I have $x$ uniform machines that are identical, except that each runs at a different speed; machine $j$ runs at speed $s_j$. I have $n$ identical jobs. Each machine can handle one job at a time. The time to complete a job on machine $j$ will be $1/s_j$ seconds.
I want to prove that the optimal schedule that minimizes $C_\text{max}$ is in fact also an optimal schedule for minimizing $\sum_i C_i$. Here, $C_i$ is the time when job $i$ completes (under that schedule); $\sum_i C_i$ is the sum of completion times of all jobs; and $C_\text{max} = \max_i C_i$ is the time when the last job completes. In other words, I want to prove that, when minimizing $C_{max}$ with identical jobs, the optimal schedule is also the optimal schedule for minimizing $\sum C_i$.
How can I prove this?
Or, to put it another way, using the standard notation for scheduling problems, I want to prove that $$ Q|p_i=1|C_{max} \quad \equiv \quad Q|p_i=1|\sum C_i. $$
