| Abstract: | Rose, Suhov and Vvdenskaya 曾經在1998年討論過一個N個節點且完全相連的封包轉換網路架構,在該網路架構下，訊息從一個起始節點i產生,並將其傳送到一個目的節點j，並假設其傳遞過程為速率為v的Poisson過程,簡記為 。每一傳遞訊息均被分成若干個封包，且每一訊息長度均視為獨立樣本且有界。訊息在兩點之間傳遞的延遲時間可視為其所有封包由起始節點完全送達目的節點的時間總和。在傳遞過程中，任一封包可選擇一條1-path的路徑 或是以一條2-path的路徑 ,此兩種路徑被選中的機率分別為p及q。Rose等人找到了在不同平均的訊息長度及不同之v下之p 及q，該p及q可使得使點至節點的傳輸延遲時間為最小。 本文應用此封包轉換網路架構，模擬建立了一具有2N+1個節點且完全相連的交通網路,網路上的節點分別標記為1, 2, 3, …, N, …, 2N-2, 。對任一組節點(i,j)吾人假設一顧客從一個起點i產生且運送至一個終點j是一個速率為v的Poisson過程 ,每一個顧客可選擇開車（機率為p）或搭公車(機率為q),若客人選擇開車，他可以選擇一條1-path的路徑或2-path路徑,其機率分別為 及 (其中 )。假設每隔 分鐘,在一條特定的2-path路徑上發一班公車, 或 （其中1 且 ）。本篇論文的目的是要在這個M/G/1的等候網路裡導引出最佳的機率 和q使得點至點的傳輸時間最小。此外亦將討論這些機率之間的關係,在現實的生活裡公車的載客容量是有限的。當公車的載客容量是有限的情況,顯然的當公車的載客容量遞減,機率q (顧客選擇搭公車的機率)之最佳值將會遞減。
038<p type="texpara" tag="Body Text" >In 1998\QTR{it}{\hspace{0in}, Rose, Suhov and \hspace{0in}\hspace{0in}\ Vvdenskaya} discussed a fully-connected \thinspace \thinspace packet-switched network with $N$ \ nodes. Messages generated \ from a \ source $\ i$ and deliver\ to a destination $j$ \ are assumed as Poisson \ process $\Psi _{i,j}$ with rate $v.$ Each \thinspace message \thinspace is divided into smaller units called \thinspace packets. The lengths of \thinspace \thinspace messages are $i.i.d$ and \thinspace bounded. The end-to-end delay \thinspace time is the time that all \thinspace packets be transmitted completely \thinspace from \thinspace its source to its destination. Each \thinspace packet \thinspace can \thinspace choose a \thinspace direct \thinspace route $(i\rightarrow j)$ with \thinspace probability$\,\hspace{0in}$\hspace{0in}$\hspace{0in}$\hspace{0in}$\hspace{0in}$\hspace{0in}$p$ or an alternate route $(i\rightarrow k\rightarrow j)$ with probability $\,q.$ They \thinspace found the optimal policies about $\,\,p$ and $q$ \thinspace with \thinspace different \thinspace means \thinspace of \thinspace \thinspace message \thinspace lengths \thinspace and $\,\,v$ \thinspace such \thinspace that \thinspace the \thinspace end-to-end delay time is minimized. 038<p type="texpara" tag="Body Text" >Here we use the same routing principle of this \thinspace packet-switched \thinspace network to simulate a transportation network. Consider a fully-connected \thinspace traffic \thinspace network \thinspace with 2N+1 nodes, \thinspace labelled \thinspace with $1,$ $\,2,$ $\,3,...,$ $N,...,$ $2N-2,$ $\,\alpha ,$ $\,\beta ,$ $\,\gamma .$ Suppose \thinspace customers generated from \thinspace a source node $i$ and traversed \thinspace to a destination node $\,j$ \thinspace is a Poisson process $\Psi _{i,j}$ with \thinspace rate $v$ for all different $\,i,$ $\,j$ pairs. Each customer \thinspace can \thinspace choose \thinspace to drive a car or to take a bus independently with \thinspace probabilities $\,p$ and $\,q$ (=$1-p$) respectively. Customers who driving a car may drive on a direct \thinspace route ( $i\rightarrow j$ ) or on a \thinspace specific alternate \thinspace route ( $i\rightarrow k\rightarrow j\,$) \thinspace with \thinspace probability $m_{1},$ \thinspace and $\frac{m_{2}}{2N-1}$ respectively, where $m_{2}=1-m_{1}.$ Suppose for every $K_{1}$ minutes, there is a bus on a specific alternate \thinspace route $\,\alpha \rightleftharpoons n_{s}\rightleftharpoons \beta ,$ \thinspace or $\,\alpha \rightleftharpoons n_{t}\rightleftharpoons \gamma ,$\thinspace \thinspace where $1\leq n_{s}\leq N-1,$ and $N\leq n_{t}\leq 2N-2.$ The goal of this paper is to find optimal policies for probabilities $m_{1},$ $m_{2},$ $p,$ and $q$ such that the end-to-end delay \thinspace time is minimized in this M/G/1 queuing network. We can also address the relationships between these probabilities. In reality, the capacity of a bus is \thinspace finite. It is obvious that \thinspace when \thinspace the \thinspace buses \thinspace have finite \thinspace capacities, \thinspace the \thinspace optimal \thinspace probability of $\,q$ (\thinspace \thinspace the probability that customers choose to take a bus) will decrease as the capacity of the bus decreases. |
