In some real-time systems, sometimes there is a need to insert a new task at run time. If the system is fully loaded, the scheduler has to compress one or more current tasks to free some bandwidth. Naturally, finding the earliest time to start the insertion without deadline missing is interesting, which is still under investigation. Instead of this, this paper discusses how to do the compression and insertion to make the insertion as early as possible, based on the conclusions in the published literature with the earliest deadline first algorithm(EDF). By introducing the instance remaining bandwidth(RB), an algorithm called maximum remaining bandwidth compression(MRBC) is provided to choose a proper task as the compressed one. An associated theorem and its lemma are also provided, which declares that the safety adjustment can be implemented immediately upon request if the increase of the bandwidth due to insertion does not exceed the lowest utilization of individual current tasks in the system and if every task is allowed to be compressed. Another algorithm named part utilization first(PUF) is also proposed, according to which a new task is inserted with a lower speed first, then to its full utilization after a certain time.