Consider a single-server service in which customers arrive according to a Poisson process with rate $\lambda$. Successive service times are independent with common distribution.
There exists two periods in such a system, one is called idle periods when there are no customers and the other is busy periods when there are at least one customer in the system, so the length of a busy period is defined as the elapsed time from the arrival of the first customer to the departure of the last customer.
Let $B$ be the length of a busy period and let $N(S)$ be the number of arrivals during time S, where $S$ denotes the service time of the first customer. For $N(S) = 1$, which indicates another customer arrives the system while the first customer is being served, the length of the busy period can be expressed as follows $$B = S+B_1$$ where $B_1$ is the additional time from S until the system becomes empty, so I assume $B_1$ is the service time for the second customer.
Then the book says because of the memoryless property of Poisson process, $B_1$ has the same distribution as $B$ and I wonder how this hold, since $P(B<x) = P(S+B_1<x)$ and for $B_1$, it's just $P(B_1<x)$