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doi: 10.3934/jimo.2019016

## The setting and optimization of quick queue with customer loss

 1 School of Management, Hefei University of Technology, Hefei, China 2 Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei, China 3 Research Center of Industrial Transfer and Innovation Development, Hefei University of Technology, Hefei, China

* Corresponding author: Yuqian Pan

Received  June 2018 Revised  October 2018 Published  March 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China under grants 71521001, 71690235, 71471052, 71601031, 71671055, and the Natural Science Foundation of Anhui Province of China under grant 1708085MG169

At the peak of a service system, customers may hesitate and even leave in the face of unavoidable queuing. This phenomenon not only affects the customer's satisfaction, but also causes the loss of the company's revenue. This paper establishes a fluid model of customer queuing behavior for the customer losses. The goal is to reduce the customer losses, and the setting and optimization method of quick queue in random service systems is studied. We construct two queuing models, in which one includes only regular queues and the other includes both regular and quick queues. We analyze the queuing systems, and describe the different forms of the objective function based on the fluid model of customer behavior. Then we compare and analyze the impact of the adoption of quick queues on the performance of the service system during peak period, and design a calculation method to obtain the optimal value for setting the number of quick queues. Thus, the overall performance of the system is optimized. Finally, we take the setting and optimization of quick queue in the supermarket service system as an example, which verifies the validity of the proposed method, and shows the reference value of this method to management practice.

Citation: Kai Li, Yuqian Pan, Bohai Liu, Bayi Cheng. The setting and optimization of quick queue with customer loss. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019016
##### References:
 [1] F. Alizadeh, J. Eckstein, N. Noyan and G. Rudolf, Arrival rate approximation by nonnegative cubic splines, Operations Research, 56 (2008), 140-156.  doi: 10.1287/opre.1070.0443.  Google Scholar [2] K. S. Anand, M. F. Pac and S. Veeraraghavan, Quality-speed conundrum: Trade-offs in customer-intensive services, Management Science, 57 (2010), 40-56.   Google Scholar [3] B. Ata and X. Peng, An equilibrium analysis of a multiclass queue with endogenous abandonments in heavy traffic, Operations Research, 66 (2018), 163-183.  doi: 10.1287/opre.2017.1638.  Google Scholar [4] I. Atencia, A Geo/G/1 retrial queueing system with priority services, European Journal of Operational Research, 256 (2016), 178-186.  doi: 10.1016/j.ejor.2016.07.011.  Google Scholar [5] K. Chung and D. Min, Staffing a service system with appointment-based customer arrivals, Journal of the Operational Research Society, 65 (2014), 1533-1543.  doi: 10.1057/jors.2013.110.  Google Scholar [6] C. Goswami and N. Selvaraju, Phase-type arrivals and impatient customers in multiserver queue with multiple working vacations, Advances in Operations Research, 2016 (2016), Art. ID 4024950, 17 pp. doi: 10.1155/2016/4024950.  Google Scholar [7] C. Guo, Queuing game based congestion control model with impatient users over space information networks, Journal of Information & Computational Science, 12 (2015), 3319-3331.   Google Scholar [8] P. Guo, R. Lindsey and Z. G. Zhang, On the downs-thomson paradox in a self-financing two-tier queuing system, Manufacturing & Service Operations Management, 16 (2014), 315-322.   Google Scholar [9] G. Horváth, Efficient analysis of the MMAP[K]/PH[K]/1 priority queue, European Journal of Operational Research, 246 (2015), 128-139.  