Asymptotically Optimal Controls for Time-Inhomogeneous Networks
SIAM Journal on Control and Optimization
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We study the Mt /G/infinity queue where customers arrive according to a sinusoidal function λt= λ + A sin(2πt/T) and the service rate is μ. We show that the expected number of customers in the system during peak congestion can be closely approximated by (λ + A)/μ for service distributions with coefficient of variation between 0 and 1. Motivated by a result derived by Eick, Massey, and Whitt that the time lag of the peak congestion from the peak of the customer arrivals is 1/2μ for models with deterministic service times, we show that the time lag for exponential service times is closely approximated by 1/μ. Based on a cycle length of 24 hours and regardless of the values of other system parameters, these approximations are of the order of 1% accuracy for μ = 1, and the accuracy increases rapidly with increasing μ.