Networks Of Infinite-server Queues Research Articles

Superinfection and the hypnozoite reservoir for Plasmodium vivax: a general framework

A characteristic of malaria in all its forms is the potential for superinfection (that is, multiple concurrent blood-stage infections). An additional characteristic of Plasmodium vivax malaria is a reservoir of latent parasites (hypnozoites) within the host liver, which activate to cause (blood-stage) relapses. Here, we present a model of hypnozoite accrual and superinfection for P. vivax. To couple host and vector dynamics for a homogeneously-mixing population, we construct a density-dependent Markov population process with countably many types, for which disease extinction is shown to occur almost surely. We also establish a functional law of large numbers, taking the form of an infinite-dimensional system of ordinary differential equations that can also be recovered by coupling expected host and vector dynamics (i.e. a hybrid approximation) or through a standard compartment modelling approach. Recognising that the subset of these equations that model the infection status of the human hosts has precisely the same form as the Kolmogorov forward equations for a Markovian network of infinite server queues with an inhomogeneous batch arrival process, we use physical insight into the evolution of the latter process to write down a time-dependent multivariate generating function for the solution. We use this characterisation to collapse the infinite-compartment model into a single integrodifferential equation (IDE) governing the intensity of mosquito-to-human transmission. Through a steady state analysis, we recover a threshold phenomenon for this IDE in terms of a parameter R_0 expressible in terms of the primitives of the model, with the disease-free equilibrium shown to be uniformly asymptotically stable if R_0<1 and an endemic equilibrium solution emerging if R_0>1. Our work provides a theoretical basis to explore the epidemiology of P. vivax, and introduces a strategy for constructing tractable population-level models of malarial superinfection that can be generalised to allow for greater biological realism in a number of directions.

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

Noise dissipation in gene regulatory networks via second order statistics of networks of infinite server queues

RNA and protein concentrations within cells constantly fluctuate. Some molecular species typically have very low copy numbers, so stochastic changes in their abundances can dramatically alter cellular concentration levels. Such noise can be harmful through constrained functionality or reduced efficiency. Gene regulatory networks have evolved to be robust in the face of noise. We obtain exact analytical expressions for noise dissipation in an idealised stochastic model of a gene regulatory network. We show that noise decays exponentially fast. The decay rate for RNA molecular counts is given by the integral of the tail of the cumulative distribution function of the degradation time. For proteins, it is given by the slowest rate-limiting step of RNA degradation or proteolytic breakdown. This is intuitive because memory of the chemical composition of the system is manifested through molecular persistence. The results are obtained by analysing a non-standard tandem of infinite server queues, in which the number of customers present in one queue modulates the arrival rate into the next.

Hypnozoite dynamics for Plasmodium vivax malaria: The epidemiological effects of radical cure

Malaria is a mosquito-borne disease with a devastating global impact. Plasmodium vivax is a major cause of human malaria beyond sub-Saharan Africa. Relapsing infections, driven by a reservoir of liver-stage parasites known as hypnozoites, present unique challenges for the control of P. vivax malaria. Following indeterminate dormancy periods, hypnozoites may activate to trigger relapses. Clearance of the hypnozoite reservoir through drug treatment (radical cure) has been proposed as a potential tool for the elimination of P. vivax malaria. Here, we introduce a stochastic, within–host model to jointly characterise hypnozoite and infection dynamics for an individual in a general transmission setting, allowing for radical cure. We begin by extending an existing activation-clearance model for a single hypnozoite, adapted to both short- and long-latency strains, to include drug treatment. We then embed this activation-clearance model in an epidemiological framework accounting for repeated mosquito inoculation and the administration of radical cure. By constructing an open network of infinite server queues, we derive analytic expressions for several quantities of epidemiological significance, including the size of the hypnozoite reservoir; the relapse rate; the relative contribution of relapses to the infection burden; the distribution of multiple infections; the cumulative number of recurrences over time, and the time to first recurrence following drug treatment. We derive from first principles the functional dependence between within–host and transmission parameters and patterns of blood- and liver-stage infection, whilst allowing for treatment under a mass drug administration regime. To yield population-level insights, our analytic within–host distributions can be embedded in multiscale models. Our work thus contributes to the epidemiological understanding of the effects of radical cure on P. vivax malaria.

