A composition and analysis technique is developed for investigation of infinite Petri nets with regular structure, introduced for modeling networks, clusters and computing grids, that also concerns cellular automata and biological systems. A case study of a square grid structure composition and analysis is presented. Parametric specification of Petri nets, parametric representation of infinite systems for the calculation of place/transition invariants, and solving them in parametric form allowed the invariance proof for infinite Petri net models. Some additional analysis techniques based on graphs of transmissions and blockings are presented. Further generalization on multidimensional structures such as hypercube and hypertorus have been implemented. Generators of Petri net models have been developed and put on GitHub for public use. Complex deadlocks are disclosed and a possibility of network blocking via ill-intentioned traffic revealed. Quality of service in modern networks and numerical parameters of blocking networks with disguised ill-intentioned traffic are investigated using reenterable models in the form of colored Petri nets.