论文标题
与预订的随机网络的平均现场分析
Mean field analysis of stochastic networks with reservation
论文作者
论文摘要
通过新的数学模型分析了大型分布式系统中预订的问题。一个典型的应用程序是基于车站的汽车共享系统，可以将其描述为一个封闭的随机网络，在该网络中节点是站点，客户是汽车。用户可以预留汽车和停车位。 在论文中，我们研究了保留停车位和汽车对所有用户有效的系统的演变。 当车站的数量$ n $以相同的速度增加时，给出了基本随机网络的渐近行为。该分析涉及在订单$ n^2 $的状态空间上的马尔可夫进程。非常值得注意的是，描述车站的演变的状态过程，其尺寸为$ n $，尽管不是马尔可夫的分布，但它却收敛到非均匀的马尔可夫进程。我们证明了这种平均场收敛。我们还使用组合参数证明，当保留和拾起汽车之间的时间足够小时，平均场限制具有独特的平衡度量。该结果扩大了只能保留停车位的情况。
The problem of reservation in a large distributed system is analyzed via a new mathematical model. A typical application is a station-based car-sharing system which can be described as a closed stochastic network where the nodes are the stations and the customers are the cars. The user can reserve the car and the parking space. In the paper, we study the evolution of the system when the reservation of parking spaces and cars is effective for all users. The asymptotic behavior of the underlying stochastic network is given when the number $N$ of stations and the fleet size increase at the same rate. The analysis involves a Markov process on a state space with dimension of order $N^2$. It is quite remarkable that the state process describing the evolution of the stations, whose dimension is of order $N$, converges in distribution, although not Markov, to an non-homogeneous Markov process. We prove this mean-field convergence. We also prove, using combinatorial arguments, that the mean-field limit has a unique equilibrium measure when the time between reserving and picking up the car is sufficiently small. This result extends the case where only the parking space can be reserved.
