# Ship Scheduling Algorithm Based on Markov-Modulated Fluid Priority Queues
## 2 Materials and Methods  
As a bottleneck in port efficiency, anchorage scheduling is a typical NP-hard problem whereby it is necessary to optimize the mooring sequence and service priority dynamically under the uncertainty of ship arrival and service durations and multi-priority constraints. From the perspective of research and development, this field has formed two major research paradigms: mathematical analytical modeling methods and intelligent algorithm methods. Scheduling research based on mathematical analytical modeling methods laid an important theoretical foundation for this field. Early stochastic programming provided a potential solution for the high-dimensional cargo transportation resource allocation problem under a dynamic information environment by utilizing the natural structure of transportation problems [12]. By using the LP [13] and IP [14] mathematical programming techniques, a systematic theoretical framework for anchorage resource optimization was provided by constructing objective functions containing constraints, such as constraints on the ship arrival timing and service duration. Later, queuing theory was also applied to ship scheduling methods. Queuing theory provides systematic solutions to port congestion problems by analyzing ship arrival and service patterns, and it proposes policy recommendations to improve the port efficiency and competitiveness [15]. The M/M/1 and M/D/1 mixed queuing systems constructed by [PERSON] et al. not only provide analytical optimization paths with deterministic service time modeling, but also effectively reduce the computational complexity [16]. [PERSON] et al. introduced a general service time distribution based on the M/G/1 model, which significantly improved the adaptability to fluctuations in the actual operation time [17]. [PERSON] et al. 's M/G/1/1 queuing model achieves theoretical efficiency optimization with fixed priorities and provides a reliable quantitative analysis tool for dealing with ship scheduling problems under multi-source input scenarios [18]. The method based on intelligent algorithms is also helpful for anchorage scheduling. Heuristic algorithms such as genetic algorithms [19] and particle swarm optimization [20] realize anchorage scheduling optimization by simulating natural evolution mechanisms, and they have feasibility in small and medium-sized scenes. The improved simulated annealing algorithm, ISA-CO, proposed by [PERSON] provides a new idea for anchorage scheduling. Its strategy of integrating crossover operators and local search can not only optimize the anchoring order and resource allocation but also effectively avoid premature convergence [21]. [PERSON] proved that FCFS provides the optimal throughput under a certain compatibility graph in FCFS fluid model research, which provides important theoretical support for anchorage scheduling strategy optimization [22]. The SDQ test platform proposed by [PERSON] reduces load imbalances through flow queue and path optimization, providing a technical reference for ship anchoring path planning and resource balance allocation in anchorage scheduling [23]. Although the traditional method has achieved some results in anchorage dispatching, it still has obvious shortcomings. Mathematical analytical modeling methods have limited adaptability to the randomness of ship arrivals and often face the problem of a rapid increase in computational complexity under multi-priority constraints, making it difficult to meet the real-time requirements of large-scale dynamic scheduling. Intelligent algorithms lack theoretical convergence proof, complex dynamic scenes make it easy to fall into local optima, the multi-priority hard constraint modeling ability is insufficient, the parameter dependence is strong, and scene migration is difficult. In recent years, the application of artificial intelligence technology in anchorage dispatching has shown new potential.