Occasionally someone needs to incorporate in an integer programming model the notion of something changing state for a predefined span of time. The typical characterization I've seen is as follows:

- we have a sequence of binary variables $x_i\in\{0, 1\}, i\in\{1,\dots,N\}$ that indicate the state of something; and
- if the state changes from 0 to 1, it must remain 1 for exactly $K$ consecutive periods (with a possible exception of we reach the horizon limit $N$ before we see the $K$-th unit value).

The answer is to rethink our choice of variables: rather than focusing on binary variables representing the system state, we focus on binary decision variables indicating whether or not a particular epoch represents the start of a string of ones. So we introduce a new set of binary variables $y_i\in\{0, 1\}, i\in\{1,\dots,N\}$, where $y_i=1$ if and only if a string of $x$ variables starting at epoch $i$ take the value 1. We enforce all this with the following constraints:

- $x_{i-1}\le 1-y_i, i\in\{2,\dots,N\}$ (the value of $x$ in the epoch immediately prior to the start of a run must be 0);
- $x_{i+K}\le 1-y_i, i\in\{1,\dots,N-K\}$ (the value of $x$ in the epoch immediately after the end of a run must be 0); and
*either*$$x_i\ge y_j \forall i\in\{1,\dots,N\}, \forall j\in \{i-K+1,\dots,i\}$$*or*$$x_i\ge \sum_{j=i-K+1}^i y_j,$$eliminating any terms with indices outside the domain $\{1,\dots,N\}$ (the $K$ values at or immediately after the start of a run must be 1).

If you do not want the time frame to end on a run of fewer than $K$ ones, constrain $y_j=0$ for $j\in \{N-K+2,\dots,N\}$. If you must have at least one zero after the final run of ones (i.e., <... 0 1 1 1 0> is okay for $K=3$ but <... 0 1 1 1> is not), constrain $y_j=0$ for $j\in \{N-K+1,\dots,N\}$. If you do not want to start a run of ones at the very beginning of the time frame (<0 1 1 1 0 ...> is okay but <1 1 1 0 ...> is not), constrain $y_1=0$.