| Abstract: | 考慮系統人數為N之封閉式不可分馬可夫等候網路。於1997年，就固定人數之網路系統，Jin、Ou和Kumar可藉由解出兩個線性規劃■和■的解，來獲得網路流量的上、下限。另外他們也藉由解出兩種類型的線性規劃（■、■、■和■）來獲得函數形式（N的函數）之流量上、下限。為了方便起見，假如T是一個線性規劃，VT則代表此線性規劃之最佳值。本文中我們證明了■ 。另一方面，若從函數形式的下限來看，當網路人數逐漸增加時，函數形式的下限也會跟著趨近■。因此當網路人數非常多時，藉由只解出■之最佳值，我們就可以獲得網路流量之下限。如果將這個特殊的線性規劃■擴充至一般的線性規劃，稱為■，我們猜測假如■是有界的，則會存在另一個線性規劃■，使得■。而這個線性規劃■可用簡單之方式，改變原來的線性規劃■來獲得。 1990年，Harrison和Wein（HW）對於兩個站台的Brownian系統提出了一個策略。我們證實該策略能用在所有兩個站台且封閉式不可分之馬可夫等候網路。HW猜測該策略是漸進最佳策略。HW也猜測該策略之漸進損失是有限的，並且提出了漸進損失的公式。1997年，Jin、Ou和Kumar只證明出HW的策略在兩個站台的系統是有效的，他們並猜測假如另外加上一特殊條件，對於任何兩個站台之平衡系統，HW的猜測是對的。但是到目前為止，關於HW策略之漸進最佳化還是沒有被證明出來。在本文中，我們可以導出有關漸進損失和這個附加條件之間的關係。對於一個特別設計的系統，我們可以證明所有非閒置的策略都是漸進最佳化。假如這樣的系統剛好又是平衡且迴歸線性的系統，那麼所有非閒置策略的漸進損失值是1。
For an irreducible closed Markovian network with N customers, Jin, Ou, and Kumar (1997) obtained pointwise bounds of the network throughput for any fixed integer N by solving linear programs ■ and ■. They also obtained other two linear programs ■ and ■ that can be used to develop the corresponding linear programs ■ and ■. Then the functional upper bound and lower bound of the network throughput can be derived from the objective values of the linear programs ■ and ■, and the optimal values of the linear programs ■ and ■ respectively. For convenience, if T is a given linear program, denote its optimum objective value by VT. We prove that ■. On the other hand, the limit of the functional lower bound as N approaches to infinity is also equal to ■. Thus the lower bound, in heavy traffic, can be obtained by solving ■ only. Extend this special linear program ■ to a general linear program, say ■, where N can be viewed as a variable. We conjecture that if ■ is bounded then there exists a limiting program ■ such that ■. This limiting program ■ can be easily obtained from modifying the original linear program ■. Harrison and Wein (HW) (1990) proposed a buffer priority policy for two-station Brownian networks. We prove that it is indeed applicable to all two-station irreducible Markovian closed networks. HW conjectured that buffer priority policy is asymptotically optimal, and the asymptotic loss always has a finite value expression. Jin, Ou, and Kumar (1997) proved that the HW-policy is efficient for all two-station systems. They conjectured that under the ''additional condition'', all of the conjectures of HW are true for balanced two-station systems. However, a proof of its asymptotic optimality has so far been unavailable. We are able to develop a relation between asymptotic loss and the ''additional condition''. For a specially constructed system, we can also prove that all non-idling policies are asymptotically optimal. In this specially constructed system, if it is a balanced re-entrant line two-station system, the value of the asymptotic loss is 1. |
