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

국문 초록 (Abstract)

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.

다국어 초록 (Multilingual 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.

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