Externalities in Queues as Stochastic Processes: The Case of FCFS M/G/1

Abstract

Externalities are the costs that a user of a common resource imposes on others. In the context of an FCFS M/G/1 queue, where a customer with service demand [Formula: see text] arrives when the workload level is [Formula: see text], the externality [Formula: see text] is the total waiting time that could be saved if this customer gave up on their service demand. In this work, we analyze the externalities process [Formula: see text]. It is shown that this process can be represented by an integral of a (shifted in time by v) compound Poisson process with a positive discrete jump distribution, so that [Formula: see text] is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of [Formula: see text] and its mean and auto-covariance functions. We also identify conditions under which a sequence of normalized externalities processes admits a weak convergence on [Formula: see text] equipped with the uniform metric to an integral of a (shifted in time by v) standard Wiener process. Finally, we also consider the extended framework when v is a general nonnegative random variable which is independent from the arrival process and the service demands. Our analysis leads to substantial generalizations of the results presented in the existing literature. Funding: This research was supported by the European Union’s Horizon 2020 research and innovation programme [Marie Skłodowska-Curie Grant Agreement 945045] and the NWO Gravitation project NETWORKS [Grant 024.002.003].