Abstract
We study the robustness of performance predictions of discrete-time finite-capacity queues by applying the framework of imprecise probabilities. More concretely, we consider the Geo/Geo/1/L model with probabilities of arrival and departure that are no longer fixed, but are allowed to vary within given intervals. We distinguish between two concepts of independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of time-homogeneous probabilities for arrival and departure, which leads us to consider a collection of stationary queues. In the second, the stationarity assumption is dropped and we allow the arrival and departure probabilities to vary from time point to time point; they may even depend on the complete history of queue lengths. We calculate bounds on the expected queue length, the probability of a particular queue length and the probability of turning on the server. For the expected queue length, both approaches coincide. For the other performance measures, we observe and discuss various differences between the bounds obtained for these two approaches. One of our observations is that ergodicity may break down due to imprecision: bounds on expected time averages of certain functions on the state space are not necessarily equal to the bounds on the expectation of that function at random instants in a steady-state queue.
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Acknowledgments
Research for this paper by Stavros Lopatatzidis, Gert de Cooman, Stijn De Vuyst and Joris Walraevens was funded through project number 3G012512 of the Research Foundation Flanders (FWO). Jasper De Bock is a PhD Fellow of the FWO and wishes to acknowledge its financial support. The authors would also like to thank three anonymous referees for their many helpful suggestions.
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Appendices
Appendix
Proofs
Proof of Theorem 1
First notice that
where the crucial third equality holds because the local probabilities of the probability trees in \(\mathscr {T}^{{\text {EI}}}\) are chosen independently of each other. By continuing in this way, we find that
The result now follows because
and because, for all \(i\in \mathbb {N}\):
Proof of Theorem 2
We provide the proof for \(\underline{E}^{{\text {EI}}}_{n}{\left( h\right) }\); the proof for \(\overline{E}^{{\text {EI}}}_{n}{\left( h\right) }\) is completely analogous. It follows from Eqs. (16) and (18) and Theorem 1 that
Therefore, the result follows—by induction—if, for any function \(h\in \mathscr {L}(\mathscr {X})\) that satisfies Eq. (17), we can show (a) that \(\underline{Q}h\) also satisfies Eq. (17) and (b) that, for all \(y\in \mathscr {X}\), the minimum in
is obtained for \(a=\underline{a}\) and \(d=\overline{d}\).
For all \(y\in \{1,\dots ,L\}\), let \(m_y:= h(y)-h(y-1)\ge 0\), where the inequality follows from Eq. (17).
We first prove (b). For \(y=0\), Eqs. (1) and (19) imply that
where the last step holds because \(m_1\ge 0\). Similarly, for \(y \in \{1,\dots ,L-1\}\), Eqs. (2) and (19) imply that
where the last step holds because \(m_y\ge 0\) and \(m_{y+1}\ge 0\).
Finally, for \(y=L\), Eqs. (3) and (19) imply that
where the last step holds because \(m_L\ge 0\). This concludes the proof of (b).
We now prove (a): \(\underline{Q}h(y+1)-\underline{Q}h(y)\ge 0\) for all \(y\in \{0,\dots ,L-1\}\). For \(y=0\), this holds because it follows from Eqs. (20) and (21) that
For \(y\in \{1,\dots ,L-2\}\), this holds because it follows from Eq. (21) that
For \(y=L-1\), this holds because it follows from Eqs. (21) and (22) that
Proof of Theorem 3
For all \(n\in \mathbb {N}\), it follows from subadditivity [27, Chap. 2.6.1(e)] that
whence
where the last inequality follows from the definition of the limit inferior. The proof for the upper expectations is completely analogous; the inequalities are reversed and subadditivity is replaced by superadditivity.
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Lopatatzidis, S., De Bock, J., de Cooman, G. et al. Robust queueing theory: an initial study using imprecise probabilities. Queueing Syst 82, 75–101 (2016). https://doi.org/10.1007/s11134-015-9458-6
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DOI: https://doi.org/10.1007/s11134-015-9458-6
Keywords
- Geo/Geo/1/L
- Imprecise probabilities
- Time-homogeneous
- Robustness
- Performance measures
- Discrete-time queueing