Automaton without variables and guards for queueing system Disclaimer: Assignment question.

I am trying to solve the below question, the condition is not to use guards and variables. Is it at all possible to do this?



My approach was to have 4 separate automatons, 1 for each of the three servers and 1 for the entire system(with states for 0, 1 and 2 customers in the system). But its not working for all the conditions.
Any help is appreciated.
One deterministic automaton is enough to model this system. 

Here are the list of states and their meanings. The alphabet is $$\{\text{a_customer_comes},\text{a_customer_leaves_server_1},\\ \text{a_customer_leaves_server_2},\text{a_customer_leaves_server_3}\}.$$ (We assume the input events happen one at a time. Otherwise, the problem might be too ill-defined enough to have a unique answer or too complex to be done as an exercise.) 

$q_{0,0,0}$ : the initial state. No customer present.
$q_{1,0,0}$ : 1 customer at server 1 and 0 customer at server 2 and 0 customer at server 3.
$q_{0,1,0}$ : 0 customer at server 1 and 1 customer at server 2 and 0 customer at server 3.
$q_{0,0,1}$ : 0 customer at server 1 and 0 customer at server 2 and 1 customer at server 3.
$q_{1,1,0}$ : 1 customer at server 1 and 1 customer at server 2 and 0 customer at server 3.
$q_{1,0,1}$ : 1 customer at server 1 and 0 customer at server 2 and 1 customer at server 3.
$q_{0,1,1}$ : 0 customer at server 1 and 1 customer at server 2 and 1 customer at server 3.  

I will let you figure out the transitions.