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:

$$\begin{aligned} \mathbf {B}^{(\ell )}&= \mu _L\left( \begin{array}{cccc} \min (2,\ell )&{} &{} &{} \\ &{} 1 &{} &{} \\ &{} &{} 1 &{} \\ &{} &{} &{} \mathbf {0} \end{array}\right) \end{aligned}$$
(1)
(2)

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.