In this paper, we present a modification of the local remaining execution-time and local time domain (LRE-TL) real-time multiprocessor scheduling algorithm, aimed at reducing the scheduling overhead in terms of task migrations. LRE-TL achieves optimality by employing the fairness rule at the end of each time slice in a fluid schedule model. LRE-TL makes scheduling decisions using two scheduling events. The bottom (B) event, which occurs when a task consumes its local utilization, has to be preempted in order to resume the execution of another task, if any, or to idle the processor if none exist. The critical (C) event occurs when a task consumes its local laxity, which means that the task cannot wait anymore and has to be scheduled for execution immediately or otherwise it will miss its deadline. Event C always results in a task migration. We have modified the initialization procedure of LRE-TL to make sure that tasks that have higher probability of firing a C event will always be considered for execution first. This will ensure that the number of C events will always be at a minimum, thereby reducing the number of task migrations.