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. For example, consider a FCFS M/G/1 queue and a customer with service demand of $x\ge 0$ minutes who arrived into the system when the workload level was $v\ge 0$ minutes. Let $E_v(x)$ be the total waiting time which could be saved if this customer gave up on his service demand. In this work, we analyse the $\mathit{\text{externalities}}\mathit{\text{process}}$ $E_v(\cdot )=\{E_v(x):x\ge 0\}$. It is shown that this process can be represented by an integral of a (shifted in time by $v$ minutes) compound Poisson process with positive discrete jump distribution, so that $E_v(\cdot )$ is convex. Furthermore, we compute the LST of the finite-dimensional distributions of $E_v(\cdot )$ as well as its mean and auto-covariance functions. We also identify conditions under which, a sequence of normalized externalities processes admits a weak convergence on $\mathcal{D}[0,\mathrm{\infty })$ equipped with the uniform metric to an integral of a (shifted in time by $v$ minutes) 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. This leads to a generalization of an existing result from a previous work of Haviv and Ritov (1998).