An Exact Continuous-Time Markov Chain Framework for Modeling and Performance Evaluation of Multi-Product Push–Pull Production Systems

Multi-product production systems require effective coordination among production, intermediate storage, downstream processing, and customer demand, particularly under stochastic operating conditions and finite capacity. Unlike existing analytical studies that mainly examine either single-product push–pull systems or multi-product manufacturing systems separately, this work develops an exact CTMC framework that jointly captures product variety, shared buffering, downstream parallelization, sequence-dependent setups, and machine unreliability. The system comprises an unreliable upstream machine with sequence-dependent setup changes, a finite intermediate buffer, a distribution center modeled as a pooled processing resource with (M) identical reliable channels, and dedicated finished-goods buffers serving product-specific demand. A high-dimensional continuous-time Markov chain is formulated, and a systematic algorithm is developed to construct the infinitesimal generator matrix and compute steady-state performance measures. Numerical experiments examine intermediate buffer capacity, downstream processing capacity, priority rules, and upstream machine reliability. Increasing buffer capacity from 0 to 20 increases total throughput from 0.5937 to 1.0988, whereas further expansion to 100 yields only 1.1606, while average work-in-process reaches 19.8524. Downstream capacity exhibits similar diminishing performance gains, while priority rules and machine reliability affect product-level and overall throughput. These findings highlight throughput–inventory trade-offs and demonstrate the framework’s applicability for evaluating alternative configurations.