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

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 x ≥ 0 arrives when the workload level is v ≥ 0, the externality E_v (x) 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 E_v (·) ={E_v (x): x ≥ 0}. 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 E_v (·) is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of E_v (·) and its mean and auto-covariance functions. We also identify conditions under which a sequence of normalized externalities processes admits a weak convergence on D[0, ∞) 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.