Time-sensitive services (TSSs) have been widely envisioned for future sixth generation (6G) wireless communication networks. Due to its inherent low-latency advantage, mobile edge computing (MEC) will be an indispensable enabler for TSSs. The random characteristics of the delay experienced by users are key metrics reflecting the quality of service (QoS) of TSSs. Most existing studies on MEC have focused on the average delay. Only a few research efforts have been devoted to other random delay characteristics, such as the delay bound violation probability and the probability distribution of the delay, by decoupling the transmission and computation processes of MEC. However, if these two processes could not be decoupled, the coupling will bring new challenges to analyzing the random delay characteristics. In this paper, an MEC system with a limited computation buffer at the edge server is considered. In this system, the transmission process and computation process form a feedback loop and could not be decoupled. We formulate a discrete-time two-stage tandem queueing system. Then, by using the matrix-geometric method, we obtain the estimation methods for the random delay characteristics, including the probability distribution of the delay, the delay bound violation probability, the average delay and the delay standard deviation. The estimation methods are verified by simulations. The random delay characteristics are analyzed by numerical experiments, which unveil the coupling relationship between the transmission process and computation process for MEC. These results will largely facilitate elaborate allocation of communication and computation resources to improve the QoS of TSSs.