行動通訊網路中剩餘時間之數學模型的研究

Abstract

在行動通訊網路中，剩餘時間（excess life）之數學模型的研究，與很多效能評估議題有重要的相關性。當手機停留在一個細胞，從使用者打出一通電話或是接到一通電話，直到這支手機離開這個細胞，這中間所經過的時間，稱之為手機停留的剩餘時間。在行動通訊網路的效能評估中，推導出手機停留之剩餘時間的機率分佈，是一件非常重要的功課。此機率分佈決定了一通電話會不會隨著手機的移動而從一個細胞被交遞（handover）到另外一個細胞，而顯著地影響了一個行動通訊網路之交遞強制斷話（force-termination）機率。行動通訊網路的模擬模型必須為非指數分佈產生其剩餘時間的隨機亂數（random number），而這並不是一件簡單而直截了當的事。本論文將呈現如何為剩餘時間之機率分佈產生隨機亂數，並且針對加瑪（gamma）分佈、帕雷多（Pareto）分佈、對數常態（lognormal）分佈、以及韋伯（Weibull）分佈，分別發展出其手機停留之剩餘時間的隨機亂數產生程序。接著我們以行動通訊網路中的兩個研究議題為例，呈現我們所發展出的剩餘時間隨機亂數產生程序，如何有效率地應用在效能評估。

第一個研究議題探討週期性位置區域更新機制。在行動通訊網路中，此機制被用於偵測一支手機是否仍與網路端保持正常聯結。第三代無線通訊的技術規格3GPP 23.012與24.008針對通用行動通訊系統UMTS，提出固定式週期性位置區域更新機制；也就是說，在兩次週期性位置區域更新之間的時間長度，一律保持為固定值。根據我們的觀察，網路端若要持續偵測一支手機是否仍與其保持正常聯結，並不一定只有透過週期性位置區域更新才能辦到。其實透過來話（incoming call）、發話（outgoing call）和一般性位置區域更新（normal location area update），網路端照樣能持續偵測一支手機是否仍與其保持正常聯結。依此，我們提出了動態式位置區域更新機制。動態意味著兩次週期性位置區域更新之間的時間長度，會動態地隨著來話、發話和一般性位置區域更新的行為變化而有所調整。我們發展出一個數學模型來與模擬模型（使用了前面所提及的剩餘時間隨機亂數產生程序）互相驗證，並比較動態式與固定式兩種週期性位置區域更新機制的效能。另外，我們的研究也提供了如何為動態式位置區域更新機制選擇參數的方針。

第二個研究議題針對小規模行動通訊網路，探討三項重要的效能評估值：塞機率（new call blocking probability）、交遞強制斷話機率（handover force-termination probability）、以及通話失敗機率（call incompletion probability）。我們認為一個細胞的交遞交通量，是與其相鄰之細胞的既有通話量有相關聯的。根據這個觀察，我們針對指數分佈的手機停留時間，推導出了精確的交遞強制斷話機率之數學方程式；接著，我們針對任意一種分佈的手機停留時間，提出了其數學逼近模型。我們將新的數學模型與模擬模型（使用了前面所提及的剩餘時間隨機亂數產生程序）互相驗證，接著將模擬模型當成基準，來比較新、舊兩種數學模型（新數學模型即此篇論文的研究成果；舊數學模型則為過去所發展）。我們的比較結果顯示出：針對小規模的行動通訊網路，新數學模型會比舊數學模型更精準地捕捉到交遞行為。

Many performance evaluation issues for cellular telecommunications are related to excess life modeling. The period when a mobile station (MS) resides in a cell (the radio coverage of a base station) is called the MS cell residence time. The period between when a call arrives at the MS and when the MS moves out of the cell is called the excess life of the MS cell residence time for that MS. In performance evaluation of a cellular telecommunications network, it is important to derive the excess life distribution from the MS cell residence times. This distribution determines if a connected call will be handed over to a new cell, and therefore significantly affects the handover force-termination probability of the network. In simulation of cellular telecommunications networks, we need to generate random numbers for the excess life of non-exponential distributions. However, generating these random numbers for non-exponential distributions is not a trivial task, which has not been addressed in the literature. We show how to generate the random numbers from the excess life distribution, and develop the excess-life random number generation procedures for MS cell residence times with gamma, Pareto, lognormal and Weibull distributions. We use two examples to show how our excess life modeling techniques can be effectively utilized in performance evaluation of cellular networks.

In the first example, we study the periodic location area update (PLAU) scheme, which is utilized in cellular telecommunications networks to detect the presence of an MS. In 3GPP Technical Specifications (TS) 23.012 and 24.008, a fixed PLAU scheme was proposed for Universal Mobile Telecommunications System (UMTS), where the interval between two PLAUs is of fixed length. We observe that MS presence can also be detected through call activities and normal location area updates (NLAUs). Therefore we propose a dynamic PLAU scheme where the PLAU interval is dynamically adjusted based on the call traffic and NLAU rate. An analytic model is developed to validate against the simulation model with the excess life modeling technique. Then we investigate the performance of dynamic and fixed PLAU schemes. Our study provides guidelines to select parameters for the dynamic PLAU scheme.

In the second example, we study the new call blocking, handover force-termination, and call incompletion probabilities for a small-scale cellular network. We show that the handover traffic to a cell depends on the workloads of the neighboring cells. Based on this observation, we derive the exact equation for the handover force-termination probability when the MS cell residence times are exponentially distributed. Then we propose an approximate model with general MS cell residence time distributions. We use the analytic model to validate against the simulation model with the excess life modeling technique. Then the analytic results are compared with a previously proposed model, where the simulation results are used as the baseline. Our comparison study indicates that the new model can capture the handover behavior much better than the old one for small-scale cellular telecommunications networks.