Asymptotic expansions for a stochastic model of queue storage
成果类型：
Article
署名作者：
Knessl, C
署名单位：
University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
2000
页码：
592-615
关键词：
摘要：
We consider an M/M/infinity queue with servers ranked as {1, 2, 3,...}. The Poisson arrival stream has rate lambda and each server works at rate mu. A new arrival takes the lowest ranked available server. We let S be the set of occupied servers and ISI is the number of elements of S. We study the distribution of max(S) in the asymptotic limit of rho = lambda/mu --> infinity. Setting P(m) = Pr [max(S) > m] we find that the asymptotic structure of the problem is different according as m = O(1) or m --> infinity, at the same rate as rho. For the latter it is furthermore necessary to distinguish the cases m/rho < 1, m/ approximate to 1 and m/rho > 1. We also estimate the average amount of wasted storage space, which is defined by E (max(S))- rho. This is the average number of idle servers that are ranked below the maximum occupied one. We also relate our results to those obtained by probabilistic approaches. Numerical studies demonstrate the accuracy of the asymptotic results.