Random walk of geometric random variables I was wondering if there's a more advanced theory that can fit with the following context:
Let $\tau_{n} = \sum_{i=1}^{n}{T_{i}}$ be a sum of iid geometric random variables with parameter $p$ and for every positive real $t$ let $\tau(t)$ be the largest index $n$ such that $\tau_{n} \leq t$. What can we say about $E[\tau(t)]$? 
 A: Looks like textbook "renewal theory" (see, for example, the Gallager book Discrete Stochastic Processes). Think of $\{T_1, T_2, T_3, \ldots\}$ as i.i.d. inter-arrival times of a process where jobs arrive over time.  Then I believe your definition of $\tau(t)$ is just the number of jobs that have arrived during the time interval $[0,t]$. Can we call this $N(t)$ instead? (I like that notation better).  So let's define $N(t) = \tau(t)$. 
Basic renewal results (mainly just the law of large numbers) can be used to show that $N(t)/t \rightarrow 1/E[T]$ with probability 1.  A  more complicated result shows that $E[N(t)]/t \rightarrow 1/E[T]$. 
In your case, the $\{T_i\}_{i=1}^{\infty}$ times are geometric random variables, which makes things a bit simpler.  The arrival process is then equivalent to a discrete time process with i.i.d. Bernoulli arrivals with probability $p$.  Every slot we independently get one arrival with probability $p$, else we get no arrival.  The expected number of arrivals in $k$ slots is $kp$. Thus: 
$$ \boxed{E[N(t)] = tp \: \: \forall t \in \{1, 2, 3, \ldots\}} $$
The discrete nature of the process ensures that there are no arrivals except at integer times, so, for example, $E[N(3.4)] = E[N(3)] = 3p$. 

If you changed $\{T_i\}_{i=1}^{\infty}$ to i.i.d. exponential random variables with rate $\lambda$, then $N(t)$ would simply be a Poisson process and $E[N(t)] = \lambda t$ for all $t \geq 0$. 
