The physics of human activities recently has been studied in the view point that they are dynamic processes of a complex system. The studies reveal that the human activities have bursty nature - occasional abrupt bursts of activity level for short periods of time, along with long periods of inactivity. Quantitative studies show that the distribution of the time, t, between two consecutive activity events exhibits a power-law behavior with universal exponents ∼τ-1.5 or ∼τ-1.0. Such universal behaviors were explained by the universality in the waitingtime distribution of tasks in model queue systems, which operate based on priority. In the models, the rates of task input are presumed to follow a Poisson-type distribution. An empirical observation of human activities, however, shows that the task arriving rate for some people also has bursty nature - the number of tasks arrive to the people follows a power-law distribution. In this paper, a new model queue system for this case is introduced and studied by analytic and numerical methods. The waiting-time distribution for the new model is found also to follow a power law, but the exponent varies according to the parameters of the model and takes other values than 1.5 or 1.0. The analytic solution is obtained via the generating function formalism, different from the biased random walk approach used in the previous studies.