On the law of the iterated logarithm in hybrid multiphase queueing systems

IF 0.7 Q4 OPERATIONS RESEARCH & MANAGEMENT SCIENCE
S. Minkevičius
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Abstract

Whitby in his book about artificial intelligence [21] states that the human brain consists of 100 networks of networks (NoN). One of the examples of NoN is a hybrid multiphase queueing system (HMQS). At first, we present the summary of works dedicated to a particular case of HMQS and multiphase queueing system (MQS, see Fig. 1). One can apply limit theorems for a waiting time of a customer and a queue length of customers to get probabilistic characteristics of MQS under various conditions of heavy traffic [2, 3]. The most fundamental example (a single-phase case, where the time intervals in between the arrivals of customers to MQS are independent identically distributed random variables and there is a single device, working independently of the output in heavy traffic) has been completely investigated by several authors [2, 8]. Iglehart [5] carefully investigated a single-device case and obtained laws of the iterated logarithm (LIL) for this case. It is surprising to note that the fundamental results of Iglehart on the queueing systems, working in heavy traffic are rarely used [4–8]. There are only a few papers on the theory of MQS in heavy traffic [10, 12, 13] with, however, no proof of LIL for the probabilistic characteristics of MQS in heavy traffic. LIL for a cumulative waiting time
混合多相排队系统的迭代对数律
Whitby在他关于人工智能的书中[21]指出,人类的大脑由100个网络的网络(NoN)组成。NoN的一个例子是混合多相排队系统(HMQS)。首先,我们对HMQS和多相排队系统(MQS,见图1)的一个具体案例进行了研究总结。我们可以应用客户等待时间和客户排队长度的极限定理来得到MQS在各种繁忙交通条件下的概率特征[2,3]。最基本的例子(单相情况下,客户到达MQS之间的时间间隔是独立的同分布随机变量,并且有一个设备,在繁忙的交通中独立于输出工作)已经被几位作者完全研究过[2,8]。Iglehart[5]仔细研究了单设备情况,并得到了这种情况下的迭代对数(LIL)定律。令人惊讶的是,Iglehart关于在繁忙交通中工作的排队系统的基本结果很少被使用[4-8]。关于大流量下MQS的理论研究论文很少[10,12,13],但没有关于大流量下MQS概率特征的LIL证明。表示累积等待时间
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来源期刊
Operations Research and Decisions
Operations Research and Decisions OPERATIONS RESEARCH & MANAGEMENT SCIENCE-
CiteScore
1.00
自引率
25.00%
发文量
16
审稿时长
15 weeks
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