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1 Correction to: Queueing Syst https://doi.org/10.1007/s11134-005-2898-7
We correct the expressions of the matrix \(\mathbf {B}^{(\ell )}\) on page 340 and the matrix \(\mathbf {L}^{(\ell )}\) on page 341 in [1]. Specifically, the following are the corrected expressions of these matrices:
for all \(\ell \ge 0\), where the definitions of the notation in the matrices are unchanged from [1] except the zero matrix \(\mathbf {0}\) in (1), whose size needs to be corrected to \(12\times 12\).
These matrices represent the transitions shown in the left panel of Figure 3 in [1]. The transition from state \((1\mathrm {H},0\mathrm {M},u\mathrm {L})\) to \((1\mathrm {H},0\mathrm {M},(u-1)\mathrm {L})\), namely the third diagonal element of \(\mathbf {B}^{(\ell )}\), was missing in the original expression. The transition rates from \((0\mathrm {H},1\mathrm {M},u\mathrm {L})\) to two states labeled with \((1\mathrm {H},1\mathrm {M},u\mathrm {L})\), namely the (2, k) element of \(\mathbf {L}^{(\ell )}\) for \(8\le k \le 11\), are \(\lambda _H\mathbf {p}^{(MH,M)}\) and \(\lambda _H\mathbf {p}^{(MH,H)}\), but erroneously were \(\lambda _M\mathbf {p}^{(MH,M)}\) and \(\lambda _M\mathbf {p}^{(MH,H)}\) in the original expression. Likewise, the transition rates from \((1\mathrm {H},0\mathrm {M},u\mathrm {L})\) to \((1\mathrm {H},1\mathrm {M},u\mathrm {L})\), namely the (3, k) element of \(\mathbf {L}^{(\ell )}\) for \(8\le k \le 11\), are \(\lambda _M\mathbf {p}^{(MH,M)}\) and \(\lambda _M\mathbf {p}^{(MH,H)}\), but were \(\lambda _H\mathbf {p}^{(MH,M)}\) and \(\lambda _H\mathbf {p}^{(MH,H)}\) in the original expression.
Reference
Harchol-Balter, M., Osogami, T., Scheller-Wolf, A., Wierman, A.: Multi-server queueing systems with multiple priority classes. Queueing Syst. 51, 331–360 (2005). https://doi.org/10.1007/s11134-005-2898-7
Acknowledgements
We thank Dr. Yee Lam Elim Thompson for pointing out the errors.
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Harchol-Balter, M., Osogami, T., Scheller-Wolf, A. et al. Correction to: Multi-server queueing systems with multiple priority classes. Queueing Syst 99, 397–398 (2021). https://doi.org/10.1007/s11134-021-09710-1
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DOI: https://doi.org/10.1007/s11134-021-09710-1