MORFOGENETICOS 6.EXPERIMENTAR 7.DESDE NUESTRO CORAZÓN
6. CONCLUSIONES DE LAS CONVIVENCIAS
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.00 1.00 2.00 3.00 4.00 5.00
Number of Servers
Pr(Redo)
Avg. ReDo Prob. Exp. ReDo Prob.
Figure 12. Impact of number of servers on probability of reprocessing. The results for Pr(Redo) at each level of number of servers were averaged across all levels of the other
Table 3. Descriptive statistics for the time series representing the expected probability of reprocessing minus the simulated probability of reprocessing. This table shows the results for all 120 design points, under a FIFO dispatch rule. The last two rows show the
upper and lower limits for the 95% confidence interval for the time series.
FIFO - All Values
Mean Delta ReDo Prob. 0.018974 Standard Deviation 0.060497 Sample Variance 0.00366 Range 0.282414 Minimum -0.139028 Maximum 0.143386 Sum 2.276855 Count 120 95% Half Width 0.010935 95% Upper Limit 0.029909 95% Lower Limit 0.008039
A second confidence interval was computed using only the 96 stable design points. The results are shown in Table 4, with other descriptive statistics for the time series. Here the 95% confidence interval contains zero. This indicates that if a large number of simulation runs were made (and statistical assumptions were met), 95% of the time the approximation could be expected to perform quite well. This, too, is not
surprising, given how closely the curves match in Figure 10, Figure 11, and Figure 12 In addition to showing the accuracy of the M/M/c approximation, Figure 7 to Figure 12 illustrate the influence of the three input variables on the probability of reprocessing. Tighter time constraints, smaller interarrival times (more heavily loaded systems) and smaller numbers of tools per workstation all appear to be associated with increased probability of reprocessing. To confirm the significance of this influence, a 3- factor ANOVA was conducted on the data. The ANOVA table is included as Table 5.
Table 4. Descriptive statistics for the time series representing the expected probability of reprocessing minus the simulated probability of reprocessing. This table shows the results for the 96 stable design points only. The last two rows show the upper and lower
limits for the 95% confidence interval for the time series.
FIFO - All Stable Values
Mean Delta ReDo Prob.-0.005009 Standard Deviation 0.040085 Sample Variance 0.001607 Range 0.209201 Minimum -0.139028 Maximum 0.070173 Sum -0.480827 Count 96 95% Half Width 0.008122 95% Upper Limit 0.003113 95% Lower Limit -0.008122
Table 5. ANOVA table for experiment TWO-STATION_1. All main effects and interactions are significant with greater than 99.9 probability.
ANOVA Table - Experiment
SIGMA_1 Variation
Degrees of Freedom
Mean Square F Statistic
.999 Pecentile F
Stat. Time Constraint Effect 0.4331 5 0.0866 1319.95 4.42
Interarrival Time Effect 2.1668 4 0.5417 8253.82 4.95
Number of Servers Effect 3.7447 3 1.2482 19019.80 5.78
TC/InterArr Interaction 0.0162 20 0.0008 12.37 2.53 InterArr/NumServers Interaction 0.2084 12 0.0174 264.58 3.02 NumServers/TC Interactions 0.0896 15 0.0060 91.05 2.78 3 Way Interaction 0.0235 60 0.0004 5.97 1.95 Residual Variation 0.0158 240 0.0001 Total Variation 6.6981 359
The ANOVA found that all three main effects were highly significant (far less than 0.1% chance of the results occurring by chance). The two and three way interaction effects were also significant, to a somewhat lesser extent. This indicates that drawing general conclusions regarding probability of reprocessing is a complex endeavor. Pr(REDO) for a workstation depends upon the magnitude of the time constraint, the utilization of the workstation, and the number of servers at the workstation. Moreover, it depends upon how these characteristics interact with one another. The single-server system, for example, is much more sensitive to changes in the other parameters than the multi-server systems.
Overall, the results of this experiment show that for systems where the
interarrival and processing times are exponential, the dispatch rule is FIFO, the service rates of the two workstation are equal, and there are no random failures, an M/M/c approximation provides a reasonable guide for estimating whether or not a time- constrained system will be stable. This approximation is particularly valuable given that the probability of reprocessing for an actual workstation varies considerably depending on the parameters of the system. This result could be strengthened by looking at a wider range of time constraint values, and possibly by looking at lower traffic systems.
