Comparisons of customer balking behavior in observable queues with N policies and geometric abandonments

ABSTRACT

This paper studies and compares customer balking behavior in some single-server observable queues with N policies and geometric abandonments. We focus on two types of N policies: the classic one and the modified one. Under the classic N policy, the server begins to take a vacation period as soon as the system becomes empty until there are N customers in the queue. On the other hand, under the modified N policy, the server takes multiple vacations until at least N customers are waiting in the queue when a vacation time ends. Abandonment opportunities occur during a whole vacation period, and customers decide independently and sequentially whether to renege. Comparing the two types of N policies, we realize that the residual time of the server's last vacation under the modified N policy indeed inhibits customers joining, whereas it may be more economic in practice if we take the whole system cost into consideration. As for the optimal N, under which the server is free of compensation, we observe that he should choose the modified N policy when the abandonment probability is much close to 1. Otherwise, he should choose the classic N policy.

KEYWORDS:

• Queueing games
• observable queues
• geometric abandonments
• $N$ policies
• balking behavior

Acknowledgments

The authors would like to thank the editor and three anonymous referees for their valuable comments and suggestions which greatly improved the final version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Although the case $n_1^c<N$ exists theoretically, the overlarge $N$ will make customers surely balk in the vacation state given no compensation. So it is generally worthless to the server to maintain the system stable in this case.

2. If $U_0^{fc}(n)=0$ only has one root denoted by $r_c$, both $n_0^{U_c}=⌈r_c⌉$ and $n_0^{L_c}=⌊r_c⌋$ are possible, especially the latter that will be shown in Section 5.

3. In order to discuss comprehensively, we assume both $n_0^{L_c}$ and $n_0^{U_c}$ exist; Otherwise, case 2 and case 3 (or case 3 and case 4) should be combined as $1\le n<n_0^{U_c}$ (or $n_0^{L_c}<n<N$).

4. In Figures 8 and 9, we omit the lower threshold $n_0^{L_m}$ for lack of space.

5. This means the case $n_0^{M_m}=N-1$ is possible, but must notice that we need to verify whether $U_{02}^{fm}(N-1)\ge 0$ instead of $U_{01}^{fm}(N-1)\ge 0$. If not, the middle threshold does not exist.

Additional information

Funding

The authors would like to thank the support from the National Natural Science Foundation of China (Nos. 71971188, 71671159), the Natural Science Foundation of Hebei Province (Nos. G2020203005, G2022203003), and the support Funded by Science and Technology Project of Hebei Education Department (No. ZD2022142).

Notes on contributors

Wei Sun

Wei Sun received her B.Sc. degree from Hebei University, Baoding, and her Ph.D. degree from Yanshan University, Qinhuangdao, China, in 2004 and 2010, respectively. She was a visiting scholar in the Department of Logistics and Maritime Studies at Hong Kong Polytechnic University from June 2009 to April 2010. Currently she is a full professor in the School of Economics and Management at Yanshan University. She has published more than 50 papers in international leading journals in the areas of operations research and applied mathematics. She has been involved in several projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, and other foundations. Her research interests include economics of queues, vacation queues and resource allocation of cloud computing.

Zhiyuan Zhang

Zhiyuan Zhang received her B.Sc. degree from Yanshan University, Qinhuangdao, in 2020 and is currently working toward the Ph.D. degree at the School of Economics and Management, Yanshan University, Qinhuangdao, China. Her research interests include game theory, decision theory, vacation queues and economics of queues.

Shiyong Li

Shiyong Li received his B.Sc. degree from Qingdao University, Qingdao, his M.Sc. degree from Yanshan University, Qinhuangdao, and his Ph.D. degree from Beijing Jiaotong University, Beijing, China, in 2004, 2007 and 2011, respectively. Currently, he is a full professor in the School of Economics and Management at Yanshan University. He is the (co)author of more than 70 papers in mathematics, technique, and management journals. He has been a principal investigator/co-investigator on several research projects supported by the National Natural Science Foundation of China, the National Education Committee Foundation of China, the China Postdoctoral Science Foundation, and other foundations. His research interests include cloud migration for enterprise applications, resource allocation of cloud/edge computing, information systems and electronic commerce, and economics of queues.