You then need to remove the string you split the file on ("*****\n*****") and split the remainder on a new line "\n". Place the result of this split in an array. If the last element of this array starts with "Server Closed", then you have a complete log entry.

#!/usr/bin/perl -w

use strict;

my @status;

{
    my ($log, @split_log, $status);
    local $/ = qq{*****\n*****};
    while ($log = <DATA>) {
        $log =~ s/\Q*****\E\n\Q*****\E$//;
        @split_log = split "\n", $log;
        $status = ($split_log[$#split_log] =~ m/^Server Closed/)
            ? "complete\n" : "incomplete\n";
        push @status, $status;
    }
}

__DATA__
*****
Server Started Monday
data1
data2
Server Closed Tuesday
*****
*****
Server Started Wednesday
data3
data4
Server Closed Friday
*****
*****
Server Started
data5
[download]

This leaves the string "*****" on the front of the first log entry. You can then do whatever processing you need on the logs either one complete log at a time, or instead of assigning the split to @split_log, create an anonymous array and push that onto @split_log. This will let you defer your processing to the end.

Nuance