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Outline. input analysis goodness of fit randomness independence of factors homogeneity of data Model 05-01. Chi-Square Test. arbitrary data grouping possibly good fit in one but bad in other groupings. Kolmogorov-Smirnov Test. advantages

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Outline

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  1. Outline • input analysis • goodness of fit • randomness • independence of factors • homogeneity of data • Model 05-01

  2. Chi-Square Test • arbitrary data grouping • possibly good fit in one but bad in other groupings

  3. Kolmogorov-Smirnov Test • advantages • no arbitrary data grouping as in the Chi-square test • goodness of fit test for continuous distributions • universal, same criterion for all continuous distributions • disadvantages • not designed for discrete distributions, being distribution dependent in that case • not designed for unknown parameters, biased goodness of fit decision for estimated parameters

  4. KS Test • F(x): underlying (continuous) distribution • Fn(x): empirical distribution of n data points • F(x) & Fn(x) being close in some sense • define Dn= supx|F(x) - Fn(x)| • if Dn being too large: data not from F(x)

  5. Idea of KS Test • continuous distribution F • Fn empirical distribution of F for n data points • F(x) = p • |Fn(x) - F(x)| ~ |Yp - p| for Yp ~ Bin(n, p) • supx |Fn(x) - F(x)| ~ supp |Yp - p|

  6. Test for Randomness Do the data points behave like random variates from i.i.d. random variables?

  7. Test for Randomness • graphical techniques • run test • run up and run down test

  8. Background • random variables X1, X2, …. (assumption Xi constant) • if X1, X2, … being i.i.d. • j-lag covariance Cov(Xi, Xi+j) cj = 0 • V(Xi) c0 • j-lag correlation j  cj/c0 = 0

  9. Graphical Techniques • estimate j-lag correlation from sample • check the appearance of the j-lag correlation

  10. Run Test • Does the following pattern of A and B appear to be random? • AAAAAAAAAAAAAAABBBBBAAAAA • Any statistical test for the randomness of the pattern? • # of permutations with 20A’s & 5B’s = 53130 • # of permutations with 5B’s together = 21 • an event of probability 0.000395

  11. Run Test for Two Types of Items • for two types of items • R: number of runs • AABBBABB: 4 runs by this 8 items • for na of item A and nb of item B • E(R) = 2nanb/(na+nb) + 1 • V(R) = • if min(na, nb) > 10, R ~ normal

  12. Run Test for Continuous Data • (43.2, 7.4, 5.4, 25.3, 27.3, 13.9, 67.5, 35.4) • sign changes:  + +  +  • 3 runs down & 2 runs up, a total 5 runs • R: number of runs, for n sample values • E(R) = (2n-1)/3 • V(R) = (16n-29)/90 • Dist(R)  normal

  13. Test for Independence Are the classifications of the random quantities independent? It is easier to simulate a system if the classifications are independent.

  14. Test for Independence • for two classifications • e.g., Is voting behavior independent of income levels • easier to simulate for independent voting opinion and income levels 2 ╳3 Contingency Table

  15. Test for Independence • independent income level and opinion • generate income level: 3 types (i.e., m types) • generate opinion: 2 types (i.e., n types) • generate an entity: 5 types (m+n types) • dependent income level and opinion • generate income level: 3 types (i.e., m types) • generate opinion: 2 types (i.e., n types) • generate an entity: 6 types (mn types) • for k factors (classifications) • independent: m1 + m2 + … + mk • dependent: m1 m2 … mk

  16. Test for Independence • H0: voting opinion and income levels are independent • H1: voting opinion and income levels are dependent

  17. Test for Independence • marginal distribution: • If H0 is true,

  18. Test for Independence • expected frequency: cell probability multiplies the total number of observations • in general, the expected frequency of any cell is:

  19. Test for Independence Observed and Expected Frequencies • d.o.f. associated with the chi-squared test is rnumber of rows cnumber of columns

  20. Test for Independence • Calculate for the r╳c Contingency Table • reject H0 if ; otherwise accept • dependent voting opinion and income levels

  21. Test for Homogeneity Are the entities of the same type?

  22. predetermined Test for Homogeneity 3 ╳3 Contingency Table

  23. Test for Homogeneity • row (or column) totals are predetermined • H0: same proportion in each row (or column) • H1: different proportions across rows (or columns) • analysis: same as the test of independence

  24. Test for Homogeneity • H0: Democrats, Republicans, and Independents give the same opinion (proportion of options) • H1: Democrats, Republicans, and Independents give different opinion (proportion of options) • = 0.05 • critical region:χ2 > 9.488 with v = 4 d.o.f. • computations: find the expected cell frequency

  25. Test for Homogeneity Observed and Expected Frequencies Decision: Do not reject H0.

  26. Model 5-1: An Automotive Maintenance and Repair Shop • additional maintenance and repair facility in the suburban area • customer orders (calls) • by appointments, from one to three days in advance • calls arrivals ~ Poisson process, mean 25 calls/day • distribution of calls: 55% for the next day; 30% for the days after tomorrow; 15% for two days after tomorrow • response missing a desirable day: 90% choose the following day; 10% leave

  27. An Automotive Repair and Maintenance Shop • service • Book Time, (i.e., estimated service time) ~ 44 + 90*BETA(2, 3) min • Book Time also for costing • promised wait time to customers • wait time = Book Time + one hour allowance • actual service time ~ GAMM(book time/1.05, 1.05) min • first priority to wait customers • customer behavior • 20% wait, 80% pick up cars later • about 60% to 70% of customers arrive on time • 30% to 40% arrive within 3 hours of appointment time

  28. Costs and Revenues • schedule rules • at most five wait customers per day • no more than 24 book hours scheduled per day (three bays, eight hours each) • normal cost: $45/hour/bay, 40-hour per week • overtime costs $120/hour/bay, at most 3 hours • revenue from customers: $78/ book hour • penalty cost • each incomplete on-going car at the end of a day: $35 • no penalty for a car whose service not yet started

  29. System Performance • simulate the system 20 days to get • average daily profit • average daily book time • average daily actual service time • average daily overtime • average daily number of wait appointments not completed on time

  30. Relationship Between Models • Model 5-1: An Automotive Maintenance and Repair Shop • a fairly complicated model • non-queueing type • Model 5-2: Enhancing the Automotive Shop Model • two types of repair bays for different types of cars • customer not on time

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