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Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues This article was downloaded by: [160.39.21.161] On: 04 October 2018, At: 11:40 Publisher: Institute for Operations Research and the Management Sciences (INFORMS) INFORMS is located in Maryland, USA Operations Research Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues Ward Whitt, Wei You To cite this article: Ward Whitt, Wei You (2018) Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues. Operations Research 66(1):184-199. https://doi.org/10.1287/opre.2017.1649 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact [email protected]. The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright © 2017, INFORMS Please scroll down for article—it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org OPERATIONS RESEARCH Vol. 66, No. 1, January–February 2018, pp. 184–199 http://pubsonline.informs.org/journal/opre/ ISSN 0030-364X (print), ISSN 1526-5463 (online) Using Robust Queueing to Expose the Impact of Dependence in Single-Server Queues Ward Whitt,a Wei Youa a Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027 Contact: [email protected], http://orcid.org/0000-0003-4298-9964 (WW); [email protected], http://orcid.org/0000-0003-0844-4194 (WY) Received: March 4, 2016 Abstract. Queueing applications are often complicated by dependence among interar- Revised: October 10, 2016; March 11, 2017 rival times and service times. Such dependence is common in networks of queues, where Accepted: May 3, 2017 arrivals are departures from other queues or superpositions of such complicated pro- Published Online in Articles in Advance: cesses, especially when there are multiple customer classes with class-dependent service- July 31, 2017 time distributions. We show that the robust queueing approach for single-server queues Subject Classifications: queues: proposed in the literature can be extended to yield improved steady-state performance approximations, networks, algorithms approximations in the standard stochastic setting that includes dependence among inter- Area of Review: Stochastic Models arrival times and service times. We propose a new functional robust queueing formulation for the steady-state workload that is exact for the steady-state mean in the M GI 1 model https://doi.org/10.1287/opre.2017.1649 and is asymptotically correct in both heavy traffic and light traffic. Simulation/ experiments/ Copyright: © 2017 INFORMS show that it is effective more generally. Funding: Support was received from the National Science Foundation [Grants CMMI 1265070 and 1634133]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2017.1649. Keywords: robust queueing • queueing approximations • dependence among interarrival times and service times • indices of dispersion • heavy traffic • queueing network analyzer 1. Introduction The RQ in Bandi et al. (2015) emphasizes the high Robust optimization is proving to be a useful approach dimension arising when we consider a network of to complex optimization problems involving signifi- queues instead of a single queue. Instead, in this paper cant uncertainty; e.g., see Bandi and Bertsimas (2012), we focus on the high dimension that occurs in a sin- Bertsimas et al. (2011), and references therein. In that gle queue when there is complex stochastic depen- context, the primary goal is to create an efficient algo- dence over time in the arrival and service processes. rithm to produce useful, practical solutions that appro- In a sequel, Whitt and You (2016), we focus on the priately capture the essential features of the uncer- high dimension that occurs in a single queue when the tainty. Bandi et al. (2015) have applied this approach deterministic arrival-rate function is time varying. For to create a robust queueing (RQ) theory, which can both problems, we find that the robust optimization be used to generate performance predictions in com- approach is remarkably effective. Here, we show that, plex queueing systems, including networks of queues with an appropriate choice of parameters, all our new as well as single queues. Indeed, they construct a full RQ solutions are asymptotically correct in the heavy- robust queueing analyzer (RQNA) to develop rela- traffic limit. Our most promising new RQ solutions tively simple performance descriptions such as those in in (18) and (28) are asymptotically correct in both light the queueing network analyzer (QNA) in Whitt (1983). traffic and heavy traffic. Our simulation experiments Our goal in this paper is to make further progress show that the new RQ solutions provide useful approx- in the same direction. We do so by introducing new imations more generally. RQ formulations and evaluating their performance. We too want to obtain useful performance descriptions for 1.1. Dependence Among Interarrival Times and complex queueing networks, but here we only con- Service Times sider a single queue. We judge our RQ formulations by Even though we only focus on one single-server queue, their ability to efficiently generate useful performance ultimately we also want to develop methods that apply approximations for the given stochastic model, which to complex networks of queues. We view the present so far has been mostly intractable. paper as an important step in that direction, because As emphasized in Bandi and Bertsimas (2012), the experience from applications of QNA has shown that a intractability is usually due to high dimension, but major shortcoming is its inability to adequately capture high dimensionality can occur in many different ways. the dependence among interarrival times and service 184 Whitt and You: Dependence in Single-Server Queues Operations Research, 2018, vol. 66, no. 1, pp. 184–199, © 2017 INFORMS 185 Figure 1. Common Queueing Network Structure That Can Induce Dependence Among Interarrival Times: Superpositions of Arrival Processes (Top) and Flow Through a Series of Queues (Bottom) Queue … Server Queue 1 Queue 2 … Queue n times at the individual queues in the network. That process is again Poisson. In other words, there are no was dramatically illustrated by comparisons of QNA to nondeterministic non-Poisson renewal departure pro- model simulations in Sriram and Whitt (1986), Fendick cesses from an M GI 1 queue; e.g., see Disney and et al. (1989), and Suresh and Whitt (1990). Konig (1985). / / Dependence among successive interarrival times at The dependence among interarrival times and ser- a queue is a common phenomenon, usually because vice times has long been recognized as a major that queue is actually part of a network of queues. difficulty in developing effective approximations for For example, arrival processes in queueing networks open queueing networks, such as in QNA in Whitt are often superpositions of other arrival processes or (1983); e.g., see Whitt (1995) and references therein. departure processes from other queues, as depicted in Refined performance approximations have been pro- Figure1. posed using second-order partial characterizations of In most manufacturing production lines, an exter- dependence, using indices of dispersion (variance-time nal (or initial) arrival process is often far less variable functions), which involve correlations among interar- than a Poisson process by design, while complicated rival times as well as means and variances; e.g., see Cox processing operations, such as those involving batch- and Lewis (1966), Heffes (1980), Heffes and Luantoni ing, often produce complicated dependence among the (1986), Sriram and Whitt (1986), Fendick et al. (1989, interarrival times at subsequent queues; e.g., see the 1991), and Fendick and Whitt (1989). Our new RQ for- example in section 3 of Segal and Whitt (1989). In both mulations will exploit these same partial characteriza- manufacturing and communication systems, depen- tions of the dependence among interarrival times and dence among successive interarrival times and among service times; see Sections 3.3 and4. Even though we successive interdeparture times at a queue often occurs only consider a single queue here, in Section6 we intro- because there are multiple classes of customers with duce a new framework in which we hope to develop a different characteristics (e.g., Bitran and Tirupati 1988). full RQNA based on the results in this paper. Multiple classes can even cause significant dependence (i) among interarrival times, (ii) among service times, 1.2. Main Contributions and (iii) between interarrival times and service times, 1. In this paper, we introduce several new RQ for- which all can contribute to a major impact on perfor- mulations for the steady-state waiting time and work- mance, as shown by Fendick et al. (1989) and reviewed load in a single-server queue, and we make useful in section 9.6 of Whitt (2002). connections to the general stationary G G 1 stochas- / / In service systems, an external customer arrival tic model and the GI GI 1 special case. In particular, / / process often is well modeled by a Poisson process, we show how to choose the RQ parameters so that because it is generated by many separate people mak- these RQ solutions all are asymptotically exact for the ing decisions independently, at least approximately, steady-state mean in the heavy-traffic limit.
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