Robust Queueing Theory

Abstract

We propose an alternative approach for studying queues based on robust optimization. We model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability. Using the generalized central limit theorem, this framework allows to model heavy-tailed behavior characterized by bursts of rapidly occurring arrivals and long service times. We take a worst-case approach and obtain closed form upper bounds on the system time in a multi-server queue. These expressions provide qualitative insights which mirror the conclusions obtained in the probabilistic setting for light-tailed arrivals and services and generalize them to the case of heavy-tailed behavior. We also develop a calculus for analyzing a network of queues based on the following key principle: (a) the departure from a queue, (b) the superposition, and (c) the thinning of arrival processes have the same uncertainty set representation as the original arrival processes. The proposed approach (a) yields results with error percentages in single digits relative to simulation, and (b) is to a large extent insensitive to the number of servers per queue, network size, degree of feedback, traffic intensity, and somewhat sensitive to the degree of diversity of external arrival distributions in the network.

Quantifying Distributional Model Risk via Optimal Transport

“…Here, f is a performance/risk measure that depends on a random element X and a decision variable b that can be chosen from an action space B. The solution to the above problem minimizes worstcase risk over a family of ambiguous probability measures P. Such an ambiguity-averse optimal choice is also referred as a distributionally robust choice, because the performance of the chosen decision variable is guaranteed to be better than OPT irrespective of the model picked from the family P. (2015) for KL-divergence and other likelihood based uncertainty sets, Pflug and Wozabal (2007); Wozabal (2012); Esfahani and Kuhn (2015); Zhao and Guan (2015); Gao and Kleywegt (2016) for Wasserstein distance based neighborhoods, Erdogan and Iyengar (2006) for neighborhoods based on Prokhorov metric, Bandi et al (2015); Bandi and Bertsimas (2014) for uncertainty sets based on statistical tests and Ben-Tal et al (2009); Bertsimas and Sim (2004) for a general overview. As most of the works mentioned above assume the random element X to be R d -valued, it is of our interest in the following example to demonstrate the usefulness of our framework in formulating and solving distributionally robust optimization problems that involve stochastic processes taking values in general Polish spaces as well.…”

Robust Multiclass Queuing Theory for Wait Time Estimation in Resource Allocation Systems

“…We remark on how our choice of service time uncertainty sets and their structure affect our results, and possible ways to calibrate the sets using data and probabilistic guarantees. For an elaborate motivation and justification based on limit theorems, we refer the interested reader to Bandi and Bertsimas (2012) and Bandi et al (2015a).…”

“…where Γ is chosen to match the percentile of interest, see Bandi et al (2015a) for details. Note that in order to estimate the average clearing time, we heuristically select Γ = 0.5, which exhibits good numerical performance.…”

Performance evaluation of law enforcement agency on crime information management using queuing network model

“…telephone exchange), manufacturing systems and service systems (petrol station, supermarket, hospital etc.). Telephone exchange is the first problems (congestion problems) solved using queueing theory and was done by Erlang (Bandi et al, 2013;Quantitative Module D, 2009). His works inspired engineers, mathematicians to deal with queueing problems using (Balsam and Marin, 2007;Adan and Resing, 2015).…”