Automaton without variables and guards for queueing system
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.