Asymptotic approximations for stationary distributions of many-server queues with abandonment

Asymptotic approximations for stationary distributions of many-server queues with abandonment
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具有放弃的多服务器队列平稳分布的渐近近似

DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
K. Ramanan
K. Ramanan
中科院分区：
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文献类型：
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作者：
W. Kang;K. Ramanan

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考虑多服务器排队系统，其中客户根据更新过程到达，并具有从两个独立的、同分布的随机变量的独立序列中得出的服务和耐心时间。顾客按照到达顺序进入服务，如果排队等待时间超过耐心时间，则视为放弃排队。具有$N$服务器的系统的状态由一个四部分进程表示，该进程由到达进程的前向循环时间、一对测量值进程组成，一个跟踪队列中客户的等待时间，另一个跟踪系统中存在的客户已处于服务状态的时间量，以及代表系统中客户总数的实值进程。在一般假设下，表明状态过程是 Feller 过程，服从平稳分布并且是遍历的。它还表明，缩放平稳分布的相关序列是紧密的，并且任何子序列都会收敛到流体极限的不变状态。特别是，这意味着当相关的流体极限具有唯一的不变状态时，平稳分布序列收敛，如 $N ightarrow infty$，到不变状态。另外，还给出了一个简单的例子来说明，无论有无遗弃情况下，$N ightarrow infty$ 和 $t ightarrow infty$ 限制不能总是互换。
A many-server queueing system is considered in which customers arrive according to a renewal process and have service and patience times that are drawn from two independent sequences of independent, identically distributed random variables. Customers enter service in the order of arrival and are assumed to abandon the queue if the waiting time in queue exceeds the patience time. The state of the system with $N$ servers is represented by a four-component process that consists of the forward recurrence time of the arrival process, a pair of measure-valued processes, one that keeps track of the waiting times of customers in queue and the other that keeps track of the amounts of time customers present in the system have been in service and a real-valued process that represents the total number of customers in the system. Under general assumptions, it is shown that the state process is a Feller process, admits a stationary distribution and is ergodic. It is also shown that the associated sequence of scaled stationary distributions is tight, and that any subsequence converges to an invariant state for the fluid limit. In particular, this implies that when the associated fluid limit has a unique invariant state, then the sequence of stationary distributions converges, as $N ightarrow infty$, to the invariant state. In addition, a simple example is given to illustrate that, both in the presence and absence of abandonments, the $N ightarrow infty$ and $t ightarrow infty$ limits cannot always be interchanged.