In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and non-exponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by discussing how our results can be used computationally and by verifying the numerical results via simulations.

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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...

We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks.