A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue
Abstract
:1. Introduction
- Models whose phases have infinite state space can be considered, as long as their joint moments (of the phase and number of busy servers) are explicitly obtained.
- Scaling limits for the number of busy servers conditional on the phase can be considered.
2. M/M/∞ Queue and Preliminary Results
3. Preliminaries for the Moment Approach
4. Main Results
- The larger the variance of the batch-size distribution, the larger the variance of the normal distribution, i.e., the variance of the CLT takes the minimum value for the constant batch size.
- The CLT is equivalent to Lemma 4, i.e., the number of busy servers is independent of the number of waiting customers and batch size under heavy traffic, although the steady state probabilities in Lemma 3 show the dependency of these random variables.
5. Examples of Symbolic Computation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
Appendix B
Appendix C
References
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i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | |
---|---|---|---|---|---|
n = 3 | 0 | # | # | # | # |
n = 4 | 0 | # | # | # | # |
n = 5 | 0 | 0 | # | # | # |
n = 6 | 0 | 0 | # | # | # |
n = 7 | 0 | 0 | 0 | # | # |
n = 8 | 0 | 0 | 0 | # | # |
n = 9 | 0 | 0 | 0 | 0 | # |
n = 10 | 0 | 0 | 0 | 0 | # |
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Nakamura , A.; Phung-Duc , T. A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue. Mathematics 2023, 11, 2088. https://doi.org/10.3390/math11092088
Nakamura A, Phung-Duc T. A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue. Mathematics. 2023; 11(9):2088. https://doi.org/10.3390/math11092088
Chicago/Turabian StyleNakamura , Ayane, and Tuan Phung-Duc . 2023. "A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue" Mathematics 11, no. 9: 2088. https://doi.org/10.3390/math11092088
APA StyleNakamura , A., & Phung-Duc , T. (2023). A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/MX/∞ Queue. Mathematics, 11(9), 2088. https://doi.org/10.3390/math11092088