Large deviations of bivariate Gaussian extrema

We establish sharp tail asymptotics for componentwise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different versus the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

Infinite-server systems with Coxian arrivals

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

Networks of infinite-server queues with multiplicative transitions

This paper considers a network of infinite-server queues with the special feature that, triggered by specific events, the network population vector may undergo a linear transformation (a ‘multiplicative transition’). For this model we characterize the joint probability generating function in terms of a system of partial differential equations; this system enables the evaluation of (transient as well as stationary) moments. We show that several relevant systems fit in the framework developed, such as networks of retrial queues, networks in which jobs can be rerouted when links fail, and storage systems. Numerical examples illustrate how our results can be used to support design problems.

On the distribution of calls in a wireless network driven by fluid traffic

This note develops a modelling approach for wireless networks driven by fluid traffic models. Introducing traffic sets that follow movement of subscribers, the wireless network with time-varying rates is transformed into a stationary network at these traffic sets, which yields that the distribution of calls over the cells of the network depends on the call length distribution only through its mean. The result is extended to a network of infinite server queues with time-varying arrival rates.

Blocking probabilities in mobile communications networks with time-varying rates and redialing subscribers

Call-blocking probabilities are among the key performance measures in mobile communications networks. For their analysis, mobile networks can be modelled as networks of Erlang loss queues with common capacity restrictions dictated by the allocation of frequencies to the cells of the network. However, due to the time-varying load offered to the cells of such networks, blocking probabilities usually cannot be obtained in closed form. The relation between networks of Erlang loss queues and networks of infinite server queues, for which the time-dependent occupancy distribution is multidimensional Poisson, suggests to use that distribution as approximate distribution for the network of Erlang loss queues. This paper extends this so-called Modified Offered Load (MOL) approximation to networks of Erlang loss queues, and also allows subscribers that find their call blocked to redial to continue their call. For GSM networks operating under Fixed Channel Allocation, it is shown that blocking probabilities are increasing in the redial rates so that the MOL approximation that is most accurate for maximal redial rates turns out to be fairly accurate for the resulting upper bound for blocking probabilities. The accuracy is explicitly evaluated in an application of the results towards blocking probabilities in a hot spot travelling along a road through a GSM network.

A Markov Renewal Based Model for Wireless Networks

We propose a queueing network model which can be used for the integration of the mobility and teletraffic aspects that are characteristic of wireless networks. In the general case, the model is an open network of infinite server queues where customers arrive according to a non-homogeneous Poisson process. The movement of a customer in the network is described by a Markov renewal process. Moreover, customers have attributes, such as a teletraffic state, that are driven by continuous time Markov chains and, therefore, change as they move through the network. We investigate the transient and limit number of customers in disjoint sets of nodes and attributes. These turn out to be independent Poisson random variables. We also calculate the covariances of the number of customers in two sets of nodes and attributes at different time epochs. Moreover, we conclude that the arrival process per attribute to a node is the sum of independent Poisson cluster processes and derive its univariate probability generating function. In addition, the arrival process to an outside node of the network is a non-homogeneous Poisson process. We illustrate the applications of the queueing network model and the results derived in a particular wireless network.

A Time-Dependent Queueing-Network Model to Describe the Life-Cycle Dynamics of Private-Line Telecommunication Services

We propose a model to help a service provider manage a family of private-line telecommunication services. The model is a time-dependent network of infinite-server queues. The queueing network is used to model switching from one service to another. The relevant time scale is quite long: service lifetimes are measured in years, while service life cycles are measured in decades. To capture changing technology and customer preferences over this extended period, the model includes time-dependent new-connection rates, time-dependent switching rates and general time-dependent service-lifetime distributions for the different services. Because of the long service lifetimes, there is typically a significant time lag between the peak arrival rate and the peak expected number of customers receiving service. Thus sales and revenue do not move together; instead sales is a leading indicator of revenue in a service life cycle. The network structure reveals how the life cycles of different services are related. We show that it is possible to reasonably fit a version of this relatively complex model to data and then analyze the model to obtain useful descriptions of system dynamics.

A note on networks of infinite-server queues

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.

A note on networks of infinite-server queues

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.