Computation of the Transient M/M/1 Queue cdf, pdf, and Mean with Generalized Q-Functions

Cantrell, P.E.

doi:10.1109/tcom.1986.1096625

Cited by 21 publications

(6 citation statements)

References 15 publications

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“…Focus now on (28). Due to the memoryless property of the exponentially distributed service time, the possible residual service time of the packet of source 2 that is under service at the arrival instant of packet 1, i for event E L 1,i is also exponentially distributed; thus, the waiting time is the sum ofM L 2,i exponentially distributed random variables, whereM L 2,i is the total number of source 2 packets in the system (either in the queue or under service) at the arrival instant of packet 1, i conditioned on the event E L 1,i [27, p. 168].…”

Section: Exact Expression For the Average Aoi Inmentioning

confidence: 99%

On the Age of Information in Multi-Source Queueing Models

Moltafet

Leinonen

Codreanu

2020

IEEE Trans. Commun.

130

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Freshness of status update packets is essential for enabling services where a destination needs the most recent measurements of various sensors. In this paper, we study the information freshness of single-server multi-source queueing models under a first-come first-served (FCFS) serving policy.In the considered model, each source independently generates status update packets according to a Poisson process. The information freshness of the status updates of each source is evaluated by the average age of information (AoI). We derive an exact expression for the average AoI for the case with exponentially distributed service time, i.e., for a multi-source M/M/1 queueing model. Moreover, we derive three approximate expressions for the average AoI for a multi-source M/G/1 queueing model having a general service time distribution. Simulation results are provided to validate the derived exact average AoI expression, to assess the tightness of the proposed approximations, and to demonstrate the AoI behavior for different system parameters.Index Terms-Information freshness, age of information (AoI), multi-source M/G/1 queueing model. I. INTRODUCTIONRecently, various services in wireless sensor networks (WSNs) such as Internet of Things and cyber-physical control applications have attracted both academic and industrial attention. In these networks, low power sensors may be assigned to send status updates about a random process to intended destinations [1]-[6]. Such a status update system can monitor, e.g., temperature of a specific environment (room, greenhouse, etc.) [1], and a vehicular status (position, acceleration,

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Section: Exact Expression For the Average Aoi Inmentioning

confidence: 99%

On the Age of Information in Multi-Source Queueing Models

Moltafet

Leinonen

Codreanu

2020

IEEE Trans. Commun.

130

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“…It is surprising that Bessel functions and the like will not occur. The numerical calculation of transient performance measures for the M/M/1 queue has been discussed by Ackroyd (1982), Cantrell (1986), Cantrell and Beall (1988), Abate and Whitt (1989) and van de Coevering (1995).…”

Section: Introductionmentioning

confidence: 99%

On the convergence to stationarity of the many-server Poisson queue

Stadje

Parthasarathy

1999

Journal of Applied Probability

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We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0∞[E(X)-E(X(t))]dt and ∫0∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

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“…In fact, according to [3], for an arbitrary initial state n 0 ∈ Z + , Bandwidth-sharing networks 99 denoting by ρ := λ/µ the traffic intensity for the standard M/M/1 queue, we have…”

Section: Let Us Mention Three Reasons For the Simple Settings Of Exammentioning

confidence: 99%

“…The first reason is to have a simple expression for P{ n Y s = 0} and, thus, for the actual rate of convergence (accuracy). In fact, according to [3], for an arbitrary initial state n 0 ∈ Z + , denoting by ρ := λ/µ the traffic intensity for the standard M/M/1 queue, we have P{ n Y s = 0} = e −(1+ρ)nµs ρ −n 0 /2 I −n 0 (2ρ 1/2 nµt) + ρ (−n 0 −1)/2 I n 0 +1 (2ρ 1/2 nµs)…”

Section: Examplesmentioning

confidence: 99%

Asymptotic Fluid Optimality and Efficiency of the Tracking Policy for Bandwidth-Sharing Networks

Avrachenkov

Piunovskiy

Zhang

2011

Journal of Applied Probability

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Optimal control of stochastic bandwidth-sharing networks is typically difficult. In order to facilitate the analysis, deterministic analogues of stochastic bandwidth-sharing networks, the so-called fluid models, are often taken for analysis, as their optimal control can be found more easily. The tracking policy translates the fluid optimal control policy back to a control policy for the stochastic model, so that the fluid optimality can be achieved asymptotically when the stochastic model is scaled properly. In this work we study the efficiency of the tracking policy, that is, how fast the fluid optimality can be achieved in the stochastic model with respect to the scaling parameter. In particular, our result shows that, under certain conditions, the tracking policy can be as efficient as feedback policies.

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Computation of the Transient M/M/1 Queue cdf, pdf, and Mean with Generalized Q-Functions

Cited by 21 publications

References 15 publications

On the Age of Information in Multi-Source Queueing Models

On the Age of Information in Multi-Source Queueing Models

On the convergence to stationarity of the many-server Poisson queue

Asymptotic Fluid Optimality and Efficiency of the Tracking Policy for Bandwidth-Sharing Networks

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