Optimization Analysis of the N Policy M/G/1 Queue with Working Breakdowns
Abstract
:1. Introduction
- (1)
- We derive several system performance measures, as well as the stability condition of this queueing model;
- (2)
- We establish a cost model to find the optimal threshold N, the optimal service rate during the normal period, and the optimal service rate during the working breakdown period under the stability condition;
- (3)
- We apply the two-stage optimization method to search for the minimum expected cost. Numerical examples are given to illustrate the effectiveness of the two-stage optimization method. Moreover, a sensitivity analysis is also performed.
2. Model Descriptions
Practical Justification of the Model
3. Steady-State Results
3.1. Steady-State Probability Equations
3.2. Probability Generating Function
3.3. Stability Condition
4. System Performance Measures
4.1. Computations for , , and
4.2. Computations for , , , and
- (1)
- Idle period I: the length of time during which the server is turned off or is removed from the system;
- (2)
- Busy period B: the length of time during which the server is turned on and in operation and customers are being served;
- (3)
- Partial breakdown period D: the length of time during which the server is broken down and customers are being served;
- (4)
- Busy cycle C: the length of time from the beginning of an idle period to the beginning of the next idle period.
4.3. Computations for and
5. Cost Optimization Analysis
5.1. Cost Function
5.2. Direct Search Method
5.3. Two-Stage Optimization Method
- Step 1.
- Set , and .
- Step 2.
- Set the initial trial solution for , convergence tolerance , inverse Hessian approximation , , and initialize by the direct search method.
- Step 3.
- Compute .
- Step 4.
- , , , ; repeat until (the Wolfe conditions).
- Step 5.
- Find the new trial solution , and according to , where is calculated from a line search method to satisfy the Wolfe conditions (see Nocedal and Wright [18]); that is,
- Step 6.
- Set and repeat Steps 3-5 if , , or , where , , and are the tolerances; otherwise, go to Step 7.
- Step 7.
- Find the minimum value , where .
5.4. Sensitivity Analysis for the Expected Cost Function
- Case 1: , , , ; select different values of 1, 3, 9, and vary from 0.6 to 0.8.
- Case 2: , , , ; choose different values of 1, 3, 9, and vary from 1.0 to 2.0.
- Case 3: , , , ; select different values of 1, 3, 9, and vary from 0.8 to 2.0.
- Case 4: , , , ; select different values of 1, 3, 9, and vary from 0.2 to 0.5.
- Case 5: , , , ; choose different values of 1, 3, 9, and vary from 3.0 to 5.0.
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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N | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|
0.4 | 339.12 | 316.37 | 326.74 | 345.78 | 368.51 | 393.24 | 419.20 | 446.00 | 473.39 | 501.23 | |
1.0 | 522.29 | 469.43 | 468.16 | 480.59 | 498.97 | 520.59 | 544.23 | 569.24 | 595.22 | 621.92 | |
2.0 | 926.11 | 921.24 | 936.73 | 958.26 | 982.65 | 1008.68 | 1035.73 | 1063.46 | 1091.67 | 1120.23 |
(0.4,0.2,0.3) | (1.0,0.2,0.3) | (2.0,0.2,0.3) | (1.0,0.25,0.3) | (1.0,0.3,0.3) | (0.4,0.2,0.4) | (0.4,0.2,0.5) | |
2 | 3 | 4 | 3 | 3 | 2 | 2 | |
3.061 | 4.277 | 5.313 | 4.336 | 4.387 | 2.944 | 2.889 | |
0.904 | 2.037 | 3.831 | 2.231 | 2.388 | 0.765 | 0.600 | |
310.70 | 448.78 | 591.93 | 458.86 | 467.79 | 302.07 | 296.21 |
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Yen, T.-C.; Wang, K.-H.; Chen, J.-Y. Optimization Analysis of the N Policy M/G/1 Queue with Working Breakdowns. Symmetry 2020, 12, 583. https://doi.org/10.3390/sym12040583
Yen T-C, Wang K-H, Chen J-Y. Optimization Analysis of the N Policy M/G/1 Queue with Working Breakdowns. Symmetry. 2020; 12(4):583. https://doi.org/10.3390/sym12040583
Chicago/Turabian StyleYen, Tseng-Chang, Kuo-Hsiung Wang, and Jia-Yu Chen. 2020. "Optimization Analysis of the N Policy M/G/1 Queue with Working Breakdowns" Symmetry 12, no. 4: 583. https://doi.org/10.3390/sym12040583