Many messages from MAILER-DAEMON@example.com are stuck in mail queue on Plesk server

Symptoms

A lot of messages from MAILER-DAEMON@example.com are stuck in the queue (Tools & Settings > Mail Server Settings > Mail Queue):

The number of such messages is growing exponentially.

The recipient's server (johndoe@example.org) is not reachable:

# telnet example.org 25
Trying 203.0.113.2...
telnet: connect to address 203.0.113.2: Connection timed out

Cause

Outgoing spam was sending (or being sent at the moment) to the addresses that do not exist or are not reachable at the moment.

Resolution

Warning: The below steps are temporary measures. To manage outbound spam protection settings, refer to the following documentation:
Protection From Outbound Spam

As a workaround, automatic cleaning may be set up as a scheduled task:

1. Log in to Plesk

2. In Tools & Settings > Scheduled Tasks > Add Task. Fill in as follows:

   CONFIG_TEXT: Active
   Task type: Run a command
   Command: /usr/local/psa/admin/sbin/mailqueuemng --clean
   Run: Hourly (set any time)
   Description:  it flushes the mail queue 

   The second task:

   CONFIG_TEXT: Active
   Task type: Run a command
   Command: mailq|gawk '/MAILER-DAEMON/ { print $1 }'|/usr/sbin/postsuper -d -
   Run: Hourly (set it after two-three minutes past the first one)
   Description: delete bounces

Mail Queue will be cleaned once in an hour.