Consider a line of equally spaced sensors and a disturbance which travels unidirectionally along the line at a fixed speed such that the disturbance takes time $\tau$ to travel between adjacent sensors. When the disturbance reaches a sensor, the sensor is triggered to send an alarm to a central server. However, the sensor must wait for its next assigned transmission slot to send the alarm, which occurs periodically every $\sigma$ slots. Different sensors have different transmission offsets within the period in general, although two sensors can be assigned the same transmission offset without interference. The alarm that a sensor sends includes the identity of the sensor, but not a timestamp indicating when the disturbance reached the sensor.
The basic question is, if the server receives alarms at slots $r_1, r_2, … , r_n$ from sensors $1, 2, … , n$ respectively, what is the maximum likelihood estimate of $\tau$? We can assume that $\tau$ is constrained to be an integer number of slots.
Here is a summary of my work on the problem to date:
If the server receives an alarm for sensor $m$ during slot $r_m$, then the disturbance must have reached sensor $m$ in one of the preceding $\sigma$ slots. Therefore, if $T_m$ is a random variable representing the slot during which the disturbance reached sensor $m$:
$$r_m -\sigma \leq T_m \leq r_m-1$$
and:
$$r_m -\sigma - (m-1) \tau \leq U_m \leq r_m - 1- (m-1) \tau$$
where:
$$U_m = T_m – (m-1) \tau$$
The likelihood $L(\tau)$ of a given value of $\tau$ is determined by the normalised overlap of the intervals represented by $U_m \forall m$. Specifically:
$$L(\tau)=\frac1\sigma\left[min(r_1 – 1, r_2 – 1-\tau,…,r_n – (n-1)\tau) – max(r_1 – \sigma, r_2 – \sigma-\tau,…,r_n – \sigma - (n-1)\tau)+1\right]$$
Using numerical simulation techniques, if I select the value of $\tau$ that maximizes $L(\tau)$, this value does converge to the true value as $n$ increases.
However, I am unsure whether it is possible to derive a general algebraic expression for the maximum likelihood estimate for $\tau$ in terms of $r_1, r_2, … , r_n$. For $n=2$, there is a trivial solution as follows (note that the triggered sensors $i$ and $j$ are not necessarily adjacent):
$$ \tau = \frac{r_j-r_i}{j-i}$$
But what about for $n \gt 2?$ Not being a mathematician, I cannot easily determine whether this problem lends itself to further analysis. If it is obvious to an expert that this cannot be solved algebraically in the general case, that would be useful information for me.