The asymptotic variance rate of the output process of finite capacity birth-death queues

Nazarathy, Yoni; Weiss, Gideon

doi:10.1007/s11134-008-9079-4

Cited by 27 publications

(36 citation statements)

References 34 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: 68%

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: 68%

The asymptotic variance of departures in critically loaded queues

Hanbali

Mandjes

Nazarathy

et al. 2011

Advances in Applied Probability

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show abstract

“…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: 82%

“…Therefore, we see that, for λt > 0, 17) where the third inequality and last line follow from A(t) = λt ≤ λt (at time 0 the queue 4 is uniformly integrable; see [22,Theorem 4.1]. Hence, both terms on the right-hand side of (4.18) are uniformly integrable, and, hence, so is Q.…”

Section: Have Q(t) ≤ W (T)/b + 1 and W (T) = W A(t) − (T − τ A(t) ) ≤mentioning

confidence: 88%

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.

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“…If a QBD process is modeled by a DTMC, and each state in the DTMC is associated with a cost, then the sum of the total cost during a given period T asymptotically approaches the Normal distribution [13]. Therefore, considering the energy consumption, ε i , at each state i as the cost, the total energy consumption during T , approaches the Normal distribution.…”

Section: Asymptotic Energy Consumption Distributionmentioning

confidence: 99%

“…respectively, where ε is the vector of ε i , i ∈ S, and vector β is the solution of the following set of equations [13]:…”

Section: Asymptotic Energy Consumption Distributionmentioning

confidence: 99%

Stochastic Analysis of Energy Consumption in Wireless Sensor Networks

Wang

Vuran

Goddard

2010

2010 7th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks (SECON)

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Abstract-Limited energy resources in increasingly sophisticated wireless sensor networks (WSNs) call for a comprehensive crosslayer analysis of energy consumption in a multi-hop network. For reliability analysis in such networks, the statistical information about energy consumption and lifetime is required. However, traditional energy analysis approaches only focus on the average energy consumed. In this paper, instead, we provide a stochastic analysis of the energy consumption in a random network environment. Accordingly, a comprehensive cross-layer analysis framework, which employs a stochastic queueing model, is developed. Using this framework, the distribution of energy consumption for nodes in WSNs during a given time period is found. We show that when the time duration is long, the energy consumption asymptotically approaches the Normal distribution. This distribution of energy consumption is then utilized to investigate the distribution of node lifetime and network lifetime. The developed analysis framework is generic and is parameterized for many WSN protocols, including an anycast protocol as a case study. Comprehensive simulations and testbed experiments are provided to validate the developed model. The cross-layer framework is also used to identify relationships between the distribution of energy consumption and network parameters, such as network density, duty cycle, and traffic rate. To the best of our knowledge, this is the first comprehensive work to investigate probabilistic distribution of energy consumption in WSNs.

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The asymptotic variance rate of the output process of finite capacity birth-death queues

Cited by 27 publications

References 34 publications

The asymptotic variance of departures in critically loaded queues

The asymptotic variance of departures in critically loaded queues

The asymptotic variance of departures in critically loaded queues

Stochastic Analysis of Energy Consumption in Wireless Sensor Networks

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