Admission Control for A Single Server Waiting Time Process in Heavy Traffic
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908454743965696 |
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| author | Xie, Bowen Yin, Haoyu |
| author_facet | Xie, Bowen Yin, Haoyu |
| contents | We address a single server queue control problem (QCP) in heavy traffic originating from Lee and Weerasinghe (2011). The state process represents the offered waiting time, the customer arrival has a state-dependent intensity, and the customers' service and patience times are i.i.d with general distributions. We introduce an infinite-horizon discounted cost functional consisting of a control cost generated from the use of control and a penalty for idleness cost. Our primary goal is to tackle the QCP, taking into account a non-trivial control cost and a non-increasing cost function resulting from the control mechanisms in the waiting time. Under mild assumptions, the heavy traffic limit of the QCP yields a stochastic control problem described by a diffusion process, which we call a diffusion control problem (DCP). We find the optimal control of the associated DCP by incorporating the Legendre-Fenchel transform and a formal Hamilton-Jacobi-Bellman (HJB) equation. Then, we ``translate'' this optimal strategy to the QCP, of which we obtain an asymptotically optimal policy. Apart from theoretical results, we also examine the REINFORCE algorithm, a Reinforcement learning (RL) approach, for solving stochastic controls motivated by recent literature. We highlight the advantages and limitations of simulation from theoretical results and data-driven algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_05674 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Admission Control for A Single Server Waiting Time Process in Heavy Traffic Xie, Bowen Yin, Haoyu Optimization and Control Probability 60K25 (Primary) 90B22, 93E20, 90B18 (Secondary), 93B70 We address a single server queue control problem (QCP) in heavy traffic originating from Lee and Weerasinghe (2011). The state process represents the offered waiting time, the customer arrival has a state-dependent intensity, and the customers' service and patience times are i.i.d with general distributions. We introduce an infinite-horizon discounted cost functional consisting of a control cost generated from the use of control and a penalty for idleness cost. Our primary goal is to tackle the QCP, taking into account a non-trivial control cost and a non-increasing cost function resulting from the control mechanisms in the waiting time. Under mild assumptions, the heavy traffic limit of the QCP yields a stochastic control problem described by a diffusion process, which we call a diffusion control problem (DCP). We find the optimal control of the associated DCP by incorporating the Legendre-Fenchel transform and a formal Hamilton-Jacobi-Bellman (HJB) equation. Then, we ``translate'' this optimal strategy to the QCP, of which we obtain an asymptotically optimal policy. Apart from theoretical results, we also examine the REINFORCE algorithm, a Reinforcement learning (RL) approach, for solving stochastic controls motivated by recent literature. We highlight the advantages and limitations of simulation from theoretical results and data-driven algorithms. |
| title | Admission Control for A Single Server Waiting Time Process in Heavy Traffic |
| topic | Optimization and Control Probability 60K25 (Primary) 90B22, 93E20, 90B18 (Secondary), 93B70 |
| url | https://arxiv.org/abs/2212.05674 |