本篇論文提供了一個明確可求得的解，解決一固定速率來到的顧客，進入系統與否所應該採取的最佳策略，能使得系統本身不過載，亦無使用不足。為了達到此目的，有一套規則決定是否要讓到達的顧客進入系統。決策的目標函數是一個PDP(piecewise deterministic processes)的特例，已經有相關文獻指出解是存在且唯一的;但是要解決其所需滿足的微分差分方程，似乎是很困難的。在這篇文章裡，我們首先對最佳解的性質作一些推導，經由這些性質加上利用由後面往前推演的方式，本文可以找到最好的決策。最後提出了其他在直觀上合理決策作模擬比較。

This paper provides an explicit solution to the problem of regulating the stream of customers arriving at a facility so that they do not overload or under-utilize the facility. To do this, a rule by which some arriving customers are sent elsewhere must be decided upon. Although techniques based on optimal control of piecewise deterministic processes can be applied and necessary conditions of optimality such as the dynamic programming verification theorem exists, explicit solutions are difficult, if not impossible, to obtain. In this paper, we first give certain characterizations of the optimal solution and then use the backward method to obtain explicit solution for certain cases. Numerical comparisons with certain intuitive control policies are also made.

Rishel, R., “Controlled Continuous Time Markov Processes,” in Handbooks in Operations Research and Management Science, vol. 2, ed. by Heyman, D.P. and Sobel, M. J., 1990, North-Holland.

Davis, M.H.A. (1984). “Piecewise-deterministic Markov Processes: A General Class of Non-diffusion Models,” Journal of Royal Statistical Society, Series B, 46, 353-388.