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

Abstract

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...

Missing link survival analysis with applications to available pandemic data

“…Our data do not show stationarity and for this reason it requires other approaches. The paper Goldenshluger and Koops (2018) is an example where non-constant intensities of incoming claims are considered as in our paper, but this is done under smoothness conditions on γ that our approach does not need.…”

Dispersal density estimation across scales

“…Example 1: Service time estimation in M/G/∞ queuing models Dating back to Brown (1970), certain M/G/∞ queuing models are embedded in our approach, see in particular the recents results of Goldenshluger (2016Goldenshluger ( , 2018; Goldenshluger and Koops (2019).…”

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“…[107], [33], [27], [67], [62], [63], [64], [53], [111], [112], [68], [24], [102] Inference with Discrete Sampling: This paradigm focuses on cases where systems are sampled discretely over time.…”

“…Extended models include the time inhomogeneous case studied in [64] by Goldenshluger and Koops. Further, in a discrete time setting, in [53], Edelman and Wichelhaus, considered parameter estimation for two-node networks of infinite server queues with geometric arrivals and general service times.…”