Customers arrive in a bank according to the Poisson process $N = \{ N(t); t\geq 0\}$ having rate $\lambda.$ There is only one teller in the bank who can serve only one customer. Thus any arrival that occurs while the teller is busy serving a customer is lost. A arrival which occurs while the teller is idle is accepted and the teller begins serving the customer. The service times form a sequence of $iid$ variables $T_1, T_2, \ldots$ independent of $N.$ Set $$Z(t) = \begin{cases} 1 & \text{if a teller is serving a customer at time } t \\ 0 & \text{if a teller is idle at time } t\end{cases}$$ For $t\geq 0,$ set $$m(t) = P\{Z(t) = 1\}.$$ What is the renewal equation for $m$?
My take is the following: \begin{align*} m(t) = E[N(t)] & = \int_{0}^{\infty} E[N(t)\big|T_1 = s ] f_T(s) ds\\ & = \int_0^t [1+m(t-s)] f_T(s)ds, \text{ if } s<t\\ & = F_T(t) + \int_0^t m(t-s)f_T(s)ds \end{align*} I don't know whether this is correct. I need help, any explicit solution is welcomed.