连接数据库报too many clients already疑惑

数据库在连接时失败并报出too many clients already，原因是连接过多，超过PG库设置的最大连接数。但是我在这里存在一个疑惑，它这个报错触发的条件，我查看了PG12.1的源码，部分如下

按照这个逻辑，too many clients already应该是regular backends, autovac workers, walsenders
和background workers的子进程数量和大于或者等于 2 * (MaxConnections + autovacuum_max_workers + 1 + max_wal_senders + max_worker_processes)的值才会输出的。所以我的疑惑是，究竟是clients的连接超过了max_connections，还是按这个逻辑来的，抑或是我少看了哪一部分或者理解有误。

    /*
     * If we're going to reject the connection due to database state, say so
     * now instead of wasting cycles on an authentication exchange. (This also
     * allows a pg_ping utility to be written.)
     */
    switch (port->canAcceptConnections)
    {
        case CAC_STARTUP:
            ereport(FATAL,
                    (errcode(ERRCODE_CANNOT_CONNECT_NOW),
                     errmsg("the database system is starting up")));
            break;
        case CAC_SHUTDOWN:
            ereport(FATAL,
                    (errcode(ERRCODE_CANNOT_CONNECT_NOW),
                     errmsg("the database system is shutting down")));
            break;
        case CAC_RECOVERY:
            ereport(FATAL,
                    (errcode(ERRCODE_CANNOT_CONNECT_NOW),
                     errmsg("the database system is in recovery mode")));
            break;
        case CAC_TOOMANY:
            ereport(FATAL,
                    (errcode(ERRCODE_TOO_MANY_CONNECTIONS),
                     errmsg("sorry, too many clients already")));
            break;
        case CAC_WAITBACKUP:
            /* OK for now, will check in InitPostgres */
            break;
        case CAC_OK:
            break;
    }
    return STATUS_OK;
}
这里匹配到CAC_TOOMANY所以输出了sorry, too many clients already

    /*
     * Don't start too many children.
     *
     * We allow more connections here than we can have backends because some
     * might still be authenticating; they might fail auth, or some existing
     * backend might exit before the auth cycle is completed.  The exact
     * MaxBackends limit is enforced when a new backend tries to join the
     * shared-inval backend array.
     *
     * The limit here must match the sizes of the per-child-process arrays;
     * see comments for MaxLivePostmasterChildren().
     */
    if (CountChildren(BACKEND_TYPE_ALL) >= MaxLivePostmasterChildren())
        result = CAC_TOOMANY;
    return result;
}
这里在满足(CountChildren(BACKEND_TYPE_ALL) >= MaxLivePostmasterChildren())，才返回了这个结果。所以这个条件，应该是触发打印出sorry, too many clients already的条件

/*
 * Count up number of child processes of specified types (dead_end children
 * are always excluded).
 */
static int
CountChildren(int target)
{
    dlist_iter    iter;
    int            cnt = 0;
    dlist_foreach(iter, &BackendList)
    {
        Backend    *bp = dlist_container(Backend, elem, iter.cur);
        if (bp->dead_end)
            continue;
        /*
         * Since target == BACKEND_TYPE_ALL is the most common case, we test
         * it first and avoid touching shared memory for every child.
         */
        if (target != BACKEND_TYPE_ALL)
        {
            /*
             * Assign bkend_type for any recently announced WAL Sender
             * processes.
             */
            if (bp->bkend_type == BACKEND_TYPE_NORMAL &&
                IsPostmasterChildWalSender(bp->child_slot))
                bp->bkend_type = BACKEND_TYPE_WALSND;
            if (!(target & bp->bkend_type))
                continue;
        }
        cnt++;
    }
    return cnt;
}
/*
 * MaxLivePostmasterChildren
 *
 * This reports the number of entries needed in per-child-process arrays
 * (the PMChildFlags array, and if EXEC_BACKEND the ShmemBackendArray).
 * These arrays include regular backends, autovac workers, walsenders
 * and background workers, but not special children nor dead_end children.
 * This allows the arrays to have a fixed maximum size, to wit the same
 * too-many-children limit enforced by canAcceptConnections().  The exact value
 * isn't too critical as long as it's more than MaxBackends.
 */
int
MaxLivePostmasterChildren(void)
{
    return 2 * (MaxConnections + autovacuum_max_workers + 1 +
                max_wal_senders + max_worker_processes);
}