## Service-level differentiation in many-server service systems: A solution based on fixed-queue-ratio routing (2007)

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Venue: | OPERATIONS RESEARCH |

Citations: | 30 - 18 self |

### BibTeX

@ARTICLE{Gurvich07service-leveldifferentiation,

author = {Itay Gurvich and Ward Whitt},

title = {Service-level differentiation in many-server service systems: A solution based on fixed-queue-ratio routing},

journal = {OPERATIONS RESEARCH},

year = {2007},

volume = {29},

pages = {567--588}

}

### OpenURL

### Abstract

Motivated by telephone call centers, we study large-scale service systems with multiple customer classes and multiple agent pools, each with many agents. For the purpose of delicately balancing service levels of the different customer classes, we propose a family of routing controls called Fixed-Queue-Ratio (FQR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified propor-tion of the total queue length. We show that the proportions can be set to achieve desired service-level targets for all classes; these targets are achieved asymptotically as the total arrival rate increases. The FQR rule is a special case of the Queue-and-Idleness-Ratio (QIR) family of controls which in a pre-vious paper where shown to produce an important state-space collapse (SSC) as the total arrival rate increases. This SSC facilitates establishing asymptotic results. In simplified settings, SSC allows us to solve a combined design-staffing-and-routing problem in a nearly optimal way. Our analysis also establishes a diminishing-returns property of flexibility: Under FQR, very moderate cross-training is sufficient to make the call center as efficient as a single-pool system, again in the limit as the total arrival rate increases.

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Citation Context ...ses and a single server (also known as the V model), and generalized to more complicated networks by Mandelbaum and Stolyar [15]. A parallel to [15] in a many-server setting has been provided by Atar =-=[3]-=-, who characterizes a family of controls that achieve asymptotically optimal performance in the QEDGurvich and Whitt: Service-Level Differentiation in Service Systems Article submitted to Operations ... |

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Citation Context ... classes). The proportions pi in turn are chosen to depend on the specified SL constraints. The FQR rule is a special case of the Queueand-Idleness-Ratio (QIR) family of controls that we introduce in =-=[12]-=-. A consequence of [12] is that FQR makes the separate queue lengths asymptotically proportional to the total queue length. In other words, FQR produces a very important state-space collapse (SSC), ca... |

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Citation Context ... problems of design, staffing and routing together, but the only previous work we are aware of that establishes asymptotic optimality for all three problems has been done by Bassamboo et. al. [9] and =-=[10]-=-. They consider a staffing problem for an SBR system and aim at minimizing a cost function which reflects the costs of waiting times, abandonments and customer rejections but not QoS constraints. They... |

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Citation Context ...e arises naturally from economic considerations. We will be considering the QED regime throughout this paper. The simplified aggregate approach to staffing is also a central idea in Wallace and Whitt =-=[30]-=-, which develops a simulation-based iterative algorithm for staffing an SBR call center that starts by choosing an initial total number of agents by acting as if the call center were a single-class si... |

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Citation Context ...st, then, to examine the performance of FQR in settings with different forms of model uncertainty. That would parallel recent work on model uncertainty by Atar [7], Bassamboo et al. [9, 10] and Whitt =-=[34]-=-. Acknowledgments: The authors are grateful to Avi Mandelbaum and Mor Armony for fruitful discussions, and to Zohar Feldman and Ohad Perry for contributions to the simulation, including the use of the... |

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Citation Context ...he three problems of design, staffing and routing together, but the only previous work we are aware of that establishes asymptotic optimality for all three problems has been done by Bassamboo et. al. =-=[9]-=- and [10]. They consider a staffing problem for an SBR system and aim at minimizing a cost function which reflects the costs of waiting times, abandonments and customer rejections but not QoS constrai... |

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Citation Context ...d Whitt [23] also addressed these three problems together, but the only previous work we are aware of that establishes asymptotic feasibility or optimality for all three problems is Bassamboo et. al. =-=[4, 5, 6]-=-. They establish asymptotic optimality for the problem of minimizing costs associated with waiting, abandonments and customer rejections. In [6] they also consider abandonment constraints but not tail... |

18 | Queue-and-idleness-ratio controls in manyserver service systems
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Citation Context ...tions pi in turn are chosen to depend on the specified SL constraints. The FQR rule is a special case of the Queue-and-Idleness-Ratio (QIR) family of controls that we introduced in our previous paper =-=[19]-=-. A consequence of our analysis in [19] is that FQR makes the separate queue lengths asymptotically equal to the fixed proportions of the total queue length. In other words, FQR produces a very import... |

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Citation Context ...e. That can be expected because waiting times are known to be of order 1/ √ λ in the QED regime. Extensive experience shows that these QED approximations exhibit excellent performance; see [11], [4], =-=[27]-=-. We will substantiate that through a numerical example in §3. Here is the first supporting theorem of this section: Theorem 2.1 (asymptotic feasibility and state-space collapse for the V model) If SC... |

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Citation Context ...established asymptotic optimality for the single-class, single-pool M/M/N queue. The asymptotic framework is the now-familiar many-server heavy-traffic limiting regime, introduced by Halfin and Whitt =-=[14]-=-, which is also known as the Quality-and-Efficiency-Driven (QED) regime. In the QED regime the arrival rate and numbers of servers both increase, while the service-time distribution remains unchanged.... |

