Abstract
We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.
Highlights
In queueing theory it is commonly assumed that the customer arrival process is a Poisson process
In this paper we have considered a network of infinite-server queues where the input process is a Cox process that allows much modeling flexibility; the arrival rate is represented as a linear combination of the components of a multivariate generalized shot-noise process
We have derived various distributional properties of the multivariate shot-noise process, subsequently exploiting them to obtain the joint transform of the numbers of customers, at consecutive time epochs, in an infinite-server queue with, as input process, such a Cox process
Summary
In queueing theory it is commonly assumed that the customer arrival process is a Poisson process. While the arrival process of visitors to the Web site may well be a Poisson process, its rate may jump up due to some (external) event, decay gradually, only to jump up again because of another event Such an example formed one of the motivations for [2,7,8], which all study infinite-server queues with an overdispersed arrival process. Cox input processes driven by a multivariate Lévy subordinator are quite relevant for the above-mentioned example of a Web site with a visit rate that jumps up because of a certain event and subsequently decays again They allow one to take into account multiple, possibly related, events that affect the visit rate of the Web site.
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