### Table 3 A comparison of the modified-offered-load (MOL) approximation with exact values of the peak delay probability and peak mean waiting time as a function of the number of servers for the model with arrival- rate function (t) = 20 + 10 sin(0:2t) and mean service time ES = 1. The exact infinite-server peak mean 29.81 is used as the offered load for MOL. number of delay probability mean waiting time

1995

"... In PAGE 13: ... Table3 compares the MOL approximation with exact values for the peak delay probability and the peak mean waiting time before beginning service. (Recall that the mean waiting time equals the mean number in queue divided by the number of servers.... ..."

### Table 3 A comparison of the modified-offered-load (MOL) approximation with exact values of the peak delay probability and peak mean waiting time as a function of the number of servers for the model with arrival- rate function (t) = 20 + 10 sin(0:2t) and mean service time ES = 1. The exact infinite-server peak mean 29.81 is used as the offered load for MOL. number of delay probability mean waiting time

1995

"... In PAGE 13: ... Table3 compares the MOL approximation with exact values for the peak delay probability and the peak mean waiting time before beginning service. (Recall that the mean waiting time equals the mean number in queue divided by the number of servers.... ..."

### Table 3: Parameter of the parallel web server model

"... In PAGE 16: ... Transitions f2s, s2f, s2st and st2s represent the changes of the network state. The parameters of the model are given in Table3 . Transitions shortsrv and fastsrv have infinite server semantic.... ..."

### Table 3. Parameter of the parallel web server model

"... In PAGE 8: ... Transitions CUBED7, D7BECU, D7BED7D8 and D7D8BED7 represent the changes of the network state. The parameters of the model are given in Table3 . Transi- tions D7CWD3D6D8D7D6DA and CUCPD7D8D7D6DA have infinite server semantic.... ..."

### 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.... ..."

Cited by 2

### Table 5.13: Summary of results of accuracy tests of each algorithm for multiple class separable queueing networks

1997

Cited by 2

### Table 2 The actual lags in peak congestion for three performance measures as a function of the number of servers, s, for the arrival-rate function 20 + 10 sin(0:2t). number lag in lag in lag in

1995

"... In PAGE 12: ... (For an exponential distribution, Se is distributed the same as S.) Table2 displays the actual lags in the peak for the three performance measures asafunctionofs.Whensis large, the actual lags are very close to the infinite-server lag.... ..."

### Table 2 The actual lags in peak congestion for three performance measures as a function of the number of servers, s, for the arrival-rate function 20 + 10 sin(0:2t). number lag in lag in lag in

1995

"... In PAGE 12: ... (For an exponential distribution, Se is distributed the same as S.) Table2 displays the actual lags in the peak for the three performance measures asafunctionofs.Whensis large, the actual lags are very close to the infinite-server lag.... ..."

### 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.... ..."

Cited by 2

### 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.... ..."

Cited by 3