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Citation Context ...uire different analysis. It is of interest, then, to examine the performance of FQR in settings with different forms of model uncertainty. That would parallel recent work on model uncertainty by Atar =-=[7]-=-, Bassamboo et al. [9, 10] and Whitt [34]. Acknowledgments: The authors are grateful to Avi Mandelbaum and Mor Armony for fruitful discussions, and to Zohar Feldman and Ohad Perry for contributions to... |

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Citation Context ...he case for more general SBR models, so that the ability to interchange the limits remains an open problem. For an example of techniques that may apply, see Gamarnik and Zeevi [16], Budhiraja and Lee =-=[13]-=-, and Gurvich and Zeevi [21]. Here we settle for the weaker notion of feasibility expressed in Theorem 3.3 and in Theorem 4.2 below. Theorem 3.3 implies that one can search for the feasible β through ... |

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Citation Context ...t only minimal cross-training provides the required flexibility. The diminishing-returns property of cross-training for call centers has been proved under certain conditions by 2sAks¸in and Karaesmen =-=[1]-=-. The approach in Wallace and Whitt [30] has two shortcomings, which we address here. First, that approach requires an iterative simulation algorithm to adjust staffing levels and skill assignments in... |

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Citation Context ...e the simple aggregate approach to staffing work. Several recent papers have proposed this simplified aggregate approach to staffing. Theoretical support is contained in Armony [2] and Armony et. al. =-=[4]-=-. These papers establish asymptotic optimality of that staffing approach with appropriate routing for special classes of models as the total arrival rate increases. The first paper considers models wi... |

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Citation Context .... It should be mentioned that within the context of single-server stations, several papers have tackled the problem of service-level constraint satisfaction. The most relevant is probably Van-Meighem =-=[29]-=- which embeds the constraint-satisfaction problem into the convex-holding-cost setting of his paper [28], rather than dealing with it directly. The analysis in this paper relies heavily on our previou... |

5 |
A multi-class fluid model for a contact center with skill-based routing
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Citation Context ...erring to the subset of J with elements j such that ¯νj > 0. The solution to the mathematical program (16) can be regarded as a first-order deterministic fluid approximation for the SBR system, as in =-=[33]-=-. From that point of view, given a selected solution (¯ν, ¯x), we would then use ¯ν to provide an initial estimate of the staffing and ¯x to provide an initial estimate of the appropriate routing. For... |

3 |
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Citation Context ...rems 4.2 and 4.3 continue to hold after the asymptotic expressions for the � model are replaced with those for a � model with abandonment. These expressions can be obtained from Armony and Mandelbaum =-=[3]-=-. If all service rates are equal, i.e, µj ≡ µ, j ∈ J , then more can be said; see Remark 4.3. adding designated-service constraints. In practice, it is natural to require that most customers receive t... |

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2 |
Scheduling flexible servers with convex delay costs in many-server service systems
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Citation Context ...s, but the treatment of the augmented hydrodynamic model equations (see §5 of [19]) used to prove state-space collapse become significantly more complicated. The following is proved in Theorem 4.4 of =-=[20]-=-. Theorem 5.1 (state-space collapse under FWR) In addition to the conditions of Theorem 3.1, assume that ˆ Q λ i (0) = 0 for all λ and all i ∈ I. Then, in addition to the results of Theorem 3.1, Ti ˆQ... |

2 |
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Citation Context ... models, so that the ability to interchange the limits remains an open problem. For an example of techniques that may apply, see Gamarnik and Zeevi [16], Budhiraja and Lee [13], and Gurvich and Zeevi =-=[21]-=-. Here we settle for the weaker notion of feasibility expressed in Theorem 3.3 and in Theorem 4.2 below. Theorem 3.3 implies that one can search for the feasible β through simulation while keeping the... |

2 | Asymptotic optimality of queue-ratio routing for many-server service systems
- Gurvich, Whitt
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(Show Context)
Citation Context ...d Whitt [23] also addressed these three problems together, but the only previous work we are aware of that establishes asymptotic feasibility or optimality for all three problems is Bassamboo et. al. =-=[4, 5, 6]-=-. They establish asymptotic optimality for the problem of minimizing costs associated with waiting, abandonments and customer rejections. In [6] they also consider abandonment constraints but not tail... |

2 | Staffing telephone call centers subject to service-level constraints: An approximate approach via constraint dualization. Working paper - Bassamboo, Zeevi - 2008 |

2 |
Managing quality of service in call centers via queue-ratio routing: Asymptotic analysis and simulation-based optimization. Working Paper
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(Show Context)
Citation Context ...d Whitt [19] also addressed these three problems together, but the only previous work we are aware of that establishes asymptotic feasibility or optimality for all three problems is Bassamboo et. al. =-=[4, 5, 6]-=-. They establish asymptotic optimality for the problem of minimizing costs associated with waiting, abandonments and customer rejections. In [6] they also consider abandonment constraints but not tail... |

1 |
Constraint satisfaction in call-centers: an algorithm based on the FQR control. Working paper
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- 2007
(Show Context)
Citation Context ...the asymptotic optimality results are not for the most general models, the general feasibility result in this paper and the simulation-based solution algorithm that we propose in the subsequent paper =-=[15]-=-, show that our FQR-based solution can be applied in very general settings to obtain an extremely simple and practical solution with only minor compromise in terms of cost optimality. Related Literatu... |

1 | 2006a. Dynamicrouting and admission control in high volume service systems: Asymptotic analysis via multi-scale fluid limits. Queueing Systems 51(3–4 - Bassamboo, Harrison, et al. |

1 | Heavy-trafficlimits for queues with many exponential servers - Halfin, Whitt - 1981 |