Nonparametric estimation of the service time distribution in the M/G/∞ queue

Goldenschluger, Alexander

doi:10.1017/apr.2016.67

Cited by 23 publications

(20 citation statements)

References 35 publications

(39 reference statements)

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“…where U (t) and I(t) are the empirical counterparts of U (t) and I(t) obtained by using (6). Note that an estimate for E[D] can be obtained directly through (2) by using the empirical counterparts of λ q and E [B].…”

Section: Inference Analysismentioning

confidence: 99%

“…The autocovariance function of the process (Q(t)) t≥0 is (11) (see e.g. [23], [6]). Differentiation yields…”

Section: Inference Analysismentioning

confidence: 99%

“…Indeed, there has been a growing interest in recent years (e.g. [5], [6]) in statistical inference problems where partial measurements of queues can be used to infer various queueing parameters. Of the available approaches relevant to the estimation of service distributions, we select three where the required data is amenable to collection by simple counters acting on the raw packet arrival information, based respectively on busy periods (BP), queue length (QL), and "arrivals-to-departures" (AD) information.…”

Section: Introductionmentioning

confidence: 99%

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Skampling for the Flow Duration Distribution

Antunes

Pipiras

Veitch

2017

2017 29th International Teletraffic Congress (ITC 29)

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This paper concerns the problem of estimating the Internet flow duration distribution from indirect measurements due to network constraints. The aim is to estimate the distribution from observing: the possible superpositions (collisions) of sampled flow durations, the flow arrivals-to-departures times without identification of sampled flows and the number of sampled flows in progress. For each type of data available, we present estimators of the flow duration distribution, formulating the problem in queueing system terms. We also propose data streaming algorithms using sampling and sketching (through counters) to obtain the considered partial information from flows. At the core of this skampling (i.e. sampling and sketching) approach is the ability to tune the flow sampling probability for "optimal" flow load onto sketch entries (queues). Finally, we present numerical results comparing the different estimators of the flow duration distribution using two real Internet traces.

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Section: Inference Analysismentioning

confidence: 99%

“…The autocovariance function of the process (Q(t)) t≥0 is (11) (see e.g. [23], [6]). Differentiation yields…”

Section: Inference Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Skampling for the Flow Duration Distribution

Antunes

Pipiras

Veitch

2017

2017 29th International Teletraffic Congress (ITC 29)

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“…First, a similar type of statistical inference questions in queueing theory was studied before, but then for the homogeneous case, i.e. the M/G/∞ system; see, e.g., [7], [5], [22], [14], [15] and references therein. The analysis in case of the M t /G/∞ system is vastly different.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

Goldenshluger

Koops

2019

Stochastic Systems

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This paper provides a mathematical framework for estimation of the service time distribution and the expected service time of an infinite-server queueing system with a non-homogeneous Poisson arrival process, in the case of partial information, where only the number of busy servers are observed over time. The problem is reduced to a statistical deconvolution problem, which is solved by using Laplace transform techniques and kernels for regularization. Upper bounds on the mean squared error of the proposed estimators are derived. Some concrete simulation experiments are performed to illustrate how the method can be applied and to provide some insight in the practical performance.Infinite-server queue First we provide some background on the M/G/∞ queueing model, which is well studied and could be considered as a standard model. In such queues there are arrivals according to a homogeneous Poisson process, each customer is served independently of all other customers and customers do not have to queue for service, because there is an infinite number of servers. The model has a wide variety of applications in e.g. telecommunication, road traffic and hospital modeling. It can be used as an approximation to M/G/n systems, where n is relatively large with respect to the arrival rate, but the model is also interesting in its own right. For example, it can be interpreted outside queueing theory as a model for the size of a population. In this paper we will use queueing terminology (servers, customers, etc.), but these terms could be adjusted according to the application. For example, 'customers' could be cars travelling between two locations and their 'service time' could be the travel time. In many applications it is seen that the arrival rate is not constant, but it varies over time. We will provide some examples below. This observation motivates studying the M t /G/∞ queue, where the arrival rate is assumed to be a nonhomogeneous Poisson process. This model is still particularly tractable (cf. [12]) and amenable for statistical analysis, as shown in this paper. Statistical queueing problemsQueueing theory studies probabilistic properties of random processes in service systems on the basis of a complete specification of system parameters. In this paper we are interested in inverse problems when unknown characteristics of a system should be inferred from observations of the associated random processes. Typically such observations are incomplete in the sense that individual customers cannot be tracked as they go through the service system. The importance of such statistical inverse problems with incomplete observations was emphasized in [3].The service time distribution and its expected value are important performance metrics of the infinite-server queue (note that waiting times are identical to service times in infinite-server queues). Our goal is to estimate these characteristics of the M t /G/∞ system from observations of the queue-length process. Specifically, let {τ j , j ∈ Z} be arrival epochs constituting a realizati...

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“…The idea of reconstructing the service time distribution from correlation structure of the queue-length process was also exploited by [23] for a discrete-time queue model. Recently [13] constructed a local polynomial estimator of G based on (1.2) and investigated its worst-case accuracy over a suitable class of target distributions.…”

Section: Introduction Background and Motivationmentioning

confidence: 99%

The $M/G/\infty$ estimation problem revisited

Goldenshluger¹

2018

Bernoulli

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The subject of this paper is the M/G/∞ estimation problem: the goal is to estimate the service time distribution G of the M/G/∞ queue from the arrival-departure observations without identification of customers. We develop estimators of G and derive exact non-asymptotic expressions for their mean squared errors. The problem of estimating the service time expectation is addressed as well. We present some numerical results on comparison of different estimators of the service time distribution.

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Nonparametric estimation of the service time distribution in the M/G/∞ queue

Cited by 23 publications

References 35 publications

Skampling for the Flow Duration Distribution

Skampling for the Flow Duration Distribution

Nonparametric Estimation of Service Time Characteristics in Infinite-Server Queues with Nonstationary Poisson Input

The $M/G/\infty$ estimation problem revisited

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