This work employs an approximate Markov chain to analyze that a parallel packet switch (PPS) can emulate a first-come first-served (FCFS) output-queued (OQ) packet switch only operating slower than the external line rate. The PPS comprises multiple packet switches operating independently and in parallel. The PPS class is characterized by deployment of parallel center-stage switches with memory buffers running slower than the external line rate. Each lower speed packet switch operates at a fraction of the external line rate R. For example, each packet switch can operate at internal line rate R/ K, where K is the number of center-stage switches. This work presents a proposed novel Markov chain model successfully exhibits these performance characteristics for throughput, cell delay and cell drop rate. This novel Markov chain model extends from our previous paper [Chia-Lung Liu, Woei Lin, Performance analysis of the sliding-window parallel packet switch, in: IEEE 40th International Conference on Communications 2005, ICC 2005, May 16–20, 2005; Chia-Lung Liu, Woei Lin, Evaluation and analysis of the sliding-window parallel packet switch, in: IEEE 19th International Conference on Advanced Information Networking and Applications, AINA, vol. 2, 2005, pp. 355–358]. Simulation comparison demonstrates that the chains are accurate for practical network loads. With the proposed model, major finding are that: (1) it is theoretically possible for the throughput and cell drop rates of a PPS to emulate those of a FCFS-OQ packet switch when each lower speed packet switch operates at a rate of approximately R/ K; and, (2) this work also demonstrates that it is theoretically possible for the cell delay of a PPS to emulate that of FCFS-OQ packet switch when each lower speed packet switch operates at a rate of approximately (3R/cell delay of FCFS-OQ switch). Additionally, this work develops and investigates a novel PIAO PPS which distributes cells or variable-length packets to center-stage switches and uses multiplexers with push-in arbitrary-out (PIAO) queues.