Altman and Shimkin (1998) discuss the case of a processor sharing system with Poisson arrivals with rate $\lambda$ and constant total service rate $\mu$. Let $x$ be the number of customers present in the system. Defining $V(x)$ as the service time, the memoryless property of the system implies that $V(x)$ equals the expected time until the next transition plus the expected remaining time after that transition. Then
$$ V(x) = \frac{1}{\mu + \lambda} + \frac{\mu}{(\mu + \lambda)}\frac{x}{(x+1)}V(x-1) + \frac{\lambda}{(\mu + \lambda)}V(x+1) $$
I understand that $\frac{1}{\mu + \lambda}$ is the expected time until the next transition, and $\frac{\lambda}{\mu + \lambda}, \frac{\mu}{\mu + \lambda}$ to be the probability of a new arrival and departure to and from the system, respectively. However, I fail to understand why the second term, accounting for the probability of a departure from the system and the expected service time after that transition, is multiplied by the term $\frac{x}{x+1}$.
I am new to queuing theory, and I have been looking around for any notes that present service time in this way and failed to find much. I really need to work with this formula so if anyone has any interpretations that would make sense of this term I would very much appreciate this.