A matrix-infinite-product-form solution for upper block-Hessenberg Markov chains: Quasi-algorithmically constructing the stationary distribution without any additional conditions.

This paper presents a new matrix-infinite-product-form (MIP-form) solution for the stationary distribution in upper block-Hessenberg Markov chains (upper BHMCs). The existing MIP-form solution requires a certain parameter set $(\boldsymbol{v},b,{\mathbb C})$ that satisfies both a Foster-Lyapunov drift condition and a convergence condition (Masuyama, Queueing Syst., Vol. 92, Nos. 1--2, 2019, pp. 173--200). On the other hand, the new MIP-form solution does not require such a parameter set $(\boldsymbol{v},b,{\mathbb C})$, and converges to the stationary distribution without any other conditions. Moreover, the new solution is constructible $quasi$-$algorithmically$, that is, constructed by iterating infinitely many time a recursive procedure of finite complexity per iteration.