Correction to: Queueing Syst (2021) 97:101–124 https://doi.org/10.1007/s11134-020-09675-7

The errors

Unfortunately, we have discovered several errors in [2]:

  1. (i)

    Lemma 5 in Sect. 4 is incorrect. A counterexample is given in Sect. 2 below.

  2. (ii)

    Theorem 5 in Sect. 5 is incorrect. It would be correct if we could replace \(t \ge -M_a\) by \(t \ge 0\) in the condition (39) in Theorem 4, but we are not free to do so, because the condition \(t \ge -M_a\) is required by the increasing convex stochastic order used in Theorem 4.

  3. (iii)

    The presentation of Lemma 3 is incorrect, but this is fixable, as explained in Sect. 3.

  4. (iv)

    Proposition 1 is incorrect, but this is fixable. This proposition becomes correct if the condition \(g(0) = 0\) is added, as holds in the intended Erlang example (\(E_k\) for \(k \ge 2\)). The correction is needed because (57) in [2] is missing the term g(0)h(t).

These errors have serious implications. The error in Lemma 5 invalidates the proofs of Theorems 1 and 3. The error in Theorem 5 invalidates the proof of Theorem 2. Thus, Theorems 1–3 become conjectures remaining to be proved or disproved.

The error in the proof of Theorem 1 invalidates the proof of Theorem 8, which invalidates the proof of Theorem 7. However, we have obtained new results, which provide a new proof of Theorem 7, as explained in Sect. 4 below.

Counterexample to Lemma 5

We will work with the two-point distributions as defined in Sect. 2.1 of [2]. Assume that the mean is \(m = 1\), the upper limit of the support is \(M = 5\) and the squared coefficient of variation is \(c^2 = 1\). Let \(X_0\) and \(X_{u}\) be random variables with the extremal two-point cdf’s \(F_0\) and \(F_u\), respectively. Then, \(P(X_0 = 2) = 1/2 = P(X_0 = 0)\), while \(P(X_u = 5) = 1/17\) and \(P(X_u = 3/4) = 16/17\). It is known that \(X_0 \le _{3-cx} X_u\), as stated in (34) of [2]. Since \(E[X_0] = E[X_u] = 1\) and \(E[X_0^2] = E[X_u^2] = 2\), we also have \(X_0 \le _{2,2} X_u\). However, contrary to Lemma 5 in [2], the ordering \(Y_0 \equiv (X_0 - 3/4)^{+} \le _{2,2} (X_u - 3/4)^{+} \equiv Y_u\) fails to hold. This is easy to see, because \(Y_0\) and \(Y_u\) are the two-point distribution with \(P(Y_0 = 0) = 1/2 = P(Y_0 = 5/4)\), while \(P(Y_u = 0) = 16/17\) and \(P(Y_u = 17/4) = 1/17\), so that we have a reverse ordering of the means: \(E[Y_0] = 5/8 > 1/4 = E[Y_u] = E[X_u] - 3/4\). For the counterexample to the ordering under consideration, note that \(Y_0 + t \ge 0\) and \(Y_u + t \ge 0\) for all \(t \ge 0\),

$$\begin{aligned} E[(Y_0 + t)^2]= & {} t^2 + 5t/4 + O(1) \quad \text{ and }\quad \\ E[(Y_u + t)^2]= & {} t^2 + t/2 + O(1) \quad \text{ as }\quad t \rightarrow \infty , \end{aligned}$$

so that \(E[(Y_0 + t)^2] > E[Y_u + t)^2]\) for all t sufficiently large. This contradicts the claim of Lemma 5.

