Ohira and Sawatari [Phys. Rev E 58, 193 (1998)] introduced a simple model for a packet-switching network which was extended by Solé and Valverde [Physica A 289, 595 (2001)]. Both models used Poisson-like traffic sources. Solé and Valverde demonstrated that long-range dependence (LRD) in autocorrelation behavior can be seen in the queue length dynamics at a given node. Actual network traffic sources are known to exhibit long-range autocorrelation. To simulate the real case more closely, we have studied the effect of introducing LRD behavior at an earlier stage. We replaced the Poisson-like sources with LRD sources, modeled using chaotic maps. As was seen in the previous models, a phase transition occurs as the traffic load on a network is increased and the network changes to a congested state where the time taken for delivery of packets increases dramatically and throughput collapses. The paper reports extensive numerical results from our simulations using both Poisson and LRD sources. It demonstrates the natural network-induced LRD when sources are purely Poisson and shows strong enhancement when LRD sources are added. The model is adapted to include congestion control mechanisms and their impact is considered.