A call center waiting time is independently distributed $\ \sim \exp( \theta ) $, and after some $\ a $ minutes of waiting, the call gets disconnected. After the call gets disconnected, the client tries again as many times as needed until success.
Let $\ Y_i $ be the number of calls the client $\ i$ has made.
I need to find MLE for $\ \theta $, so I thought first I'll figure out how $\ Y_i $ is distributed. To my understanding to have success of the k-th call the customer will have to call $\ k-1 $ times without success, meaning $\ 1-(1-e^{-\theta a} ) ^{k-1} $ and then succeed with probability $\ 1-e^{\theta a} $
$$\ P\{Y_i = k \} = (e^{-\theta a})^{k-1} \cdot (1-e^{-\theta a}) $$
So its like $\ Y_i \sim Geom(p) $ where success $\ p = 1-e^{-\theta a} $
And then $$\ L(\theta, Y) = \prod_{i=1}^n (e^{-\theta a})^{Y_i - 1}\cdot(1-e^{-\theta a} ) = \\ (1-e^{-\theta a})^n \ \prod_{i=1}^n \ (e^{-\theta a})^{Y_i} \cdot e^{\theta a} = \\ (1-e^{-\theta a})^n \cdot e^{\theta a(n-\sum Y_i)} $$
But I couldn't proceed from here and extract $\ \theta $, so I guess I'm wrong.