Tightening the time constraint, not surprisingly, increases the probability of reprocessing. Loading a system more heavily leads to longer queues, and also increases the probability of reprocessing. Having fewer servers in each workstation also leads to higher values of average queue time, and hence also increases the probability of reprocessing. The remaining experiments successively remove some of the simplifying assumptions, to see whether or not the same conclusions hold in more general cases.
Experiment TWO-STATION_2
The goals of the second experiment were to determine: 1) whether LIFO systems perform in general better or worse than FIFO systems under a time constraint; 2)
whether the other input factors have the same effect for LIFO systems that they have for FIFO systems; and 3) whether the M/M/c approximation works equally well for LIFO systems. As in the first experiment, the input variables were time constraint, interarrival time, and number of servers at the furnace and press workstations. The only difference from the first experiment was that lots were processed at the press workstation in LIFO order. Note that varying the dispatch rule at the furnace would not have influenced the reprocessing probability, because the time constraint does not come into play until after jobs are finished processing at the furnace. There were 120 total design points in this experiment. Each was replicated three times. The results for each level of each input variable were averaged across all values of other input variables, to investigate the overall effect of each variable on predicted and simulated reprocessing probability. The results are shown in Figure 13 to Figure 15. The results for the FIFO system are also shown on the same graph, for the purpose of comparison.
As in the FIFO case, the highest traffic point is unstable for all levels of the other factors, and the inaccuracy of the approximation for those values distorts the results. Therefore, the highest interarrival time was eliminated from the experiments, and the graphs were re-generated from the remaining 96 design points (4 levels of interarrival time, 6 levels of time constraint, and 4 levels of number of servers). The results are shown in Figure 16 to Figure 18.
Impact of Time Constraint on Pr(Redo)
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100Time Constraint
Pr(Redo)
LIFO Avg. ReDo Prob. Exp. ReDo Prob. FIFO Avg. ReDo Prob.
Figure 13. Impact of time constraint on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged across
Impact of Interarrival Time on Pr(Redo)
0.000 0.100 0.200 0.300 0.400 0.500 0.600 1.000 1.200 1.400 1.600 1.800 2.000 2.200Interarrival Time
Pr(Redo)
LIFO Avg. ReDo Prob Exp. ReDo Prob. FIFO Avg. ReDo Prob
Figure 14. Impact of interarrival time on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged across
Impact of Number of Servers on Pr(Redo)
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.00 1.00 2.00 3.00 4.00 5.00Number of Servers
Pr(Redo)
LIFO Avg. ReDo Prob. Exp. ReDo Prob. FIFO Avg. ReDo Prob.
Figure 15. Impact of number of servers on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged
Impact of Time Constraint on Pr(Redo)
Highest Traffic Eliminated
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.100
Time Constraint
Pr(Redo)
LIFO Avg. ReDo Prob. Exp. ReDo Prob. FIFO Avg. ReDo Prob.
Figure 16. Impact of time constraint on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged across
Impact of Interarrival Time on Pr(Redo)
Highest Traffic Eliminated
0.000 0.100 0.200 0.300 0.400 0.500 0.600 1.200 1.400 1.600 1.800 2.000 2.200
Interarrival Time
Pr(Redo)
LIFO Avg. ReDo Prob Exp. ReDo Prob. FIFO Avg. ReDo Prob
Figure 17. Impact of interarrival time on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged across
Impact of Number of Servers on Pr(Redo)
Highest Traffic Eliminated
0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.00 1.00 2.00 3.00 4.00 5.00
Number of Servers
Pr(Redo)
LIFO Avg. ReDo Prob. Exp. ReDo Prob. FIFO Avg. ReDo Prob.
Figure 18. Impact of number of servers on probability of reprocessing for FIFO and LIFO systems. The results for Pr(Redo) at each level of time constraint were averaged
across all levels of the other variables for which the furnace was stable.
From the graphs, it is clear that changing from a FIFO to a LIFO dispatch rule at the press workstation did not have a major impact. However, the values for Pr (REDO) from the LIFO experiments were in general slightly larger than those from the FIFO experiments. A one-sided t-test was used to test the null hypothesis that the mean Pr
alternate hypothesis that the LIFO experiments had a higher mean. The results of the test (with α = 0.05) are summarized in Table 6. According to the t-test, the null hypothesis would be rejected. The conclusion would be that the FIFO and LIFO experiments had a different mean, with the LIFO system performing slightly worse than the FIFO system. The probability of the observed difference in means occurring randomly is essentially zero.
Table 6. Results of a one-sided t-test comparing the probability of reprocessing under FIFO and LIFO. According to the test, the null hypothesis that FIFO and LIFO systems
have the same mean is rejected.
