### Table 2 Queueing Network Model

"... In PAGE 18: ... Finally notice that PEPS requires exactly the same amount of time even though the reachable state space ranges from large to small. Table2 shows the results for the queueing network example. The descriptor is com- posed of 3 tensor terms without functions, and 3 tensor terms with a single functional factor when 2 is constant.... ..."

### Table 6 Queueing Network Model

"... In PAGE 17: ... A grouping of the automata according to queue (A1 and A2) and (A31 and A32); The second grouping allows for the possibility of a reduction in the state space of the joint automata, (A31 and A32), since the priority queue is now represented by a single automaton. The results obtained are presented in Table6... ..."

### Table 4 Queueing Network Model

"... In PAGE 23: ... A grouping of the automata according to queue (A1 and A2) and (A31 and A32); The second grouping allows for the possibility of a reduction in the state space of the joint automata, (A31 and A32), since the priority queue is now represented by a single automaton. The results obtained are presented in Table4 and Figure 8. The last column of Table 4 shows the results for 1 group, thus the sparse method.... In PAGE 23: ... The results obtained are presented in Table 4 and Figure 8. The last column of Table4 shows the results for 1 group, thus the sparse method.... ..."

### Table 5: The average queue lengths of the queueing network

2002

"... In PAGE 28: ... The iterations, along with the refined Brownian estimates, are given in Table 8 to 10. The case n = 1 corresponds the original Brownian model whose results are shown in Table5 to 7. By observing the numerical results in Table 8 to 10, we can see that the above iterative procedure provides a slightly better Brownian model for performance evaluation compared to the original Brownian model, especially in System No.... ..."

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### Table 6: The average throughput rates of the queueing network

2002

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### Table 3. Parameters for generating single class separable queueing networks

2000

"... In PAGE 9: ... We chose single class separable queueing networks because it is feasible to obtain the exact solution of such networks. The parameters used to generate the random networks in each of these experiments are given in Table3 . The detailed experimental results for these ve experiments are presented elsewhere [16].... ..."

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### Table 1. Parameters for generating multiple class separable queueing networks

2000

"... In PAGE 8: ...1 Experiments for Multiple Class Separable Queueing Networks In the rst set of experiments, two thousand random networks were generated and solved by each of the three approximate algorithms and the exact MVA algorithm for each number of classes from one to four. The parameters used to generate the random networks are given in Table1 . The mean absolute relative errors in throughput, response time, queue length, and the maximum absolute relative errors in queue length are shown in Table 2.... In PAGE 9: ... By applying the statistical hypothesis testing procedure [16], we can further conclude that, on the average, the accuracy of all three approximate MVA algorithms decreases as the number of classes increases, and the FL algorithm is the most accurate while the PE algorithm is the least accurate algorithm among the three algorithms for multiple class separable queueing networks with su ciently small population. However, these conclusions are only valid for small networks with a small number of classes, and small customer populations as governed by the parameters in Table1 . Although we would like to have experimented with larger networks with more classes and larger populations, the execution time of obtaining the exact solution of such networks prevented us from doing so.... In PAGE 9: ... 4.2 Experiments for Single Class Separable Queueing Networks In order to gain some insight into the behavior of the three approximate MVA algorithms for larger networks than those whose parameters are speci ed in Table1 , the second set of experiments was performed. This involved ve ex- periments for single class separable queueing networks.... ..."

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### Table 2: Comparison of estimator sample variances for queueing network model.

1998

"... In PAGE 23: ...fter 106 transitions. We let v = (0; 0; 0; 0; 0; 0; 0; 0) in all cases. In the rst part (i.e., the rst three rows) of Table2 , we vary the choice for the state w over those states that have exactly 3 customers at one station and none at each of the other stations. In the second part (i.... In PAGE 23: ...e., the last ve rows) of Table2 , we vary the choice for w over those states that have exactly n1 customers at station 1 for n1 = 3; 7; 10; 12; 15, and no customers at any of the other stations. In each replication we computed the standard and permuted estimates of the time-average variance constant, and then we calculated the sample variances of these estimates over the 1,000 independent replication.... In PAGE 23: ...01 percent), so the ratio of sample variances is essentially also our estimate of the relative e ciency (Glynn and Whitt 1992) of the permuted estimator. Now we consider the rst part of Table2 . Recall that these rows correspond to states w having exactly 3 customers at a particular station k, and no customers at any other station, where k = 1; 2; 3.... In PAGE 24: ...Table 2: Comparison of estimator sample variances for queueing network model. rst part of Table2 seems to indicate that the amount of variance reduction increases as the steady-state probability of the state w increases. In the second part of Table 2, recall that the di erent rows correspond to states in which there is some number of customers at the rst station and no customers at any of the other stations.... In PAGE 24: ... rst part of Table 2 seems to indicate that the amount of variance reduction increases as the steady-state probability of the state w increases. In the second part of Table2 , recall that the di erent rows correspond to states in which there is some number of customers at the rst station and no customers at any of the other stations. Of these states, the steady-state probabilities of those having many customers at station 1 are higher than those with fewer customers at station 1, and the second part of Table 2 shows that the variance reduction seems to increase for state w having more customers at station 1.... In PAGE 24: ... In the second part of Table 2, recall that the di erent rows correspond to states in which there is some number of customers at the rst station and no customers at any of the other stations. Of these states, the steady-state probabilities of those having many customers at station 1 are higher than those with fewer customers at station 1, and the second part of Table2 shows that the variance reduction seems to increase for state w having more customers at station 1. The results in the two parts of Table 2 agree with the suggestions given in Calvin and Nakayama (1997,1998a) that one should try to choose a state w that is visited often to maximize the variance reduction.... In PAGE 24: ... Of these states, the steady-state probabilities of those having many customers at station 1 are higher than those with fewer customers at station 1, and the second part of Table 2 shows that the variance reduction seems to increase for state w having more customers at station 1. The results in the two parts of Table2 agree with the suggestions given in Calvin and Nakayama (1997,1998a) that one should try to choose a state w that is visited often to maximize the variance reduction. Finally, we see in Table 2 that permuting can result in a signi cant decrease in the variance (up to about one order of magnitude).... In PAGE 24: ... The results in the two parts of Table 2 agree with the suggestions given in Calvin and Nakayama (1997,1998a) that one should try to choose a state w that is visited often to maximize the variance reduction. Finally, we see in Table2 that permuting can result in a signi cant decrease in the variance (up to about one order of magnitude). 8 Conclusions In this paper we introduced permuted estimators for various performance measures.... ..."

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### Table C.1 Characteristics of the NCD Queueing Network Problem. symmetric

in Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains

1998

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