Optimal Control of the M/G/1 and G/M/1 Queueing Systems with a Removable Server

• Main Authors: Jau-Chuan Ke, 柯昭川
• Other Authors: Kuo-Hsiung Wang
• Format: Others
• Language: en_US
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/70428949168136232742

博士 === 國立中興大學 === 應用數學系 === 88 === In this dissertation, we deal with the optimal control of a single removable server in M/G/1 and G/M/1 queueing systems operating under the N policy in which the server may be turned on at arrival epochs or off at service completion epochs. The server begins service only when the number of customers in the system reaches a certain number, say $N$ $(N \ge 1)$. The supplementary variable technique and the probability generating function technique are used to develop the exact steady-state results for the N policy M/G/1 and G/M/1 queueing systems with infinite capacity. Examples are presented to calculate the steady-state probability distribution of the number of customers in the N policy M/G/1 queueing system for three different service time distributions, including exponential, 2-stage Erlang and 2-state hyperexponential distributions. We provide two special cases in the N policy G/M/1 queueing system, such as the ordinary G/M/1 queueing system and the N policy M/M/1 queueing system. We use the maximum entropy principle to develop the approximate steady-state results for the N policy M/G/1 and G/M/1 queueing systems with infinite capacity. We perform comparative analysis between some exact results and the corresponding approximate results in the N policy M/G/1 queueing system for two different service time distributions, such as exponential and 3-stage Erlang distributions. We also provide comparative analysis between some exact results and the corresponding approximate results in the N policy G/M/1 queueing system for the exponential interarrival time distribution. We study the N policy M/G/1 and G/M/1 queueing systems with finite capacity $L$. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining service (or nterarrival) time, to establish the steady-state probability distributions of the number of customers in two finite queueing systems. To illustrate analytically for the two recursive methods, we present examples of different service time distributions, such as exponential, 3-stage Erlang and deterministic distributions, in the N policy M/G/1 queueing system and exponential interarrival time distribution in the N policy G/M/1 queueing system. We provide the numerical results of system characteristics for different service (or interarrival) time distributions in the N policy M/G/1 and G/M/1 queueing systems, including exponential, 2-stage hyperexponential, 4-stage Erlang and deterministic time distributions.