I assume the state number corresponding to the total number of customers in the waiting room
and service room.
S(0) ~ Exp(λ)
The only way to leave this state is that a new pair of customer arrive (jump to state 2)
So you will need to wait Exp(λ) time for them, which is a property of Possion process.
S(1) ~ Exp(λ+μ)
The way to leave this state is that either finish the service with this customer
(jump to state 0) or wait a new customer arrive (jump to state 2)
So the the waiting time is the minimum of the waiting time of these two events
and you can show that the minimum of these two exponential time ~ Exp(λ+μ)
S(2) ~ Exp(μ)
The only way to leave this state is to finish the serive with the current customer
and start the service with the customer in the waiting room (jump to state 1)
P.S. The pdf of Exp(λ) is f(s) = λe^(-λs), s > 0, λ > 0