risk of adopting inferior solution with fewer arrivals. Both approaches introduce dependency in subruns. A third approach: compensate for the number of entities. sr mq #ent sr mq #ent sr mq #ent 5, , , , , subruns sortedcombined

Sensitivity analysis Simulation model based on assumptions; both modeling and parameters. Assess dependency of simulation result on assumptions. Simulate with modified parameters and compare. Sensitive parameters / uncertain assumptions: show various outcomes. avql sol Aavql sol B current demand % increase % increase