Infinite Server Research Articles

Transient solution of an M/M/? queue with system's additional tasks and impatient customers

This paper studies the impatient behaviour in an infinite server queue with additional tasks assigned to the system. Whenever the system becomes empty, the system as a whole is assigned a secondary task of duration U whose distribution is exponential. Any arrival during the period U becomes impatient due to the unavailability of service facility. Each individual waiting customer activates an independent impatience timer of duration T which is exponentially distributed. When the system comes back after the completion of U, before T expires, the waiting customers are simultaneously taken for service and they leave the system after the completion of service. If T expires before the completion of task U, the customers abandon the system and never to return. The transient system size probabilities of this model are derived explicitly for both single and multiple task cases. The time-dependent mean and variance of system size are also derived. Further, numerical simulations are also presented to analyse the effect of system indices.

Queues Driven by Hawkes Processes

Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through self-excitation of the arrival process. In this paper, we analyze an infinite server queueing system in which the arrivals are driven by the self-exciting Hawkes process and in which service follows a phase-type distribution or is deterministic. In the phase-type setting, we derive differential equations for the moments and a partial differential equation for the moment-generating function; we also derive exact expressions for the transient and steady-state mean, variance, and covariances. Furthermore, we also derive exact expressions for the autocovariance of the queue and provide an expression for the cumulant moment–generating function in terms of a single ordinary differential equation. In the deterministic service setting, we provide exact expressions for the first and second moments and the queue autocovariance. As motivation for our Hawkes queueing model, we demonstrate its usefulness through two novel applications. These applications are trending internet traffic and arrivals to nightclubs. In the web traffic setting, we investigate the impact of a click. In the nightclub or club queue setting, we design an optimal control problem for the optimal rate to admit club-goers.

Performance Optimization for Timed Weighted Marked Graphs Under Infinite Server Semantics

This paper deals with the performance optimization of resource allocation systems with the aim of maximizing the system's throughput under a given budget for acquiring resources. Resources are assumed to be renewable, i.e., they are not consumed by the operations and become available again after they have been released. The systems under consideration are modeled by a subclass of timed Petri nets called deterministic timed weighted marked graphs. In addition, we take into account infinite server semantics, i.e., the degree of self-concurrency of each transition is infinite. We propose an approach that provides an optimal solution, but has a high computational cost. For this reason, we also present two different approaches that can find suboptimal solutions with a reduced computational cost. The performances of the proposed approaches are compared by means of numerical simulations.

On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues

This paper considers a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported (IBNR) claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in G/G/infinity queues with correlated batch arrivals. We study the long term behavior of this process as well as its moments. Asymptotic expressions and bounds for the quantities of our interest, and also convergence result for the distribution of this process after renormalization, are studied, when interarrival times and time delays are light tailed. Next, assuming exponentially distributed delays, we derive some explicit and numerically feasible expressions for the limiting joint moments. In such case, for an innite server queues with renewal arrival process, we obtain limiting results on the expectation of the workload, and the covariance of queue size and workload. Finally, some queueing theoretic applications are provided MSC classification: 60G50, 60K30, 62P05, 60K25

Transient analysis of a discrete-time infinite server queue with system disaster

This paper studies a discrete-time infinite server queue subject to system disaster. The exact time-dependent probabilities of the number of customers present in the system are obtained using generating functions, continued fractions and confluent hypergeometric functions. Further, the results are extended to obtain closed form expression for busy period distribution and steady state system size probabilities. Numerical illustrations are provided to visualise the effect of system size probabilities, both in steady state and transient state, and busy period distribution for different parameter values.

Sufficient Conditions and Sensor Placement for Structural Fault Diagnosis in a class of Timed Continuous Petri Nets

This work addresses the characterization of the fault diagnosability property in Timed Continuous Petri Nets (TCPN) under infinite server semantics. In the approach herein presented, the potential faults are included in the model, however, their occurrence time, magnitude and location are unknown. Two novel results are presented. In the first one, sufficient conditions that guarantee the diagnosability in TCPN substructures are provided. These conditions are based on the net structure, avoiding the analysis of all linear systems describing the net dynamics. The second result is a methodology to place a reduced number of sensors in such a way that a class of TCPN becomes diagnosable. In order to show the applicability of the reported results, an illustrative example is presented.

