The asymptotic variance of departures in critically loaded queues

Abstract

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load equals 1, and prove that the asymptotic variance rate satisfies, where λ is the arrival rate, and c 2 a and c 2 s are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

The variance of departure processes: puzzling behavior and open problems

“…This type of phenomena has been termed BRAVO (Balancing Reduces Asymptotic Variance of Outputs). It was first observed for M/M/1/K in [11], and has recently been established for GI/G/1 queues under quite general conditions [2].…”

“…Quite surprisingly (1) is not the correct guess. The following has been shown in [11] for M/M/1/K and related birth death queues and later in [2] for the M/M/1:…”

“…Some more recent studies ( [2], [10] and [11]) have brought back attention to elementary models. In this respect, the asymptotics of the variance curve have been analyzed.…”

Heavy-Traffic Limit of theGI/GI/1 Stationary Departure Process and Its Variance Function

“…There is a recent related result for the M/GI/1 queue in Hautphenne et al (2015, Proposition 6), as part of an investigation into the BRAVO (balancing reduces asymptotic variance of outputs) effect; see Nazarathy (2011), Hanbali et al (2011), the earlier in Berger andWhitt (1992, Theorem 4.1), andWilliams (1992). For the stationary M/GI/1 model, a two-term asymptotic expansion is developed for the variance function V d, ρ (t) ≡ Var(D ρ (t)) as t → ∞ for fixed ρ < 1.…”

“…There has since been a substantial literature on that case; see Hanbali et al (2011), Karpelevich and Kreinin (1994), and Whitt (2002). As can be seen from Whitt (2002, Sections 9.3 and 13.5), for the queue length, the key map is the reflection map ψ applied to a potential net-input function x,…”