In a certain system, a customer must first be served by server $1$ then by server $2.$ The service times at server i are exponential with rate ${}_i,i=1,2.$ An arrival finding server $1$ busy waits in line for that server. Upon completion of service at server $1,$ a customer either enters service with server $2$ if that server is free or else remains with server $1($blocking any other customer from entering service$)$ until server $2$ is free. Customers depart the system after being served by server $2.$ Suppose that when you arrive there is one customer in the system and that the customer is being served by server $1.$ What is the expected total time you spend in the system?

Hint: Write $T=W_1+S_1+W_2+S_2$ where $T$ is the total time you are in the system, $S_i$ is your service time at server i and $W_i$ is your waiting time until that server $i$ is free, for $i=1,2.$