Externalities in the M/G/1 queue LCFS-PR versus FCFS

Consider a stable M/G/1 system in which, at time t = 0 , there are exactly n customers with residual service times equal to v₁, v₂, … , v_n . In addition, assume that there is an extra customer c who arrives at time t = 0 and has a service requirement of x. The externalities which are created by c are equal to the total waiting time that others will save if her service requirement is reduced to zero. In this work, we study the joint distribution (parameterized by n, v₁, v₂, … , v_n, x) of the externalities created by c when the underlying service distribution is either last-come, first-served with preemption or first-come, first-served. We start by proving a decomposition of the externalities under the above-mentioned service disciplines. Then, this decomposition is used to derive several other results regarding the externalities: moments, asymptotic approximations as x→ ∞ , asymptotics of the tail distribution, and a functional central limit theorem.