Correction to: Queueing Syst (2021) 97:101–124 https://doi.org/10.1007/s11134020096757
The errors
Unfortunately, we have discovered several errors in [2]:

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

(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.

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

(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 twopoint 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 twopoint 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 \(X_0 \le _{3cx} X_u\), 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 twopoint 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 \(t \ge 0\),
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
where
\(\Gamma \) is a cdf of a realvalued random variable Y with a continuous positive density function over the interval [a, b]. 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
For \(h(x) \equiv x\) in condition (i), we have \(h^{(1)} (x) = 1\) for all x, so that
so that, by the condition on \(\Gamma \),
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
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
\(\square \)
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 steadystate 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).
References
Chen, Y., Whitt, W.: Extremal models for the \(GI/GI/K\) waitingtime tailprobability decay rate. Oper. Res. Lett. 48, 770–776 (2020)
Chen, Y., Whitt, W.: Extremal \(GI/GI/1\) queues given two moments: exploiting Tchebycheff systems. Queueing Syst. 97, 101–124 (2021)
Chen, Y., Whitt, W.: Extremal higher cumulants for \(GI/GI/1\) queues given two moments: exploiting Tchebycheff systems. Working paper, Columbia University (2022)
Karlin, S., Studden, W.J.: Tchebycheff Systems; With Applications in Analysis and Statistics, vol. 137. Wiley, New York (1966)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, Y., Whitt, W. Correction to: Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems. Queueing Syst (2022). https://doi.org/10.1007/s11134022097970
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11134022097970