# Incentive Contracts for a Queueing System with a Strategic Server: A Principal-Agent Perspective

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Service Pricing of Queueing System

#### 2.2. Service Effort of Queueing System

#### 2.3. Incentive Contract of Queueing System

## 3. Problem Description and Notations

## 4. Optimal Decisions under Exogenous Price

#### 4.1. Centralized Decision-Making

**Proposition 1.**

#### 4.2. Unit-Price Contract

**Proposition 2.**

**Corollary 1.**

**Corollary 2.**

#### 4.3. Cost-Sharing Contract

**Proposition 3.**

**Corollary 3.**

**Corollary 4.**

**Corollary 5.**

## 5. Optimal Decisions under Endogenous Price

#### 5.1. Centralized Decision-Making

**Proposition 4.**

#### 5.2. Unit-Price Contract

**Proposition 5.**

**Corollary 6.**

**Corollary 7.**

#### 5.3. Cost-Sharing Contract

**Proposition 6.**

**Corollary 8.**

**Corollary 9.**

## 6. Results Analysis

## 7. Numerical Results

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Proposition 1.**

**Proof of Proposition 2.**

**Proof of Corollary 1.**

**Proof of Corollary 2.**

**Proof of Proposition 3.**

**Proof of Corollary 3.**

**Proof of Corollary 4.**

**Proof of Corollary 5.**

**Proof of Proposition 4.**

**Proof of Proposition 5.**

**Proof of Corollary 6.**

**Proof of Corollary 7.**

**Proof of Proposition 6.**

**Proof of Corollary 8.**

**Proof of Corollary 9.**

## References

- Guo, P.; Zhang, Z.G. Strategic queueing behavior and its impact on system performance in service systems with the congestion-based staffing policy. Manuf. Serv. Oper. Manag.
**2013**, 15, 118–131. [Google Scholar] [CrossRef] - Sunar, N.; Tu, Y.; Ziya, S. Pooled vs. dedicated queues when customers are delay-sensitive. Manag. Sci.
**2021**, 67, 3785–3802. [Google Scholar] [CrossRef] - Ata, B.; Shneorson, S. Dynamic control of an M/M/1 service system with adjustable arrival and service rates. Manag. Sci.
**2006**, 52, 1778–1791. [Google Scholar] [CrossRef] - Dimitrakopoulos, Y.; Burnetas, A.N. Customer equilibrium and optimal strategies in an M/M/1 queue with dynamic service control. Eur. J. Oper. Res.
**2016**, 252, 477–486. [Google Scholar] [CrossRef] - Tan, B.; Khayyati, S. Supervised learning-based approximation method for single-server open queueing networks with correlated interarrival and service times. Int. J. Prod. Res.
**2022**, 60, 6822–6847. [Google Scholar] [CrossRef] - Musalem, A.; Olivares, M.; Yung, D. Balancing agent retention and waiting time in service platforms. Oper. Res. 2023; accepted. [Google Scholar]
- Albana, A.S.; Frein, Y.; Hammami, R. Effect of a lead time-dependent cost on lead time quotation, pricing, and capacity decisions in a stochastic make-to-order system with endogenous demand. Int. J. Prod. Econ.
**2018**, 203, 83–95. [Google Scholar] [CrossRef] - Kim, J.; Randhawa, R.S. The value of dynamic pricing in large queueing systems. Oper. Res.
**2018**, 66, 409–425. [Google Scholar] [CrossRef] - Feldman, P.; Segev, E. The important role of time limits when consumers choose their time in service. Manag. Sci.
**2022**, 68, 6666–6686. [Google Scholar] [CrossRef] - Jiang, H.; Pang, Z.; Savin, S. Performance-based contracts for outpatient medical services. Manuf. Serv. Oper. Manag.
**2012**, 14, 654–669. [Google Scholar] [CrossRef] - Legros, B. The principal-agent problem for service rate event-dependency. Eur. J. Oper. Res.
**2022**, 297, 949–963. [Google Scholar] [CrossRef] - Wang, Z.; Yang, L.; Cui, S.; Ülkü, S.; Zhou, Y.P. Pooling agents for customer-intensive services. Oper. Res. 2022; accepted. [Google Scholar]
- Armony, M.; Roels, G.; Song, H. Pooling queues with strategic servers: The effects of customer ownership. Oper. Res.
**2021**, 69, 13–29. [Google Scholar] [CrossRef] - Lin, C.A.; Shang, K.; Sun, P. Wait Time–Based Pricing for Queues with Customer-Chosen Service Times. Manag. Sci. 2022; accepted. [Google Scholar]
- Jia, H.; Shi, C.; Shen, S. Online learning and pricing for service systems with reusable resources. Oper. Res. 2022; accepted. [Google Scholar]
- Güler, M.G.; Bilgiç, T.; Güllü, R. Joint inventory and pricing decisions when customers are delay sensitive. Int. J. Prod. Econ.
**2014**, 157, 302–312. [Google Scholar] [CrossRef] - Moshe, S.; Oz, B. Charging more for priority via two-part tariff for accumulating priorities. Eur. J. Oper. Res.
**2023**, 304, 652–660. [Google Scholar] [CrossRef] - Liu, J.; Chen, J.; Bo, R.; Meng, F.L.; Xu, Y.; Li, P. Increases or discounts: Price strategies based on customers’ patience times. Eur. J. Oper. Res.
**2023**, 305, 722–737. [Google Scholar] [CrossRef] - Zhang, Y.; Wang, J.; Wang, F. Equilibrium pricing strategies in retrial queueing systems with complementary services. Appl. Math. Model.
**2016**, 40, 5775–5792. [Google Scholar] [CrossRef] - Yang, L.; Guo, P.; Wang, Y. Service pricing with loss-averse customers. Oper. Res.
**2018**, 66, 761–777. [Google Scholar] [CrossRef] - Zhang, Z.G.; Yin, X. Information and pricing effects in two-tier public service systems. Int. J. Prod. Econ.
