In this paper we consider Interval-Timed Petri nets (ITPN), an extension of Timed Petri nets in which the discrete time delays of transitions are allowed to vary within fixed intervals including possible zero durations. These nets will be analyzed for the first time under maximal step semantics with auto-concurrency. This matches well with the reality of time critical systems which could be modeled and analyzed with our model. Full algebraic representations of the semantics are proposed. We introduce time-dependent state equations for a sequence of global firing steps of ITPNs which are analogous to the state equation for a firing sequence in standard Petri nets and we prove its correctness using linear algebra. Our result delivers a necessary condition for reachability which is also a sufficient condition for non-reachability of an arbitrary marking in an ITPN.