arrival rate in single server with general service time distribution.

Customers arrive at a single-server station with Poisson rate $\lambda$. A customer enters the bank if the server is available; otherwise, the customer leaves. The service times of successive customers are independent and have a common distribution $G$ and mean $\mu$. What is the rate at which the customer enters the system?

I am unable to figure out the answer. I assume that renewal reward process is to be applied here with regeneration happening at every service completion. Could anyone please help. Thanks in advance!

• I've never done one of these problems, but it would seem that the sum of two poisson random variables is another poisson variable. E.g $X= X_{1} + X_{2}$ with mean $\lambda = \lambda_{1} + \lambda_{2}$. It seems idealized but they enter the computer when with rate $\lambda + \mu$ – Shogun Jul 6 '18 at 23:42
• answer was give as λ/(1+λμ) . this is logic I could assume to arrive at the answer given . since arrival happens after each service completion (as customer who enters when server is busy leaves the system), inter-arrival for an accepted customer would be service time μ + 1/λ (exponential arrival has memoryless property), therefore arrival rate is λ/(1+λμ). – ranya Jul 7 '18 at 17:22

In your question it is implicitly assumed that the system operates in the steady state. For the steady state to exist the system's load (denote it by $\rho$) must be less than one i.e. $\rho=\lambda \mu<1$.
The system you are asking about, when operating in the steady state, follows the cycle $empty \rightarrow busy \rightarrow empty \rightarrow busy \rightarrow ...$. Thus the proportion of time that the system is empty is $${(mean \,\, empty \,\, time) \over (mean \,\, empty \,\, time) + (mean \,\, busy \,\, time)}= {{1 \over \lambda} \over {1 \over \lambda} + \mu}.$$
Thus the arrival rate of customers which find the system empty is $\lambda \times {{1 \over \lambda} \over {1 \over \lambda} + \mu}$.
p.p.s. in Kendall's notation, you are asking about the $M/G/1/0$ queueing system. In any classic queueing theory book, there are many results on this queue.