Emergence of various types of services has brought about explosive growth of traffic as well as diversified traffic characteristics in the cellular networks. To have a comprehensive understanding of the influences caused by various traffic status is vital for the deployment of the next-generation wireless networks. In this paper, we develop a mathematical analytical model by utilizing queuing theory and stochastic geometry where the randomness of the traffic and the geographical locations of the interferers can be captured. We derive the b-th moments of the conditional success probability and the closed-form expressions of the meta distribution for the cell-center users (CCUs) and the cell-edge users (CEUs), respectively. Fixed-point equations are then formulated to obtain the exact value of the meta distribution by taking the random arrival traffic into consideration and the impact of the random arrival traffic on the queue status is revealed. In addition, the mean local delays for CCUs nad CEUs are derived and the corresponding regions for CCUs and CEUs where the mean local delays maintain finite are obtained. Finally, the impact of the critical network parameters on the meta distribution and the mean local delay is investigated with the numerical results.