多佇列系統中佇列選擇法則的分析與研究 顏豪緯、陳木松
E-mail: 324686@mail.dyu.edu.tw
摘 要
日常生活中只要有人、事、物等待接受服務就可能形成排隊的現象，這類排隊等候服務就會形成佇列系統，而佇列理論即 是對於佇列內等待被服務的人/事/物(或統稱為事件)分析事件的等待時間、佇列長度、或事件有效進入率等的研究。 佇列 問題的研究隨著不同的需求可衍生出不同的形態，例如(I)將相同屬性的事件歸類於同一佇列，將有利於資源的分配與調度
，因此單佇列問題將轉換為多佇列模式的研究。(II)現今佇列問題大都假設容量無限，然而真實系統大多是屬於有限容量，
因此有限容量佇列問題的研究較具有應用價值，而且有限容量佇列模式的極限行為應類似於無限容量的佇列模式。(III)若 佇列內的事件有時效性的需求，則系統必須具有強健的排程能力，以預防或降低事件逾時服務的機率。 針對以上所述本論 文主要探討多佇列的問題，並依無限與有限容量佇列模式分析求解佇列參數，另外本論文也提出具有學習能力的訊息排程 控制器，以提高在時變系統時佇列的排程能力。關於無限容量的模式(q-M/G/1/inf/FCFS/EDF)的研究，本文以最早截止 期限優先(EDF)為佇列選擇規則(QSR)，並藉由定義等待時間的機率密度函數，計算佇列訊息的等待時間及訊息逾時服務率
。另外根據相對截止期限的極限行為，以EDF為佇列選擇規則的佇列問題可等效於以先到先服務(FCFS)或優先服務(PRI)的 佇列問題。 關於有限容量模式 (q-M/G/1/Ki/FCFS/QSR)的研究，將以多種佇列選擇規則(QSR)分析其排程的優劣。這一 部份的研究以多維度聯合狀態轉移，及以嵌入式馬可夫鏈(EMC)或馬可夫鏈(MC)求解佇列的狀態機率。針對EMC模式共有 四個求解步驟，包括(1)訊息抵達機率、(2)佇列轉移機率、(3)離開佇列瞬間的狀態機率、與(4)任意時間的狀態機率等。針 對MC模式則需要二個步驟，即(1)佇列轉移機率與(2)任意時間的狀態機率。由模擬實驗得知，本方法可正確的分析佇列 以FCFS, PRI, WFQ, RR等為佇列選擇規則(QSR)的佇列參數。 另一部份的研究是以具有學習能力的訊息排程控制器(MSC) 為佇列選擇規則，以滿足時效性的需求並求解相關的佇列參數。MSC屬於閉迴路的控制流程並提供事前學習(Type I)或事 後學習(Type II)調整內部參數，以適應時變系統達到預防或降低訊息逾時服務發生的目的。本論文提出三種建構MSC的方 法，包括輻射基底函數網路(RBFN)、模糊神經網路(NFN)、與關聯向量機(RVM)等。由模擬實驗得知訊息排程控制器比傳 統的QSR更具強韌性且有較低的訊息逾時服務率。 最後本文也將針對當佇列負載趨近於無窮時，推導包括以MSC, EDF, FCFS, WFQ, RR, PRI等為QSR的訊息等待時間上界。由模擬實驗得知本文的方法能夠準確的估測不同QSR的等待時間上 界。
關鍵詞 : 多佇列有限容量、狀態機率、佇列選擇規則、訊息排程控制器 目錄
封面內頁 簽名頁 中文摘要...iii 英文摘
要...v 誌謝...vii 目 錄...viii 圖目錄...xii 表目 錄...xvi 符號說明...xviii 第一 章...1 1.1 研究動機...1 1.2 佇 列選擇規則(QSR)簡介...6 1.3 研究目的 ...7 1.4 論文架構 ...8 第二章 佇列系統的探
討...10 2.1 馬可夫鏈與佇列系統...10 2.2 單 佇列模型...13 2.2.1 M/G/1/ 模型...13 2.2.2 M/M/1/K 模型...14 2.2.3 M/G/1/K 模
型...16 2.2.4 單佇列系統的等效模式...18 2.3 多佇列模型...20 2.4 多佇列模型的佇列選擇規
則...22 第三章 多佇列無限容量模式的系統分析...28 3.1 文獻回顧...28 3.2 以機率分佈分析EDF的等待時
間...32 3.3 EDF的等效模式...36 第四章 多佇列模 式的系統分析...38 4.1 q-M/G/1/Ki模式...38 4.1.1 訊息抵達機率(Message Arrival Probability, MAP)...43 4.1.2 佇列轉移機率(Queue Transition Probability, QTP)...45 4.1.3 q-M/G/1/Ki的狀態轉移方程式(STE)...47 4.1.4 q-M/G/1/Ki的狀態平衡方程式(SBE)...47 4.1.5 q/M/G/1/Ki -個案分析:以2個佇列為
例...53 4.2 q-M/M/1/Ki模式...58 4.2.1 狀態平衡方程 式(SBE)...59 4.2.2 q/M/M/1/Ki -個案分析:以2個佇列為
例...60 4.3 馬可夫鏈與嵌入式馬可夫鏈的狀態平衡方程式...62 第五 章 訊息排程控制器...66 5.1 MSC-MQP的閉迴路控
制...66 5.2 選擇輸入變數...67 5.3 Type I 與Type II學習模式...68 5.4 輻射基底函數網路(Radial Basis Function Network, RBFN)...71 5.4.1 參數學習...72 5.4.2 架構學 習...73 5.5 模糊類神經網路(Neuro-Fuzzy Network,
NFN)...75 5.5.1 前向推論...76 5.5.2 反向參數學
習...78 5.5.3 架構學習...79 5.6 關聯向量 機(Relevance Vector Machine, RVM)...81 第六章 多佇列有限容量模式的等待時間上
界...86 6.1 MSC等待時間的上界評估...86 6.2 EDF
、FCFS、與PRI等待時間的上界...93 6.3 WFQ與RR等待時間的上
界...94 第七章 模擬實驗...96 7.1 多佇列無 限容量...96 7.1.1 以為變量 ...97 7.1.2 以Di為變量...99 7.2 多佇列有限容量...101 7.2.1 以為變量 ...102 7.2.2 以Ki為變
量...110 7.2.3 q-M/G/1/Ki的px;s與 的關係...115 7.3 訊息排程控制器的逾時服務率...117 7.3.1 固定
值...117 7.3.2 變動值...123 7.4 佇列參 數的上界...128 7.4.1 的佇列參數...128 7.4.2 探討第四章與第六章的公式在的ㄧ致性...131 第八章 結論與展
望...134 參考文獻...137 附錄 A...144 附錄 B...145 附錄 C...146 附錄 D...148 附錄 E...150 附錄 F...151 附錄 G...152 附錄 H...159 參考文獻
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