The variance of departure processes: puzzling behavior and open problems

Nazarathy, Yoni

doi:10.1007/s11134-011-9249-7

Cited by 11 publications

(8 citation statements)

References 17 publications

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“…This critically loaded regime, in which the mean inter-arrival time equals the mean service time, is relevant from a practical standpoint (as in many real-life situations queues are saturated or close to saturation). Moreover, it is mathematically interesting since it leads to counterintuitive results in line with the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect observed previously in finite-capacity birth-death queues [22], see also [21]. We now describe the contribution of our work in more detail.…”

Section: Introductionmentioning

confidence: 64%

The asymptotic variance of departures in critically loaded queues

Hanbali

Mandjes

Nazarathy

et al. 2011

Advances in Applied Probability

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We consider the asymptotic variance of the departure counting processD(t) of the GI/G/1 queue;D(t) denotes the number of departures up to timet. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) /t= λ(1–2/π)(ca2+cs2), where λ is the arrival rate, andca2andcs2are 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 ϱ 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 ofD(t) for anyt.

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Section: Introductionmentioning

confidence: 64%

The asymptotic variance of departures in critically loaded queues

Hanbali

Mandjes

Nazarathy

et al. 2011

Advances in Applied Probability

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“…Further, the discovery of the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) has motivated research on the variability of departure processes of queues, particularly in critically loaded regimes. Recent papers on this topic are [1], [8], [9], [23] and [24].…”

Section: Introductionmentioning

confidence: 99%

“…It was later analyzed for M/M/1 queues and more generally GI/G/1 queues in [1]. BRAVO was numerically conjectured for GI/G/1/K queues in [23], and observed for multi-server M/M/s/K queues in the many-server scaling regime in [9].…”

Section: Introductionmentioning

confidence: 99%

The intercept term of the asymptotic variance curve for some queueing output processes

Hautphenne

Kerner

Nazarathy

et al. 2015

European Journal of Operational Research

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We consider queueing output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve, indicating the variance of the number of served customers over a time interval. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to the results by finding explicit expressions for the second order approximation of the variance curve, namely the y-intercept of the linear asymptote.For M/M/1/K queues our results are based on the Drazin inverse of the generator. It turns out that by viewing output processes as MAPs (Markovian Arrival Processes) and considering the Drazin inverse, one can obtain explicit expressions for the y-intercept, together with some further insight regarding the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs). For M/G/1 queues our results are based on a classic transform of D.J. Daley. In this case we represent the y-intercept of the variance curve in terms of the first three moments of the service time distribution.A further performance measure that we are able to calculate for both models, is the asymptotic covariance between the queue length and the number of arrivals or departures. In addition we shed light on a classic conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve. Keywords: Queueing, M/M/1/K Queue, M/G/1 Queue, MAPs, Characterization of the M/M/1 Queue, Output Processes. 1 The skewness coefficient of a random variable X is E X−E[X] √ Var(X) 3 .

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“…This critically loaded regime, in which the mean interarrival time equals the mean service time, is relevant from a practical standpoint (as in many real-life situations queues are saturated or close to saturation). Moreover, it is mathematically interesting since it leads to counterintuitive results in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect observed previously in finite-capacity birth-death queues [17]; see also [16].…”

Section: Resultsmentioning

confidence: 83%

The asymptotic variance of departures in critically loaded queues

et al. 2011

View full text Add to dashboard Cite

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.

show abstract

The variance of departure processes: puzzling behavior and open problems

Cited by 11 publications

References 17 publications

The asymptotic variance of departures in critically loaded queues

The asymptotic variance of departures in critically loaded queues

The intercept term of the asymptotic variance curve for some queueing output processes

The asymptotic variance of departures in critically loaded queues

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