Correcting Lemma 3

Lemma 3 is important because it provides a way to apply the theory of Tchebycheff (T) systems from [4], as briefly reviewed in [1] and Section 3 of [2]. However, in the statement of Lemma 3 insufficient care was given to the support of the random variable Y with distribution \(\Gamma \) appearing in (22) of [2]. The support of Y should be chosen so that the integrand \(\phi (u)\) appearing in (21) of [2] is not identically 0 for any subinterval of \([0, M_a]\). Hence, the support of Y should be changed from \([0,\infty )\) to a more general interval, i.e., (22) should be replaced by

$$\begin{aligned} \phi (u) \equiv \int _{a}^{b} h((y-u)^{+}) \, \mathrm{d}\Gamma (y) = h(0)\Gamma (u) + \int _{u+}^{b} h(y-u) \, \mathrm{d}\Gamma (y), \quad 0 \le u \le M_a, \end{aligned}$$
(1)

where

$$\begin{aligned} -\infty \le a \le 0 < M_a \le b \le \infty , \end{aligned}$$
(2)

\(\Gamma \) is a cdf of a real-valued random variable Y with a continuous positive density function over the interval [ab]. Then, in Lemma 3 of [2] we should replace (25) by (2) above. The proof also needs to be adjusted accordingly. In particular, the revised proof is:

Proof

First, observe that the finite mgf condition implies that all integrals are finite. In each case, we can apply Lemmas 1 and 2 of [2] with (1) and (2). To do so, we apply the Leibniz rule for differentiation of an integral with (1). Using (2), we have

$$\begin{aligned} \phi (u)= & {} \int _{a}^{b} h ((y-u)^{+}) \, \mathrm{d}\Gamma (y) = \int _{u}^{b} h (y-u) \, \mathrm{d}\Gamma (y) + h(0) \Gamma (u) \quad \text{ and }\quad \nonumber \\ \phi ^{(1)} (u)= & {} -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y) - h(0)\gamma (u) + h(0) \gamma (u)\nonumber \\&= -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y). \end{aligned}$$
(3)

For \(h(x) \equiv x\) in condition (i), we have \(h^{(1)} (x) = 1\) for all x, so that

$$\begin{aligned} \phi ^{(1)} (u) = -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y) = -\int _{u}^{b} \, \mathrm{d}\Gamma (y) = -(1 - \Gamma (u)), \end{aligned}$$
(4)

so that, by the condition on \(\Gamma \),

$$\begin{aligned} \phi ^{(2)} (u) = \gamma (u) > 0 \quad \text{ and }\quad \phi ^{(3)} (u) = \gamma ^{(1)} (u) < 0 \quad \text{ for }\quad 0 \le u \le M_a. \end{aligned}$$
(5)

From the form of \(\phi ^{(3)} (u)\) in (5), we see that the condition on \(\gamma \) is necessary as well as sufficient. We also see that the UB and LB are switched if instead \(\gamma ^{(1)} (u) > 0\).

Turning to \(h(x) = x^2\) in condition (ii), we use \(h^{(1)} (0) = 0\) and \(h^{(2)} (x) = 2\) for all x with the second line of (3) to get

$$\begin{aligned} \phi ^{(2)} (u) = \int _{u}^{b} h^{(2)} (y-u) \, \mathrm{d}\Gamma (y) = 2\int _{u}^{b} \, \mathrm{d}\Gamma (y) = 2 (1 - \Gamma (u)) > 0, \end{aligned}$$
(6)

so that \(\phi ^{(3)} (u) = -2 \gamma (u) < 0\) for \(0 \le u \le M_a\).

Conditions (iii) and (iv) are both special cases of condition (v), which implies that

$$\begin{aligned} \phi ^{(3)} (u) = -\int _{u}^{b} h^{(3)} (y-u) \, \mathrm{d}\Gamma (y) < 0.~~~ \end{aligned}$$
(7)

\(\square \)

Application of Lemma 3 to the higher cumulants

In [3], we have applied the corrected Lemma 3 in [2] to develop new extremal results for the higher cumulants of the steady-state waiting time that provide corrected proofs of Theorems 7 and 8 in [2]. These bounds for higher cumulants are interesting and important because they clearly demonstrate the value of Lemma 3 in [2] and highlight its limitation for treating the mean. In particular, the decreasing pdf condition in Lemma 3 (i) prevents positive results for the mean that we now obtain for the higher cumulants from Lemma 3 (ii) and (iii).