Calculate session concurrency for each user in SQL

Question

I have a table of logins and logouts by user.

the table looks like this but has a few hundred thousand rows:

Sorry, my example is in Python.But im interested in SQL solution for this interesting problem.

data = [['aa', '2020-05-31 00:00:01', '2020-05-31 00:00:31'],
        ['bb','2020-05-31 00:01:01', '2020-05-31 00:02:01'],
        ['aa','2020-05-31 00:02:01', '2020-05-31 00:06:03'],
        ['cc','2020-05-31 00:03:01', '2020-05-31 00:04:01'],
        ['dd','2020-05-31 00:04:01', '2020-05-31 00:34:01'],
        ['aa', '2020-05-31 00:05:01', '2020-05-31 00:07:31'],
        ['bb','2020-05-31 00:05:01', '2020-05-31 00:06:01'],
        ['aa','2020-05-31 00:05:01', '2020-05-31 00:08:03'],
        ['cc','2020-05-31 00:10:01', '2020-05-31 00:40:01'],
        ['dd','2020-05-31 00:20:01', '2020-05-31 00:35:01']]


df_test = pd.DataFrame(data,  columns=['user_id','login', 'logout'], dtype='datetime64[ns]')

I was able to solve this problem in a hacky way using a for loop in Python.

Basically, this code calculates how many users were logged in at the same time for each session (session being each row)

Here is my solution. it gives the result that i need.

# create a new column for simultaneous
df_test['simultaneous'] = 0

start_time = time.time()

# loop through dataframe and check condition
for i in df_test.index:
    login, logout = df_test.loc[i,'login'], df_test.loc[i,'logout']
    this_index = df_test.index.isin([i])
    df_test.loc[i, 'simultaneous'] = int(sum(
        (df_test[~this_index]['login'] <= logout) & (df_test[~this_index]['logout'] >= login)
    ))
print("--- %s seconds ---" % (time.time() - start_time))

. Any type of SQL implementation of concurrency is ok. I would like to compare python's and SQL runtimes for this.

Thanks in advance!