Analytical solution to mixed distribution fit to failure time data -- Lambert W perhaps? I have a set of $n$ device failure times $\{t_i>0\}$ for $i=1...n$ and $N-n$ devices which have not yet failed.  Using maximum likelihood I am attempting to find a closed-form analytical solution to fit the data to the following cumulative distribution function:
$$F(t|\lambda,p)=p(1-e^{-\lambda t})$$ 
where $0<p<1$ is the asymptotic fraction of units to eventually fail and $\lambda>0$ the sub-population failure rate. The likelihood for this MLE attempt is given by:
$$\mathscr{L} = (1-F(t_n))^{N-n}\prod_{i=1}^n f(t_i)$$ and
$$ \ln{ \mathscr{L}} = (N-n)\ln{\left(1 - p+pe^{-\lambda t_n}\right)} + n\lambda p  -\lambda\sum_{i=1}^n t_i$$
with ${p}{d}{f}$ of  $f(t)=dF/dt=\lambda p e^{-\lambda t}$. Here we take $\nabla_{\lambda p}\mathscr{L}=0 $ or $ \nabla_{\lambda p}\ln \mathscr{L}=0$ to solve for $p$ and $\lambda$ at max likelihood (or log likelihood).  I've just recently learned a smidgen about the Lambert $W$ function and was hoping that someone with a more nimble mind than mine might be able to derive a closed form solution using this and/or other cleverness.
 A: Lambert W function can not help:
\begin{cases}
\dfrac{\mathrm d\mathcal L}{\mathrm dp} = -\dfrac {(N-n)\left(1-e^{-\lambda t_n}\right)}{1-p+pe^{-\lambda t_n}}+n\lambda = 0,\\
\dfrac{\mathrm d\mathcal L}{\mathrm d\lambda} = -\dfrac {(N-n)pt_ne^{-\lambda t_n}}{1-p+pe^{-\lambda t_n}}+np - \sum\limits_{t=1}^n t_i = 0
\end{cases}
\begin{cases}
p= \dfrac1{1-e^{-\lambda t_n}} - \dfrac{n\lambda}{N-n}\\
\dfrac {(N-n)pt_n\left(1-\dfrac{1}{p+\dfrac{n\lambda}{N-n}}\right)}{1-\dfrac p{p+\dfrac{n\lambda}{N-n}}} = np - \sum\limits_{t=1}^n t_i,
\end{cases}
\begin{cases}
p= \dfrac1{1-e^{-\lambda t_n}} - \dfrac{n\lambda}{N-n}\\
\dfrac {(N-n)pt_n\left(p-1+\dfrac{n\lambda}{N-n}\right)}{\dfrac{n\lambda}{N-n}} = np - \sum\limits_{t=1}^n t_i,
\end{cases}
\begin{cases}
\dfrac{n\lambda}{N-n} = \dfrac{(N-n)t_np(1-p)}{\left((N-n)t_n-n\right)p+\sum\limits_{t=1}^n t_i}\\
p= \dfrac1{1-e^{-\lambda t_n}} - \dfrac{n\lambda}{N-n}\tag1
\end{cases}
I am not able to find a closed analytical solution of the system $(1)$.
At this time, it can be solved using iteration method or by eliminating of one unknown.