LIFO vs. FIFO - All Obs. Values LIFO FIFO
Mean 0.28866 0.27942
Variance 0.01964 0.01872
Observations 120 120
Pearson Correlation 0.99899 Hypothesized Mean Difference 0
df 119
t Stat 14.33072
P(T<=t) one-tail 0.00000 t Critical one-tail 1.65776
Although the t-test showed a difference in means for LIFO and FIFO, the graphs indicate that this difference is small relative to the magnitude of the factor effects. Therefore, the ANOVA was not repeated for the LIFO data. It is clear that the factors all remain significant under LIFO, and that the factors cause Pr (REDO) to move in the same direction as before. To confirm the graphical results regarding the appropriateness of the approximation, the time series {di} was computed, where for each design point d represented the expected probability of reprocessing minus the simulated probability of
reprocessing. A 95% confidence interval for d was then computed. The results are shown in Table 7. The fact that the confidence interval contains zero indicates that if a large number of simulation runs were made (and statistical assumptions were met), 95% of the time the difference between the predicted and simulated results would be very small. Thus, the M/M/c approximation continues to work well for the LIFO system.
Table 7. Descriptive statistics for the time series representing the expected probability of reprocessing minus the simulated probability of reprocessing. This table shows the results for all 120 design points, under a LIFO dispatch rule. The last two rows show the
upper and lower limits for the 95% confidence interval for the time series.
LIFO - All Values
Mean Delta ReDo Prob. 0.009726 Standard Deviation 0.062142 Sample Variance 0.003862 Range 0.280343 Minimum -0.149775 Maximum 0.130568 Sum 1.167169 Count 120 95% Half Width 0.011233 95% Upper Limit 0.020959 95% Lower Limit -0.001506 Experiment TWO-STATION_3
In the third experiment, the assumption of exponential service at both workstations was relaxed. The goal of the experiment was to see how the M/M/c approximation performed when the system no longer bore a direct relationship to a Jackson network. When the service times at the two workstations were both constant, the observed number of reprocessed jobs was always zero. This occurred despite
variability in the system introduced by exponential external arrival times. On closer reflection, this behavior makes sense. The service rates at the two workstations are equal. Consider first the single server case. Because service times at the furnace are constant, the time between arrivals to the press can never be less than the furnace service time. But since the press can always process a job in a time equal to the furnace service time, there is never any queueing for the press. Hence, there are never any reprocessed jobs. In the multi-server case, since the number of servers at each workstation is
balanced, each furnace can be effectively linked up with a single press machine, and the same lack of queueing holds. Since the probability of reprocessing is zero for all design points, the constant furnace/constant press results will not be discussed further at this time. Note that for systems with failures, and for systems where the two workstations are not so balanced in terms of service time and number of servers, the probability of reprocessing would probably not always be zero.
For systems in which either the furnace or press service time is constant, and the other service time is exponential, there is a positive probability of reprocessing.
However, because the systems have less variability than systems where both service times are exponential, the M/M/c approximation tends to overestimate the probability of reprocessing. The 120 design points from experiment TWO-STATION_1 were each repeated four times for this experiment, with the changes indicated in Table 8, for a total of 480 new design points.
Table 8. Additional factors and labels for experiment TWO-STATION_3
Experimental Condition I II III IV
Dispatch Rule FIFO FIFO LIFO LIFO
Furnace Distribution Constant Expo. Constant Expo. Press Distribution Expo. Constant Expo. Constant
Each design point was replicated three times. The results for each level of the three original input variables were averaged across all values of other input variables, to investigate the overall effect of each variable on predicted and simulated reprocessing probability. The results are shown in Figure 19 to Figure 21. The M/M/c probability of reprocessing is also shown on the figures. Although the simulation results from
experiments TWO-STATION_1 and TWO-STATION_2 (exponential service at both workstations, with FIFO and LIFO service) are not shown (for clarity) they would be very near to the approximation results at all points except the highest interarrival time point on Figure 20.
Not surprisingly, the M/M/c approximation performs poorly when one of the service time distributions is not exponential. The system with exponential service at the furnace and constant service at the press bears some resemblance to an M/D/c system, except that reprocessed jobs cause the arrival process to the press to no longer be Markovian. Unfortunately, closed-form exact results are not available for the waiting time distribution of the M/D/c or M/G/c systems. Some numerical results are available for the M/G/1 system [see, for example, Gross and Harris, p. 275]. However, solving the available equations requires performing repeated integration for the various experimental
conditions. Since such results are not easily included in static capacity models, they are not explored further here.