SPDE Limits of Many Server Queues
SPDE Limits of Many Server Queues
复制标题
许多服务器队列的 SPDE 限制
DOI：
10.1214/11-aap821
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
K. Ramanan
中科院分区：
文献类型：
--
作者：
H. Kaspi;K. Ramanan
考虑一个多服务器排队系统，在该系统中，服务时间独立且分布相同的客户按到达顺序进入服务。系统的状态由描述系统中客户总数的过程以及跟踪服务中客户年龄的测量值过程表示，从而导致对动态的马尔可夫描述。在适当的假设条件下，建立了服务器数量趋于无穷时状态过程序列（居中和缩放）的功能中心极限定理。描述系统总数的极限过程被证明是一个具有恒定扩散系数的伊藤扩散，它对服务分布不敏感。年龄过程序列（中心和缩放）的极限显示为希尔伯特空间值扩散，该扩散也可以表征为与伊藤扩散耦合的随机偏微分方程的唯一解。进一步证明了极限过程是半鞅并具有强马尔可夫性。
A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.