while (<INPUT>)  {
        chomp($_);
        # Look for title line
        if ($_ =~/Start -/) {
            my @sdk_line = split(/\t/, $_);
            push(@temp, substr($sdk_line[1], 18));            
        }
    }
    while (<INPUT>)  
        {
            chomp($_);        
            # find process time for that title
            if ($_ =~/Process/) {
                print split(/\t/, $_), "\n";
                my @sdk_line = split(/\t/, $_);
                push(@temp , $sdk_line[2]);            
            
            }
        }
[download]

The second while loop will never be executed, because the first one exhausts all line in the INPUT file.

Also please indent your code in a way that makes it obvious to see which loop ends where.

I have hardcoded my script to parse only 3 files. How do I do it for multiple/unknown number of files.

By iterating over the file names, open each, process it, and close it again. Something like this:

for my $fn (@file_names){
    open my $handle, '<', $fn or die "Can't open `$fn' for reading: $!
+";
    # do something with the file here
    # and write to output file
    close $fn;
}
[download]