doi: 10.1016/j.ejor.2015.03.004.  Google Scholar [10] M. Jain, A. Bhagat and C. Shekhar, Double orbit finite retrial queues with priority customers and service interruptions, Applied Mathematics & Computation, 253 (2015), 324-344.  doi: 10.1016/j.amc.2014.12.066.  Google Scholar [11] T. James, K. Glazebrook and K. Lin, Developing effective service policies for multiclass queues with abandonment: asymptotic optimality and approximate policy improvement, INFORMS Journal on Computing, 28 (2016), 251-264.  doi: 10.1287/ijoc.2015.0675.  Google Scholar [12] A. J. E. M. Janssen, J. S. H. V. Leeuwaarden and B. Zwart, Refining square-root safety staffing by expanding erlang C, Operations Research, 59 (2011), 1512-1522.  doi: 10.1287/opre.1110.0991.  Google Scholar [13] Z. Jing, J. J. Hasenbein, D. P. Morton and V. Mehrotra, Staffing call centers under arrival-rate uncertainty with Bayesian updates, Operations Research Letters, 46 (2018), 379-384.  doi: 10.1016/j.orl.2018.04.003.  Google Scholar [14] O. Jouini and A. Roubos, On multiple priority multi-server queues with impatience, Journal of the Operational Research Society, 65 (2014), 616-632.   Google Scholar [15] C. Kim, V. I. Klimenok and A. N. Dudin, Priority tandem queueing system with retrials and reservation of channels as a model of call center, Computers & Industrial Engineering, 96 (2016), 61-71.  doi: 10.1016/j.cie.2016.03.012.  Google Scholar [16] S. B. Li and W. J. Xing, Improving operational efficiency of double-queuing system in large supermarket, Operations Research and Management Science, 26 (2017), 61-67.   Google Scholar [17] Y. Liu and W. Whitt, Stabilizing customer abandonment in many-server queues with time-varying arrivals, Operations Research, 60 (2012), 1551-1564.  doi: 10.1287/opre.1120.1104.  Google Scholar [18] H. Wang, T. L. Olsen and G. Liu, Service capacity competition with peak arrivals and delay sensitive customers, Omega, 77 (2017), 80-95.  doi: 10.1016/j.omega.2017.06.001.  Google Scholar [19] R. Wang, O. Jouini and S. Benjaafar, Service systems with finite and heterogeneous customer arrivals, Manufacturing & Service Operations Management, 16 (2014), 329-480.  doi: 10.1287/msom.2014.0481.  Google Scholar [20] J. Wang, O. Baron and A. Schellerwolf, M/M/c queue with two priority classes, Operations Research, 63 (2015), 733-749.  doi: 10.1287/opre.2015.1375.  Google Scholar [21] A. Weerasinghe, Diffusion approximations for G/M/n+GI queues with state-dependent service rates, Mathematics of Operations Research, 39 (2014), 207-228.  doi: 10.1287/moor.2013.0587.  Google Scholar [22] W. Whitt, The steady-state distribution of the M$_{t}$/M/$\infty$ queue with a sinusoidal arrival rate function, Operations Research Letters, 42 (2014), 311-318.  doi: 10.1016/j.orl.2014.05.005.  Google Scholar [23] W. J. Xing, S. B. Li and L. He, Simulation model of supermarket queuing system, Control Conference IEEE, 2015, 8819–8823. Google Scholar [24] M. Yu and A. S. Alfa, Algorithm for computing the queue length distribution at various time epochs in DMAP/G$^{(1, a, b)}$/1/N queue with batch-size-dependent service time, European Journal of Operational Research, 244 (2015), 227-239.  doi: 10.1016/j.ejor.2015.01.056.  Google Scholar [25] W. Zhan and L. Dai, Massive random access of machine-to-machine communications in LTE networks: Modeling and throughput optimization, IEEE Transactions on Wireless Communications, 17 (2018), 2771-2785.  doi: 10.1109/TWC.2018.2803083.  Google Scholar [26] S. Ziani, F. Rahmoune and M. S. Radjef, Customers' strategic behavior in batch arrivals M$^{2}$/M/1 queue, European Journal of Operational Research, 247 (2015), 895-903.  doi: 10.1016/j.ejor.2015.06.040.  Google Scholar

