Add serial number column based on another column in mysql

I have a table code with the following table structure. id is auto incrementing id, the code is a random unique 15 digit code and customer_id is the customer's id. The table has close to 0.13 million rows

 id      code    customer_id 
 1     123...4      1            
 2     124...5      1            
 3     128...3      1            
 4     234...1      2            
 5     678...3      2            
 6     567...8      1            

I have added a column serial_number in table and I want to add a serial_number which will increment for each value of respective customer like below based on the customer_id for existing values. How do I do this in mysql?

 id    code       customer_id  serial_number
 1     123...4      1            1
 2     124...5      1            2
 3     128...3      1            3
 4     234...1      2            1
 5     678...3      2            2
 6     567...8      1            4     

I tried this approach but my mysql connection is getting timed out:

MySQL: Add sequence column based on another field

CREATE temporary table serial_number ( id bigint, serial_number int);
INSERT INTO serial_number ( id, serial_number )
    SELECT id, 
      (SELECT count(*) + 1 FROM code c 
      WHERE c.id < code.id AND c.customer_id = code.customer_id) as    serial_number 
    FROM code;

UPDATE test INNER join serial_number ON code.id = serial_number.id SET   code.serial_number = serial_number.serial_number;

Error Code: 2013. Lost connection to MySQL server during query