Harmonic Periods

In real-time scheduling, predictability is very important, i.e., we would like to know the whole schedule before we really run the tasks. Rate-monotonic scheduling is very popular in real-time scheduling for periodic tasks, where tasks with shorter periods have higher priority. However, it is still dicult to know the start time and finish time of each task and they might be dierent in each period, especially for tasks with low priority, i.e. long period. The hyperperiod, the least common multiple of all periods,is usually too big to be practical to describe the whole schedule. However, if the task periods are harmonic, i.e. are multiples, it is possible to find the start time and finish time of each task quickly because the schedule becomes more regular.

Figure 1 shows that periodic tasks T1, T2, T3, T4 with execution times 1, 1, 3, 1 and periods 2, 4, 16, 32 respectively are schedulable, each task finishes execution in its period, using the Rate-Monotonic scheduling algorithm since T1, T2, T3, T4 finish execution at time 1, 2, 12, 16 respectively. T3 is preempted by T1 and T2 at time 4 and time 8 and resume at time 7 and time 11.

All the input numbers are positive integers, < 500000, separated by a space or new line. The first line is the number of task sets. Then, the task sets are listed set by set. Each task set is listed by a line of the number of tasks, <= 100, and lines of task execution time and period pairs, execution time < period. The periods are harmonic, not sorted, and are different in a task set.

For each task set, find and print out the finish time of the task with the largest period using rate-monotonic scheduling, if schedulable; otherwise print out -1.

3
4
1 2
1 4
3 16
1 32
3
1 4
4 8
256 1024
3
1 2
3 8
1 4

16
1024
-1

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