On monotonicity and continuity: computing problematic configurations in timed continuous Petri nets

This work is concerned with the computation of problematic configurations in timed continuous Petri nets under the infinite server semantics. Those net structural objects are related to possible bad/paradoxical behaviors of systems, such as non-monotonicities and discontinuities of the equilibrium throughput. The calculation of problematic configurations is a computationally complex task because their number grows exponentially with the input cardinality of join transitions. This work introduces a branch and bound type algorithm to perform such computation and, as an improvement of the calculation, it proposes to reduce the net into a smaller one using three different types of reduction rules that preserve the set of problematic configurations. The transformed net system allows to better understand why undesired behaviors may appear.

Queuing theory-based innovation diffusion modelling incorporating change in adoption rate

Adoption has always been an important process to discuss among marketers. Major work in the field of innovation adoption has been based on theory of first purchase by consumers. Of late attention has also been given to multi-stage nature of diffusion process. There are practitioners who have verified adoption as multi-stage process (depending on awareness and motivation). Researchers have lately also understood the value of change in marketing strategy and other factors that often lead to change in the rate of adoption. In this paper, we have made use of this stage wise approach of market penetration along with change point concept, have developed a methodical approach based on infinite server queuing theory and predicted sales for consumer durables. Experimental results estimated on sales of two different consumer durables show that present proposal can depict the change in adoption rates and predict the behaviour of the product in more accurate manner.

On the Distributions of Infinite Server Queues With Batch Arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as (or, in some cases, bulk) arrival queueing systems. In this work we study the affect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the service distribution of its order statistics. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the Harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

Transient analysis of a discrete-time infinite server queue with system disaster

This paper studies a discrete-time infinite server queue subject to system disaster. The exact time-dependent probabilities of the number of customers present in the system are obtained using generating functions, continued fractions and confluent hypergeometric functions. Further, the results are extended to obtain closed form expression for busy period distribution and steady state system size probabilities. Numerical illustrations are provided to visualise the effect of system size probabilities, both in steady state and transient state, and busy period distribution for different parameter values.

Fork-join queueing~networks with an arbitrary topology: the infinite server case

Fork-join queueing~networks with an arbitrary topology: the infinite server case

Customizing exponential semi-Markov decision processes under the discounted cost criterion

The uniformization technique is a widely used method for establishing the existence of optimal policies with certain monotonicity properties. This technique converts a semi-Markov decision process with exponential sojourn times (ESMDP) into an equivalent discrete-time Markov decision process by defining some fictitious jumps. This study proposes a new device, called customization, which can convert a given ESMDP into another equivalent ESMDP whose formulation possibly simplifies mathematical analysis. The customization technique uses the fictitious jump idea to establish the equivalence under deterministic stationary policies just like the uniformization technique. However, it allows the transition rates of the new ESMDP to be different. Moreover, it can be applied even when the transition rates of the initial ESMDP are unbounded. This flexibility can be very useful in analyzing the problems where the uniformization is not applicable or not so helpful. We analyze a complex optimal replacement problem and an infinite server queueing problem with unbounded transition rates to demonstrate the applicability and advantages of customization.

Stationary distributions of the R[X]/R/1 cross-correlated queue

ABSTRACTWe consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.

Homothecy, bifurcations, continuity and monotonicity in timed continuous Petri nets under infinite server semantics

This work studies how equilibrium markings and throughputs change in Timed Continuous Petri Net (TCPN) systems as transition firing rates vary. In particular, it analyzes the bifurcations of the former, and the discontinuities and non-monotonicities of the latter; specifically, using structural objects of the net, such as P-semiflows, T-semiflows, and configurations, among others, the following properties can be obtained. For Join Free TCPN systems, a sufficient structural condition guaranteeing that the equilibrium markings do not bifurcate when firing rates vary, is derived. A dual result is obtained for Choice Free TCPN systems. For Mono-T-Semiflow TCPN systems, the equilibrium throughput is investigated; using a time-scale (a homothetic) property it is proven that a discontinuity of the equilibrium throughput implies its non-monotonicity, even if not evident at first glance. This is a connection of two timed behavioral properties of the equilibrium throughput. Moreover, a sufficient structural condition, parametrized by the equilibrium markings, ensures its continuity under firing rate variations. It is also proven that the monotonicity of the equilibrium throughput can be characterized by the previous structural condition. The convergence of the marking evolution of TCPN systems to its equilibrium markings is also discussed.

