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This question is regarding a simple problem, which can be posed for different physical situations. I am posing for a particular physical situation, but the idea is to exposit the general mathematical problem.

The Problem

A device is running on an ideal battery of capacity $C$ (in mAh). It has a baseline current consumption of $i_s$ mA; it always consume this current at all times. There exist several "modes" for the device, which recur at fixed intervals, continue for fixed time and consume a given amount of current during that time. Multiple modes can co-occur. We can denote these modes by $w_i$. And their respective recurrence interval, runtime and current consumption as $r_i$, $t_i$ and $i_i$.

Example:

To visually understand the problem and its modes, here is a simple example.

Let us take a device of $10 Ah$ capacity, with baseline current consumption $i_s$ = $5 mA$. It has the following modes.

\begin{array}{|c|c|c|c|} \hline Mode & r_i (s) & t_i (s) & i_i (mA) \\ \hline w_1 & 3*3600 & 1000 & 40 \\ \hline w_2 & 1.2*3600 & 2000 & 80 \\ \hline w_3 & 0.5*3600 & 500 & 200 \\ \hline \end{array}

Here is an image showing the current consumption of different modes over time: Graph showing consumption of each mode

Here is an image showing the net/total current consumption over time: Graph showing total consumption

To Find:

We have to find the total run-time of the device (i.e. its battery).

Solutions:

I have multiple solutions to this problem, some approximate (but analytical) and some numerical (iterative).

I want to know is there an analytical solution to this problem? Moreover, what tools are used to solve such problems? Can this be posed as an LP problem?

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