**2021**, 231, 107897–107932. [Google Scholar] [CrossRef] - Mai, Y.; Hu, B.; Pekeč, S. Courteous or Crude? Managing User Conduct to Improve On-Demand Service Platform Performance. Manag. Sci.
**2022**, 69, 996–1016. [Google Scholar] [CrossRef] - Gilbert, S.M.; Weng, Z.K. Incentive effects favor nonconsolidating queues in a service system: The principal–agent perspective. Manag. Sci.
**1998**, 44, 1662–1669. [Google Scholar] [CrossRef] - Shunko, M.; Niederhoff, J.; Rosokha, Y. Humans are not machines: The behavioral impact of queueing design on service time. Manag. Sci.
**2018**, 64, 453–473. [Google Scholar] [CrossRef] - Yu, Y.; Benjaafar, S.; Gerchak, Y. Capacity sharing and cost allocation among independent firms with congestion. Prod. Oper. Manag.
**2015**, 24, 1285–1310. [Google Scholar] [CrossRef] - Gopalakrishnan, R.; Doroudi, S.; Ward, A.R.; Wierman, A. Routing and staffing when servers are strategic. Oper. Res.
**2016**, 64, 1033–1050. [Google Scholar] [CrossRef] - Zhan, D.; Ward, A.R. Staffing, routing, and payment to trade off speed and quality in large service systems. Oper. Res.
**2019**, 67, 1738–1751. [Google Scholar] [CrossRef] - Liu, N.; Jaarsveld, W.; Wang, S.; Xiao, G. Managing Outpatient Service with Strategic Walk-ins. Manag. Sci. 2023; accepted. [Google Scholar]
- Benjaafar, S.; Wu, S.; Liu, H.; Gunnarsson, E.B. Dimensioning On-Demand Vehicle Sharing Systems. Manag. Sci.
**2022**, 68, 1218–1232. [Google Scholar] [CrossRef] - Li, L.; Jiang, L.; Liu, L. Service and price competition when customers are naive. Prod. Oper. Manag.
**2012**, 21, 747–760. [Google Scholar] [CrossRef] - Sun, M.; Ng, C.T.; Wu, F.; Cheng, T.C.E. Optimization of after-sales services with spare parts consumption and repairman travel. Int. J. Prod. Econ.
**2022**, 244, 108382–108392. [Google Scholar] [CrossRef] - Hoseinpour, P.; Marand, A.J. Designing a service system with price-and distance-sensitive demand: A case study in mining industry. Eur. J. Oper. Res.
**2022**, 303, 1355–1371. [Google Scholar] [CrossRef] - Yu, J.; Fang, Y.; Zhong, Y.; Zhang, X.; Zhang, R. Pricing and quality strategies for an on-demand housekeeping platform with customer-intensive services. Transp. Res. Part E Logist. Transp. Rev.
**2022**, 164, 102760–102779. [Google Scholar] [CrossRef] - Jiang, Y.B.; Seidmann, A. Integrated capacity and marketing incentive contracting for capital-intensive service systems. Decis. Support Syst.
**2011**, 51, 627–637. [Google Scholar] [CrossRef] - Jiang, Y.B.; Seidmann, A. Capacity planning and performance contracting for service facilities. Decis. Support Syst.
**2014**, 58, 31–42. [Google Scholar] [CrossRef] - Sun, P.; Tian, F. Optimal Contract to Induce Continued Effort. Manag. Sci.
**2017**, 64, 4193–4217. [Google Scholar] [CrossRef] - Taylor, T.A. On-demand service platforms. Manuf. Serv. Oper. Manag.
**2018**, 20, 704–720. [Google Scholar] [CrossRef] - Bai, J.; So, K.C.; Tang, C.S.; Chen, M.; Wang, H. Coordinating supply and demand on an on-demand service platform with impatient customers. Manuf. Serv. Oper. Manag.
**2019**, 21, 556–570. [Google Scholar] [CrossRef] - Legros, B. Agents’ Self-Routing for Blended Operations to Balance Inbound and Outbound Services. Prod. Oper. Manag.
**2021**, 30, 3599–3614. [Google Scholar] [CrossRef] - Benioudakis, M.; Zissis, D.; Burnetas, A.; Ioannou, G. Service provision on an aggregator platform with time-sensitive customers: Pricing strategies and coordination. Int. J. Prod. Econ.
**2023**, 257, 108760–108775. [Google Scholar] [CrossRef] - Jacob, J.; Roet-Green, R. Ride solo or pool: Designing price-service menus for a ride-sharing platform. Eur. J. Oper. Res.
**2021**, 295, 1008–1024. [Google Scholar] [CrossRef] - Ravula, P. Monetary and hassle savings as strategic variables in the ride-sharing market. Res. Transp. Econ.
**2022**, 94, 101184–101194. [Google Scholar] [CrossRef] - Babic, J.; Carvalho, A.; Ketter, W.; Podobnik, V. A data-driven approach to managing electric vehicle charging infrastructure in parking lots. Transp. Res. Part D Transp. Environ.
**2022**, 105, 103198–103223. [Google Scholar] [CrossRef] - Aljafari, B.; Jeyaraj, P.R.; Kathiresan, A.C.; Thanikanti, S.B. Electric vehicle optimum charging-discharging scheduling with dynamic pricing employing multi agent deep neural network. Comput. Electr. Eng.
**2023**, 105, 108555–108570. [Google Scholar] [CrossRef] - Hu, R.; Chen, L.; Zheng, L. Congestion pricing and environmental cost at Guangzhou Baiyun International Airport. J. Air Transp. Manag.
**2018**, 70, 126–132. [Google Scholar] [CrossRef] - Zhang, B.; Ye, Z.; Wang, L.L. Airport airside congestion pricing considering price discrimination between aircraft type under a Stackelberg game. Transp. Plan. Technol.
**2020**, 43, 48–61. [Google Scholar] [CrossRef] - Hum, S.H.; Parlar, M.; Zhou, Y. Measurement and optimization of responsiveness in supply chain networks with queueing structures. Eur. J. Oper. Res.
**2018**, 264, 106–118. [Google Scholar] [CrossRef] - Sagir, M.; Saglam, V. Optimization and analysis of a tandem queueing system with parallel channel at second station. Commun. Stat.
**2022**, 51, 7547–7560. [Google Scholar] [CrossRef]

