Lower Bound of Network Throughput and Validity of HW Conjectures for Irreducible Closed Markovian Queueing Network

• Main Authors: Ya-Yin Yang, 楊雅茵
• Other Authors: Kwang-Fu Li
• Format: Others
• Language: en_US
• Published: 1999
• Online Access: http://ndltd.ncl.edu.tw/handle/16951459551419532664

碩士 === 東海大學 === 數學系 === 87 === 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.