Please use this identifier to cite or link to this item: http://hdl.handle.net/11455/17511
標題: 含有單一可移動服務者之M/G/1 和 G/M/1排隊系統之最佳控制
Optimal Control of the M/G/1 and G/M/1 Queueing Systems with a Removable Server
作者: 柯昭川
Ke, Jau-Chuan
關鍵字: 比較分析
comparative analysis
G/M/1排隊系統
M/G/1排隊系統
極大熵原理
可移動服務者
遞迴方法
輔助變數技巧。
G/M/1 queueing system
M/G/1 queueing system
M/G/1 queueing system
maximum entropy principle
removable server
recursive method
supplementary variable technique.
出版社: 應用數學系
摘要: 本文旨在研究含有一可移動服務者M/G/1與G/M/1排隊系統之最佳控制,其中控制決策為N-方策。所謂N-方策,它的含意是指在系統中,如果要求服務之顧客數目累積到N個時,則服務站立即被開啟並開始提供服務;而當系統中所有顧客都被服務完時,則服務站立即被關閉,等系統中的顧客數目累積到N個時再被開啟。 首先我們利用輔助變數技巧(supplementary variable technique)和機率生成函數技巧(probability generating function technique),探討在N-方策下兩個具無限容量的M/G/1和G/M/1排隊系統,分別獲得這兩個排隊系統正確的穩定特性結果,諸如顧客數在系統中之機率分配、顧客數之期望數、顧客在隊伍中之等候時間分配、服務者忙碌與閒置週期的期望長度等。我們也利用極大熵原理(maximum entropy principle)對於這兩個排隊系統分別建立其近似之穩定特性結果,同時藉由某些在這兩個排隊系統已存在之正確結果及其相對的近似結果實施其數值比較分析。 其次,我們研究在N-方策下兩個具有限容量的M/G/1和G/M/1排隊系統,利用輔助變數技巧(supplementary variable technique)分別為這兩個排隊系統發展一有效率之遞迴方法來計算顧客數在系統中之機率分配,藉由電腦程式求得系統穩定特性數值結果,並分別對兩個排隊系統之不同服務(或到達)時間分配實施其穩定特性結果之數值比較分析。 最後,對以上四個N-方策排隊系統,我們分別定義其單位時間之穩態期望成本,藉由此成本決定控制參數N的最佳解N*,使成本函數達到最小。 關鍵字:比較分析,G/M/1排隊系統,M/G/1排隊系統,極大熵原理,可移動服務者,遞迴方法,輔助變數技巧。
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.
URI: http://hdl.handle.net/11455/17511
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