References | Service Pricing | Service Effort | Incentive Contract |
---|---|---|---|

Ata and Shneorson [3] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Li, Jiang, and Liu [30] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Sun et al. [31] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Hoseinpour and Marand [32] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Yu et al. [33] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Taylor [37] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Bai et al. [38] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Jiang, Pang, and Savin [10] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Legros [11] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

Wang et al. [12] | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | |

This study | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ |

References | Ride Sharing Platform | Electric Vehicle Charging System | Airport Congestion Pricing | Outpatient Medical Service |
---|---|---|---|---|

Bai et al. [38] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Jacob and Roet-Green [41] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Ravula [42] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Lin, Shang, and Sun [14] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Babic et al. [43] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Aljafari et al. [44] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Hu, Chen, and Zheng [45] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Zhang, Ye, and Wang [46] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

Jiang, Pang, and Savin [10] | $\phantom{\rule{1.em}{0ex}}\surd $ | |||

This study | $\phantom{\rule{1.em}{0ex}}\surd $ | $\phantom{\rule{1.em}{0ex}}\surd $ |

Notation | Description | |
---|---|---|

Parameters | $\lambda $ | The arrival rate of customers |

${\lambda}_{0}$ | The basic arrival rate of customers | |

$\mu $ | The service rate of the server | |

${\mu}_{0}$ | The basic service rate of the server | |

c | The unit service cost for serving a customer | |

g | The unit reputation income of the owner | |

k | The effort cost coefficient of the server | |

w | The sojourn time of customers | |

$\alpha $ | The marginal effect coefficient of the service price on the arrival rate | |

$\beta $ | The marginal effect coefficient of the effort on the service rate | |

Decision variables | e | The service effort of the server |

p | The service price of queueing system | |

r | The unit contract payment provided by the owner to the server | |

$\theta $ | The owner’s sharing proportion of the server’s effort cost | |

Subscripts | $xb$ | Centralized decision-making with exogenous price |

$xu$ | Unit-price contract with exogenous price | |

$xc$ | Cost-sharing contract with exogenous price | |

$nb$ | Centralized decision-making with endogenous price | |

$nu$ | Unit-price contract with endogenous price | |

$nc$ | Cost-sharing contract with endogenous price |

Scenario | Service Effort | Service Price | The Sojourn Time of Customers |
---|---|---|---|

