A variety of information communication devices have been developed to help people maintain his/her awareness of interesting information. Notification systems play an important role in delivering up-to-date, important information via such devices. We have been developing a novel push-style notification system that delivers a succession of new messages onto the mobile terminal display in a well-timed manner. This system increases the probability of the user discovering interesting information from among a stream of items shown one after another, and accessing the network to get more details if desired. Needless to say, the appropriateness of the message to be notified is important to the user. To make a push-style notification system more useful and comfortable for users, however, it is also essential to deliver the information without interrupting the user's on-going task. Furthermore, some mechanism that can recover from the user's failure to notice a message is also needed, We fold all these abilities into the one term of "Message Reachability", and propose a comprehensive and flexible push-style notification system. One of its main features is a Bayesian "Notification User Model" which takes task state, user state and user characteristic into consideration. The other is an adaptive notification control mechanism based on the user model, and a compensation mechanism to handle messages that the user missed. This paper details the user model and message control mechanisms, and concludes by showing implementation details.