In this paper, with the aid of the mathematical tool of stochastic geometry, we introduce analytical and computational frameworks for the distribution of three different definitions of delay, i.e., the time that it takes for a user to successfully receive a data packet, in large-scale cellular networks. We also provide an asymptotic analysis of one of the delay distributions, which can be regarded as the packet loss probability of a given network. To mitigate the inherent computational difficulties of the obtained analytical formulations in some cases, we propose efficient numerical approximations based on the numerical inversion method, the Riemann sum, and the Beta distribution. Finally, we demonstrate the accuracy of the obtained analytical formulations and the corresponding approximations against Monte Carlo simulation results, and unveil insights on the delay performance with respect to several design parameters, such as the decoding threshold, the transmit power, and the deployment density of the base stations. The proposed methods can facilitate the analysis and optimization of cellular networks subject to reliability constraints on the network packet delay that are not restricted to the local (average) delay, e.g., in the context of delay sensitive applications.

翻译：在本文中,在随机几何数学工具的帮助下,我们引入了用于分配三种不同延迟定义的分析和计算框架,即用户在大型蜂窝网络中成功接收数据包所需的时间;我们还对其中一种延迟分布进行无症状分析,可被视为某一网络的包丢失概率;为减轻某些情况下获得的分析配方固有的计算困难,我们根据数字转换方法、Riemann总和和和Beta分布,提出高效的数字近似值;最后,我们展示了所获得分析配方的准确性,以及针对蒙特卡洛模拟结果的相应近似值,并揭示了对若干设计参数的延迟性能,例如解码阈值、传输功率和基地台站部署密度等;拟议方法可促进移动电话网络的分析和优化,但网络延迟的可靠性受限制,而网络延迟不限于本地(平均)延迟,例如延迟敏感应用。