$xb$ | $\frac{g\beta \left({\lambda}_{0}-\alpha p\right)}{k}$ | - | $\frac{k}{(g{\beta}^{2}-k)({\lambda}_{0}-\alpha p)+k{\mu}_{0}}$ |

$xu$ | $\frac{g\beta ({\lambda}_{0}-\alpha p)}{2k}-\frac{{\mu}_{0}}{2\beta}$ | - | $\frac{k}{(\frac{g{\beta}^{2}}{2}-k)({\lambda}_{0}-\alpha p)+\frac{k{\mu}_{0}}{2}}$ |

$xc$ | $\frac{g\beta ({\lambda}_{0}-\alpha p)}{(2-{\theta}_{h})k}-\frac{(1-{\theta}_{h}){\mu}_{0}}{(2-{\theta}_{h})\beta}$ | - | $\frac{k}{[\frac{g{\beta}^{2}}{(2-{\theta}_{h})}-k]({\lambda}_{0}-\alpha p)+\frac{k{\mu}_{0}}{(2-{\theta}_{h})}}$ |

$nb$ | $\frac{g\beta \left({\Delta}_{1}+\alpha g{\mu}_{0}\right)}{2k(1+g\alpha )-\alpha {\beta}^{2}{g}^{2}}$ | $\frac{k{\Delta}_{3}-{\Delta}_{4}}{\alpha [2k(1+g\alpha )-\alpha {\beta}^{2}{g}^{2}]}$ | $\frac{2k(1+g\alpha )-\alpha {g}^{2}{\beta}^{2}}{k{\mu}_{0}(2+\alpha g)+(g{\beta}^{2}-k){\Delta}_{1}}$ |

$nu$ | $\frac{g{\beta}^{2}({\Delta}_{1}+\alpha g{\mu}_{0})-2{\Delta}_{2}}{\beta [4k(1+g\alpha )-\alpha {g}^{2}{\beta}^{2}]}$ | $\frac{2k{\Delta}_{3}+k\alpha g{\mu}_{0}-{\Delta}_{4}}{\alpha [4k\left(1+g\alpha \right)-\alpha {g}^{2}{\beta}^{2}]}$ | $\frac{4k\left(1+g\alpha \right)-\alpha {g}^{2}{\beta}^{2}}{k{\mu}_{0}(2+\alpha g)+(g{\beta}^{2}-2k){\Delta}_{1}}$ |

$nc$ | $\frac{g{\beta}^{2}({\Delta}_{1}+\alpha g{\mu}_{0})-2(1-{\theta}_{h}){\Delta}_{2}}{\beta [2(2-{\theta}_{h})k(1+g\alpha )-\alpha {g}^{2}{\beta}^{2}]}$ | $\frac{k(2-{\theta}_{h}){\Delta}_{3}+(1-{\theta}_{h})k\alpha g{\mu}_{0}-{\Delta}_{4}}{\alpha [2(2-{\theta}_{h})k(1+g\alpha )-\alpha {g}^{2}{\beta}^{2}]}$ | $\frac{2k(2-{\theta}_{h})(1+g\alpha )-\alpha {g}^{2}{\beta}^{2}}{k{\mu}_{0}(2+\alpha g)+[g{\beta}^{2}-(2-{\theta}_{h})k]{\Delta}_{1}}$ |

Parameter | c | g | k | ${\mathit{\lambda}}_{0}$ | ${\mathit{\mu}}_{0}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{\theta}}_{\mathit{h}}$ |
---|---|---|---|---|---|---|---|---|

Value | 8 | 12 | $0.4$ | 5 | 6 | $0.01$ | $0.5$ | $0.6$ |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tu, J.; Hu, X.; Huang, M.
Incentive Contracts for a Queueing System with a Strategic Server: A Principal-Agent Perspective. *Axioms* **2023**, *12*, 272.
https://doi.org/10.3390/axioms12030272

**AMA Style**

Tu J, Hu X, Huang M.
Incentive Contracts for a Queueing System with a Strategic Server: A Principal-Agent Perspective. *Axioms*. 2023; 12(3):272.
https://doi.org/10.3390/axioms12030272

**Chicago/Turabian Style**

Tu, Jun, Xiaoying Hu, and Min Huang.
2023. "Incentive Contracts for a Queueing System with a Strategic Server: A Principal-Agent Perspective" *Axioms* 12, no. 3: 272.
https://doi.org/10.3390/axioms12030272