Extraction of Exact Symbolic Stationary Probability Formulas for Markov Chains in Finite Space with Application to Production Lines. Part II: Unveiling accurate Formulas for Very Short Serial Production Lines without Buffers (Three-and Four-Stations)

Konstantinos S Boulas ¹ Georgios D Dounias ¹ Chrissoleon T Papadopoulos ^2*

• ¹ University of the Aegean, Mytilene, North Aegean, Greece
• ² Aristotle University of Thessaloniki, Thessaloniki, Greece

The exact closed-form formulas for all performance measures (i.e., throughput, maximum utilization, work in process, blocking probability for the second station, probability of the third station being idle, etc.) of short serial production lines with two, three, and four stages without buffers are presented for the first time in literature. Markov chains model these production lines. An Algorithm is used to construct their state transition diagram, from which the steady-state probabilities are extracted in the symbolic form of two polynomial ratios, using a recently introduced method that assigns probabilities on the graph anti-arborescences. Then, the underlying Markov chain can obtain every performance measure via its known definition from the literature by simple algebraic operations. The proposed Algorithm's results shed light on the well-known bowl phenomenon of production lines, and understanding the exact formula structure is used to create a simple model for throughput estimation of fully balanced short serial production lines through genetic programming. The enormous size of the exact formulas highlights the problem of the need for more computational support for production lines larger than four stages without buffers.