Heavy Traffic Limits for Unobservable Queues with Clearing Times
Heavy Traffic Limits for Unobservable Queues with Clearing Times
复制标题
具有清除时间的不可观察队列的大流量限制
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Jamol Pender
中科院分区：
文献类型：
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作者：
Jamol Pender
在许多服务系统中，如果客户被认为等待时间过长，他们可能会选择放弃该系统。在可观察的队列中，服务器始终知道放弃的客户已经离开队列。然而，在不可观察的队列中，服务器可能不知道客户已经放弃系统的事实。服务器不仅看不到放弃的客户，而且还可能花费不必要的时间来尝试定位客户以保持其在队列中的位置。我们将服务器必须用来确认客户是否已放弃队列的额外时间称为清除时间。本文研究了清空次数对单服务台丢弃不可观测队列动态特性的影响。我们的分析涉及到关于排队长度和工作负载过程的重流量极限定理的推导。这些繁重的业务量限制说明，系统中的客户数量不仅在任何时间具有由截断的高斯给出的稳定状态分布，而且清除时间还用于减少状态依赖于向原点漂移的影响或强度。此外，我们还给出了服务器尝试定位客户所花费的时间比例、阻止客户的比例和放弃客户的比例的近似值。将模拟结果与显式交通流量公式进行比较，证实了近似公式在量化疏散时间和不可观测动态的影响方面是准确的。
In many service systems, customers may choose to abandon the system if their wait is deemed excessive. In an observable queue, the server is always aware that an abandoning customer has left the queue. However, in unobservable queues, the fact that the customer has abandoned the system may be unbeknownst to the server. Not only are the abandoning customers unobservable to the server, the server may also spend unnecessary time attempting to locate the customer to preserve their position in line. We call this additional time that the server has to spend to confirm whether a customer has abandoned the queue a clearing time. In this paper, we investigate the impact of clearing times on the dynamics of single server unobservable queues with abandonment. Our analysis involves the derivation of heavy traffic limit theorems for the queue length and workload processes. These heavy traffic limits illustrate that not only the number of customers in the system at any time has a steady state distribution given by a truncated Gaussian, but also that the clearing times serve to reduce the impact or strength of the state dependent drift towards the origin. Moreover, we also develop approximations for the fraction of wasted time that the server spends trying to locate customers, the fraction of balking customers, and the fraction of reneging customers. Comparisons between simulation and our explicit heavy traffic formulas confirm that the approximations are accurate at quantifying the impact of the clearing times and unobservable dynamics.