show all references

##### References:
 [1] F. Alizadeh, J. Eckstein, N. Noyan and G. Rudolf, Arrival rate approximation by nonnegative cubic splines, Operations Research, 56 (2008), 140-156.  doi: 10.1287/opre.1070.0443.  Google Scholar [2] K. S. Anand, M. F. Pac and S. Veeraraghavan, Quality-speed conundrum: Trade-offs in customer-intensive services, Management Science, 57 (2010), 40-56.   Google Scholar [3] B. Ata and X. Peng, An equilibrium analysis of a multiclass queue with endogenous abandonments in heavy traffic, Operations Research, 66 (2018), 163-183.  doi: 10.1287/opre.2017.1638.  Google Scholar [4] I. Atencia, A Geo/G/1 retrial queueing system with priority services, European Journal of Operational Research, 256 (2016), 178-186.  doi: 10.1016/j.ejor.2016.07.011.  Google Scholar [5] K. Chung and D. Min, Staffing a service system with appointment-based customer arrivals, Journal of the Operational Research Society, 65 (2014), 1533-1543.  doi: 10.1057/jors.2013.110.  Google Scholar [6] C. Goswami and N. Selvaraju, Phase-type arrivals and impatient customers in multiserver queue with multiple working vacations, Advances in Operations Research, 2016 (2016), Art. ID 4024950, 17 pp. doi: 10.1155/2016/4024950.  Google Scholar [7] C. Guo, Queuing game based congestion control model with impatient users over space information networks, Journal of Information & Computational Science, 12 (2015), 3319-3331.   Google Scholar [8] P. Guo, R. Lindsey and Z. G. Zhang, On the downs-thomson paradox in a self-financing two-tier queuing system, Manufacturing & Service Operations Management, 16 (2014), 315-322.   Google Scholar [9] G. Horváth, Efficient analysis of the MMAP[K]/PH[K]/1 priority queue, European Journal of Operational Research, 246 (2015), 128-139.  doi: 10.1016/j.ejor.2015.03.004.  Google Scholar [10] M. Jain, A. Bhagat and C. Shekhar, Double orbit finite retrial queues with priority customers and service interruptions, Applied Mathematics & Computation, 253 (2015), 324-344.  doi: 10.1016/j.amc.2014.12.066.  Google Scholar [11] T. James, K. Glazebrook and K. Lin, Developing effective service policies for multiclass queues with abandonment: asymptotic optimality and approximate policy improvement, INFORMS Journal on Computing, 28 (2016), 251-264.  doi: 10.1287/ijoc.2015.0675.  Google Scholar [12] A. J. E. M. Janssen, J. S. H. V. Leeuwaarden and B. Zwart, Refining square-root safety staffing by expanding erlang C, Operations Research, 59 (2011), 1512-1522.  doi: 10.1287/opre.1110.0991.  Google Scholar [13] Z. Jing, J. J. Hasenbein, D. P. Morton and V. Mehrotra, Staffing call centers under arrival-rate uncertainty with Bayesian updates, Operations Research Letters, 46 (2018), 379-384.  doi: 10.1016/j.orl.2018.04.003.  Google Scholar [14] O. Jouini and A. Roubos, On multiple priority multi-server queues with impatience, Journal of the Operational Research Society, 65 (2014), 616-632.   Google Scholar [15] C. Kim, V. I. Klimenok and A. N. Dudin, Priority tandem queueing system with retrials and reservation of channels as a model of call center, Computers & Industrial Engineering, 96 (2016), 61-71.  doi: 10.1016/j.cie.2016.03.012.  Google Scholar [16] S. B. Li and W. J. Xing, Improving operational efficiency of double-queuing system in large supermarket, Operations Research and Management Science, 26 (2017), 61-67.   Google Scholar [17] Y. Liu and W. Whitt, Stabilizing customer abandonment in many-server queues with time-varying arrivals, Operations Research, 60 (2012), 1551-1564.  doi: 10.1287/opre.1120.1104.  Google Scholar [18] H. Wang, T. L. Olsen and G. Liu, Service capacity competition with peak arrivals and delay sensitive customers, Omega, 77 (2017), 80-95.  doi: 10.1016/j.omega.2017.06.001.  Google Scholar [19] R. Wang, O. Jouini and S. Benjaafar, Service systems with finite and heterogeneous customer arrivals, Manufacturing & Service Operations Management, 16 (2014), 329-480.  doi: 10.1287/msom.2014.0481.  Google Scholar [20] J. Wang, O. Baron and A. Schellerwolf, M/M/c queue with two priority classes, Operations Research, 63 (2015), 733-749.  doi: 10.1287/opre.2015.1375.  Google Scholar [21] A. Weerasinghe, Diffusion approximations for G/M/n+GI queues with state-dependent service rates, Mathematics of Operations Research, 39 (2014), 207-228.  doi: 10.1287/moor.2013.0587.  Google Scholar [22] W. Whitt, The steady-state distribution of the M$_{t}$/M/$\infty$ queue with a sinusoidal arrival rate function, Operations Research Letters, 42 (2014), 311-318.  doi: 10.1016/j.orl.2014.05.005.  Google Scholar [23] W. J. Xing, S. B. Li and L. He, Simulation model of supermarket queuing system, Control Conference IEEE, 2015, 8819–8823. Google Scholar [24] M. Yu and A. S. Alfa, Algorithm for computing the queue length distribution at various time epochs in DMAP/G$^{(1, a, b)}$/1/N queue with batch-size-dependent service time, European Journal of Operational Research, 244 (2015), 227-239.  doi: 10.1016/j.ejor.2015.01.056.  Google Scholar [25] W. Zhan and L. Dai, Massive random access of machine-to-machine communications in LTE networks: Modeling and throughput optimization, IEEE Transactions on Wireless Communications, 17 (2018), 2771-2785.  doi: 10.1109/TWC.2018.2803083.  Google Scholar [26] S. Ziani, F. Rahmoune and M. S. Radjef, Customers' strategic behavior in batch arrivals M$^{2}$/M/1 queue, European Journal of Operational Research, 247 (2015), 895-903.  doi: 10.1016/j.ejor.2015.06.040.  Google Scholar
Fluid model with balking
Queue system that does not include quick queue
Queue system that includes quick queues
The frequency $(p)$ of a customer purchasing goods $(x)$ in a supermarket in 2016
 Number of goods $x$ [1,4] [5,9] [10,14] [15,19] [20,24] $[25, +\infty)$ Frequency $p$ 0.2 0.25 0.25 0.2 0.07 0.03
 Number of goods $x$ [1,4] [5,9] [10,14] [15,19] [20,24] $[25, +\infty)$ Frequency $p$ 0.2 0.25 0.25 0.2 0.07 0.03
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