Given an irreducible Boolean matrix \(A\) of order \(n\) denote by \(D(A)\) its digraph. For any positive integer \(m\) with \(1\leq m\leq n\), the \(m\)-competition graph of \(A\) (in symbols: \(C_{m}(A)\)) is that graph which has the same vertex set as the digraph \(D(A)\), and there is an edge between distinct vertices \(x\) and \(y\) if and only if there exist \(m\) distinct vertices \(v_{1},v_{2},\dots ,v_{m}\) such that there exist walks \(x\to v_{i}\) and \(y\to v_{i}\) for \(1\leq i\leq m\) in \(D(A)\). The smallest positive integer \(q\) such that \(C_{m}(A^{q+i})=C_{m}(A^{q+r+i})\) for some positive integer \(r\) and every nonnegative integer \(i\) is called the \(m\)-competition index of \(A\). The \(m\)-competition index is a generalisation of the competition index and the index of an irreducible Boolean matrix. The main result of the paper gives the upper bound of the \(m\)-competition index of \(A\).