TY - JOUR

T1 - Externalities in Queues as Stochastic Processes

T2 - The Case of FCFS M/G/1

AU - Jacobovic, Royi

AU - Mandjes, Michel

N1 - Publisher Copyright:
© 2023 The Author(s).

PY - 2024/3

Y1 - 2024/3

N2 - 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 Ev (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 Ev (·) ={Ev (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 Ev (·) is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of Ev (·) 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.

AB - 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 Ev (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 Ev (·) ={Ev (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 Ev (·) is convex. Furthermore, we compute the Laplace-Stieltjes transform of the finite-dimensional distributions of Ev (·) 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.

KW - Gaussian approximation

KW - M/G/1

KW - congestion costs

KW - convex stochastic process

KW - externalities

UR - http://www.scopus.com/inward/record.url?scp=85188533728&partnerID=8YFLogxK

U2 - 10.1287/stsy.2022.0021

DO - 10.1287/stsy.2022.0021

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AN - SCOPUS:85188533728

SN - 1946-5238

VL - 14

SP - 1

EP - 21

JO - Stochastic Systems

JF - Stochastic Systems

IS - 1

ER -