Fluid approximation of Petri net models with relatively small populations

Fluidization is an appealing relaxation technique based on the removal of integrality constraints in order to ease the analysis of discrete Petri nets. The result of fluidifying discrete Petri nets are the so called Fluid or Continuous Petri nets. As with any relaxation technique, discrepancies among the behaviours of the discrete and the relaxed model may appear. Moreover, such discrepancies may have a comparatively bigger effect when the population of the system, the marking in Petri net terms, is “relatively” small. This paper proposes two complementary approaches to obtain a better fluid approximation of discrete Petri nets. The first one focuses on untimed systems and is based on the addition of places that are implicit in the untimed discrete system but not in the continuous. The idea is to cut undesired spurious solutions whose existence worsens the fluidization. The second one focuses on a particular situation that can severely affect the quality of fluidization in timed systems. Namely, such a situation arises when the enabling degree of a transition is equal to 1. This last approach aims to alleviate such a state of affairs, which is termed the bound reaching problem, on systems under infinite servers semantics.

In the search for the infinite servers queue with Poisson arrivals busy period distribution exponential behaviour

This paper purpose is to investigate exponential behavior conditions for the infinite servers queue with Poisson arrivals busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the infinite servers queue with Poisson arrivals busy period length distribution moments computation. In infinite servers queue with Poisson arrivals practical applications, in economic, management, and business areas, the management of the effective number of servers is essential since the physical presence of infinite servers is not viable and so it is necessary to create that condition through an adequate management of the number of servers during the busy period.

In the search for the infinite servers queue with Poisson arrivals busy period distribution exponential behaviour

The purpose of this paper is to investigate exponential behaviour conditions for the M|G|∞ queue busy period length distribution. It is presented a general theoretical result that is the basis of this work. The complementary analysis rely on the M|G|∞ queue busy period length distribution moments computation. In M|G+∞ queue practical applications - in economic, management and business areas - the management of the effective number of servers is essential since the physical presence of infinite servers is not viable and so it is necessary to create that condition through an adequate management of the number of servers during the busy period.

Queues Driven by Hawkes Processes

Many stochastic systems have arrival processes that exhibit clustering behavior. In these systems, arriving entities influence additional arrivals to occur through self-excitation of the arrival process. In this paper, we analyze an infinite server queueing system in which the arrivals are driven by the self-exciting Hawkes process and where service follows a phase-type distribution or is deterministic. In the phase-type setting, we derive differential equations for the moments and a partial differential equation for the moment generating function; we also derive exact expressions for the transient and steady-state mean, variance, and covariances. Furthermore, we also derive exact expressions for the auto-covariance of the queue and provide an expression for the cumulant moment generating function in terms of a single ordinary differential equation. In the deterministic service setting, we provide exact expressions for the first and second moments and the queue auto-covariance. As motivation for our Hawkes queueing model, we demonstrate its usefulness through two novel applications. These applications are trending internet traffic and arrivals to nightclubs. In the web traffic setting, we investigate the impact of a click. In the nightclub or Club Queue setting, we design an optimal control problem for the optimal rate to admit club-goers.

Unified approach for modeling innovation adoption and optimal model selection for the diffusion process

Purpose – Mathematical modeling of innovation diffusion is a constantly evolving field within marketing science. The diffusion process explains the dispersion of an innovation among potential buyers. Prior research on innovation diffusion has been based on modeling varied aspects of real life situations in marketing. One such aspect is studying the adoption process depending on the awareness and motivation level among the customers. Awareness is having knowledge of an innovation, whereas motivation is about the perception of an individual. In line with these aspects, the purpose of this paper is to propose a unified modeling framework for the adoption process based on the awareness and motivation about the product. Design/methodology/approach – When the market is well informed about the product, there are some people who are motivated and some, who have adopted the product earlier and shall now influence others in their buying behavior. It is very much similar to queuing system in which some units are waiting in a queue for the service, service for some units are being processed and some units have already been served. This analogous behavior between two approaches has motivated the use of infinite server queuing theory in modeling adoption of the product. Thereafter, the authors have proposed a unification scheme to model different market scenarios. Findings – From analyzing the values of comparison criteria, it was not clear that which among them is performing best. Thus there was a need for an approach which can judiciously find the optimal model. For this very purpose the authors applied distance-based approach which was capable of computing the optimal model based on the distance of attribute value from the optimal. The analysis performed on two real life sales data sets depict that model in which awareness is following logistic pattern and motivation and adoption are following a constant pattern is ranked one. Research limitations/implications – The idea has been validated on product. It would be interesting to know how the methodology works on service. Originality/value – The modeling framework discussed in this paper can be helpful to know from the available set of alternative, which among them is performing better in capturing the spread of the product in the market. The proposed framework offer some managerial guidance by highlighting the unusual aspects of diffusion process and also present an approach to judge the best among a set of different models.

Infinite Server Queuing Models with Resource Expenditures and Change-point for Software Reliability

Infinite Server Queuing Models with Resource Expenditures and Change-point for Software Reliability