  K. Igusa and G. Todorov introduced the $\varphi$ function which generalizes
the notion of projective dimension. We study the behavior of the $\varphi$
function for cyclic Nakayama algebras of infinite global dimension. We prove
that the supremum of values of $\varphi$ is always an even number. In
particular we show that the $\varphi$-dimension is $2$ if and only if the
algebra satisfies certain symmetry conditions. Also we give a sharp upper bound
for $\varphi$-dimension in terms of the number of monomial relations which
describes the algebra.
