M/M/s와 M/D/s 대기행렬의 점근 근사법 분석을 위한 시뮬레이션 연구
Simulation Studies on Asymptotic Approximations Analysis of M/M/s and M/D/s Queues
0
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
본 연구는 M/M/s와 M/D/s 대기행렬의 점근 근사법 분석을 수행한다. M/M/s 대기행렬 분석을 위해, 활용률과 고객의 시스템 대기확률이 특정값을 가질 때 서버수 s의 증가에 따라 평균 시스템 대기자의 크기가 변화하는 양상을 통해 규모의 경제를 관찰하였다. 시뮬레이션 결과, s가 증가함에 따라 유휴시간을 갖는 서버의 수가 발산함을 보여주었다. 그리고 고객의 수 k의 변화에 따른 시스템 성능을 관찰한 결과, heavy-traffic regime(활용률이 점점 증가하는 상태)을 살펴보기 위하여 활용률과 서버수의 관계식을 이용하는 것은 k가 1인 경우에만 유효함을 확인하였다. M/D/s 대기행렬의 경우 고정된 활용률과 대기확률 하에서 평균 시스템 대기자의 크기 분석을 위해 두 가지의 근사법을 이용하였다. 시뮬레이션 및 비교 분석 결과, 서버의 수가 작고 heavy-traffic인 경우에는 Cosmetatos 근사법이 좋은 성과를 보여 주지만, 서버의 수가 큰 경우에는 실제값을 과대평가하는 경향이 있음을 보여주었다. 반면에 수정 근사법(modified approximation)은 서버의 수가 증가할 때에도 시스템 안정상태에 대한 보다 정확한 근사치를 제공하였다.
Abstract
This paper deals with asymptotic approximations analysis of M/M/s and M/D/s queues. For M/M/s queue, we observe economies of scale under the fixed utilization and the fixed probability that customer waits in system, how the average system size vary according to the number of servers (s) increasing. Simulation results show that as s increases, the number of servers who are idling increases, that is, the slack diverges. In addition, through changing the waiting probability under the M/M/s system, this probability was not highly sensitive to the behavior of the system size. And, it is shown that using the relationship between the utilization and the number of servers to handle heavy-traffic regime is only appropriate for the case where the number of customers (k) is one, by observing the effect on the performance of the system with different values of k. For the M/D/s queue, two approximations are used to evaluate the expected system size under the fixed utilization and the fixed waiting probability. Simulations and comparison of these two approximations show that Cosmetatos’approximation performs quite well when s is small and traffic intensity is heavy, but it overestimates the true value for a large s. Meanwhile, the modified approximation gives good results for the steady state count of the system although s grows large.
참고문헌 (23)
- Boxma, O. J., "Approximations of the mean waiting time in an M", G, Operations Research, pp.1979, 7(6)[DOI]
- Cooper, R.B., "Introduction to Queueing Theory" (1972), Macmilan
- Cosmetatos, G. P., "Approximate explicit formulae for the average queueing time in the processes (M", D, M, pp.r), INFOR: Information Systems and Operational Research[DOI]
- Crommelin, C. D., "Delay probability formulae", Post Office Electrical Engineer's Journal, 26, pp.266~274 (1934)
- Cruz, F. R. B., "Bayesian estimation of traffic intensity based on queue length in a multi-server M", M, Communications in Statistics - Simulation and Computation, pp.2017, 46(9)[DOI]
- Halachmi, B., "A diffusion approximation to the multi-server queue", Management Science, 24, pp.522~529 (1978)
- Halfin, S., "Heavy-traffic limits for the queues with many exponential servers", Operations Research, 29(3), pp.567~588 (1981)
- Hokstad, P., "Approximations for the M", G, Operations Research, pp.1978, 26[DOI]
- Jeong, Y., "The effects of fairness and quality on the trust and loyalty in the R&D process", Journal of Service Research and Studies, 8(3), pp.115~136 (2018)
- Kimura, T., "Refining Cosmetatos'approximation for the mean waiting time in the MID", s queue, 1991, pp.42(7), 595~603
- Kimura, T., "Approximations for multi-server queues: System interpolations", Queueing Systems, 17, pp.347~382 (1994)
- Kumar, S., "Stochastic Networks", pp.2008
- Lee, J.W., "Classification of servic quality for HMR unmanned store business", Journal of Service Research and Studies, 13(2), pp.41~61 (2023)
- Miyazawa, M., "Approximation for the queue-length distribution of an M", GI, Journal of Applied Probability, pp.1986, 23[DOI]
- Maaloe, E., "Approximation formulae for estimation of waiting-time in multiple-channel queueing systems", Management Science, 19, pp.703~710 (1973)
- Nakamura, A., "Exact and asymptotic analysis of infinite server batch service queues with random batch sizes", Queueing Systems, 106, pp.129~154 (2024)
- Park, J., "Analysis of an M", M, Journal of the Korea Industrial Information Systems Research, pp.2018, 23(5)[DOI]
- Shim, J., "The effects of traditional leadership recognized by members of the telemarketing organization on orgnaizational perfomance-centered on the moderating effect of followship", Journal of Service Research and Studies, 12(3), pp.45~59 (2022)
- Smith, V. L., "Approximating the distribution of customers in M", En, Journal of Operations Research Society, pp.1985, 36[DOI]
- Takahashi, Y., "An approximation formula for the mean waiting time of an M", G, Journal of Operations Research Society of Japan, pp.1977, 20[DOI]
- Tijms, H. C., "Approximations for the steady-state probabilities in the M", G, Advances in Applied Probability, pp.1981, 13[DOI]
- Wang, X., "A study on effect of service characteristic factors of theme park on customer satisfaction and revisit intention", Journal of Service Research and Studies, 10(2), pp.43~57 (2020)
- Wu, J. S., "Refining the diffusion approximation for the G", G, Computational Mathematics and. Applications, pp.1990, 20[DOI]