In this paper, we consider a discrete-time Geo/G/1 queue controlled by the combination of the N and D policies (called ND-policy). In this system, when there are N waiting customers or the service time backlog of all waiting customers exceeds a given threshold D, whichever emerges first, the idle server immediately resumes its service. Under this policy, since the service times of the customers arriving during the idle period, conditioned on the number of these customers, are dependent, and stochastically different from the service times of the customers arriving during the busy period, the customers in the system are classified into two types. Based on this classification, we first derive the probability generating functions and means of the queue length, idle and busy periods, service time backlog, waiting time and sojourn time, where the busy period is first studied in the discrete-time queues involving the D-policy. Next, by analyzing some results and flaws in the work of Gu et al. (J Syst Sci Complex, 2016. doi: 10.1007/s11424-016-4180-y), we theoretically show the discrepancies that could arise if the conditional dependency of the service times of the customers arriving during the idle period is ignored. Finally, the numerical examples are provided to study the effects of different parameters on the mean queue length. Through an energy consumption optimization problem in wireless sensor networks, the application of our queueing model in the real world is illustrated, and the flaws that resulted from the results by Gu et al. (J Syst Sci Complex, 2016. doi: 10.1007/s11424-016-4180-y) are numerically revealed.