# Correction to: A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

• Hiroyuki Masuyama
Correction

## 1 Correction to: Queueing Systems  https://doi.org/10.1007/s11134-019-09599-x

In Introduction, below Eq. (1.3), the sentence should read “From (1.2) and (1.3), we have” instead of “From (1.2) and (1.6), we have”. In the same section, under Eq. (1.6), the sentence should read

“Note that $$_{(n)} \widehat{\varvec{\pi }}$$ is equal to $$_{(n)} \overline{\varvec{\pi }}$$ in (1.3) with $$_{(n)} \varvec{\alpha } =_{(n)} \widehat{\varvec{\alpha }};$$” instead of “Note that $$_{(n)} \widehat{\varvec{\pi }}$$ is equal to $$_{(n)} \overline{\varvec{\pi }}$$ in (1.6) with $$_{(n)} \varvec{\alpha } =_{(n)} \widehat{\varvec{\alpha }};$$”.

In Proposition 2.1, the last sentence should read “where, for any vector $$\varvec{m} := (m(i)),\;\left\|\varvec{m}\right\|$$denotes the total variation norm ofm, i.e.,$$\left\| \varvec{m} \right\| = \sum\nolimits_{i} {\left| {m(i)} \right|}$$”.

The sentence below Eq. (2.20) should read “Using (2.19) and (2.20), we rewrite (2.15) as” instead of “Using (2.10) and (2.20), we rewrite (2.15) as”.

In Remark 2.5, the third sentence should read, “Through numerical experiments, Masuyama [20] investigates such a problem for the last-column block-augmented truncation, though the function E is referred to therein as the error decay function, instead of the error bound function”.

In section “A counterexample to convergence”, the third sentence of the first paragraph should read “However, it always holds that $$\lim_{n \to \infty } {_{(n)}} \widehat{\varvec{\pi }} = \varvec{\pi}$$ for special upper block-Hessenberg Markov chains such that the $$\varvec{Q}_{k,\ell }$$ are scalars”.

Above Theorem 3.1, Eq. (3.4) should be:
$$j_{n}^{*} \in \mathop {\arg\min }\limits_{{j \in {\mathbb{M}}_{n} }} \frac{{y_{n} (j)}}{{u_{n}^{*} (j)}} .$$
(3.4)

Under Eq. (3.13), the sentence should read “It also follows from (2.14), (2.19), and (2.20) that” instead of “It also follows from (2.14), (2.10), and (2.20) that”.

In Remark 3.6, the last equation should read:
$$\breve{\varvec{y}}_n := ({\breve{y}}_n(j))_{j\in {\mathbb{M}}_n} = \varvec{v}_n + \sum _{k=0}^n \varvec{U}_{n,k}^*\varvec{w}_{k,n},$$
and find
$$j_{n}^{*} \in \mathop {{\arg\min} }\limits_{{j \in {\mathbb{M}}_{n} }} \frac{{\breve{y}}_n(j)}{{u_{n}^{*} (j)}}.$$

The correct reference 12 is given below:

12. Kontoyiannis, I., Meyn, S.P.: On the f-norm ergodicity of Markov processes in continuous time. Electron. Commun. Probab. 21, Paper no. 77, 1–10 (2016)

The original article has been corrected.