Parallel machines problem and minimal machines

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Given $n$ jobs where job $i=1..n$ starts at $a(i)$ and finish at $b(i)$ find the minimal machines that needed to finish all the jobs.
Find the a rule for a solution and prove it's correctness.
I came to a conclusion that the minimal machines that is needed is the maximum overlap at given period of time(for example at a given minute).
for example $a(i)=(0,1,3,4,7),b(i)=(2,8,5,6,9)$ we can arrange the jobs with only 3 machines.

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You can formulate the problem as finding the chromatic number of a graph with a node for each job and an edge for each pair of jobs whose time windows overlap.