Interarrival Times Research Articles

Exponential and Pareto-type bounds for the renewal function and the excess lifetime of a renewal process

In this paper, our objective is to derive two-sided bounds for the renewal function even if the interarrival times have no finite moment. By proving a relationship between the renewal function and a suitable ‘truncated' compound geometric distribution, under a truncated generalized Lundberg condition for proper renewal equations, we derive upper and lower bounds for this truncated compound geometric distribution. Using this relationship, we obtain bounds for the renewal function. Under specific truncated generalized Lundberg conditions, we obtain two-sided exponential-type bounds and Pareto-type lower bounds for the renewal function. Also, by proving a relationship between the excess lifetime of a renewal process and the df of a suitable ‘truncated deficit at ruin' of a renewal risk (or Sparre Andersen) model, we obtain two-sided exponential bounds for the df of the excess lifetime.

A deterministic fluid model for production and energy mode control of a single machine

Improving machines’ energy efficiency through dynamic energy mode control to meet demand requirements with minimal energy consumption is a promising approach. This study considers a machine operating in working, idle, off, and warmup energy modes with different energy consumption in each mode. A deterministic fluid model is developed to analyze an energy mode control policy that determines when to keep the machine working, off or idle, and switch to other modes based on the inventory/backlog level to minimize the total energy, inventory, and backlog costs. This approach facilitates the derivation of closed-form expressions for the optimal thresholds and the associated costs. This modeling approach allows us to prove that a policy that operates the machine between the working and off modes or the working and idle modes is always better than a hybrid policy that operates the machine in working, off, and idle modes simultaneously. We use the solution of the deterministic fluid model to propose an approximate policy for machines with stochastic production, warmup, and demand processes. We compare the results of the proposed approximation method with the optimal solution of a stochastic system where the production and warmup times are exponential and the demand inter-arrival times have Erlang distribution. The optimal solution for the stochastic system is determined by solving a Markovian Decision Process (MDP). Our numerical experiments show that the proposed approximation method predicts the optimal policy type for the stochastic case with a 89.3% accuracy, and the average error between the optimal cost and the cost obtained with the approximation method is 1.37% for 729 different cases tested. Furthermore, the computational efficiency of the proposed approximation is around 250 times better than the effort to determine the optimal policy using an MDP approach. We propose this approximation method where the control parameters are given in closed form as an easy-to-implement and effective policy to control energy modes to minimize the total energy, inventory, and backlog costs. Furthermore, we present the deterministic fluid modeling approach as a versatile approach to analyze energy mode control problems.

Transient GI/MSP/1/N Queue.

A non-zero correlation between service times can be encountered in many real queueing systems. An attractive model for correlated service times is the Markovian service process, because it offers powerful fitting capabilities combined with analytical tractability. In this paper, a transient study of the queue length in a model with MSP services and a general distribution of interarrival times is performed. In particular, two theorems are proven: one on the queue length distribution at a particular time t, where t can be arbitrarily small or large, and another on the mean queue length at t. In addition to the theorems, multiple numerical examples are provided. They illustrate the development over time of the mean queue length and the standard deviation, along with the complete distribution, depending on the service correlation strength, initial system conditions, and the interarrival time variance.

A Comprehensive examination of utilizing neutrosophic parameters in manufacturing industry through queueing approach

Queueing theory plays a pivotal role in analyzing the dynamics of manufacturing systems subject to random arrivals and service times. The M/G/1 queueing model, which represents a single-server queue with general distributions for inter-arrival and service times, is fundamental in this regard. However, traditional queueing models often fail to account for the inherent uncertainties encountered in real-world manufacturing scenarios. Neutrosophic theory provides a robust framework for modeling indeterminacy, ambiguity, and inconsistency in queueing parameters, thereby offering a more adaptable and precise depiction of system behavior. This research investigates the utilization of Neutrosophic sets in defining arrival rates, service times, and queue length distributions within the M/G/1 framework tailored to manufacturing settings. By employing numerical examples and comparative analyses, the study explores the effects of Neutrosophic parameters on key performance indicators such as mean waiting time, queue length distribution, and server utilization, considering distributions like Exponential, Erlang, and Deterministic. Additionally, it delves into the implications of integrating Neutrosophic parameters into queueing theory, providing valuable insights into improved decision-making and system optimization across various manufacturing operational contexts.

Space and time distribution of seismic source energy at Campi Flegrei, Italy through the last unrest phase (1.1.2000–31.12.2023)

We describe the space-time pattern of seismicity occurring on Campi Flegrei Caldera (CFC), Italy, where ground deformations and seismicity represent the drivers of its current bradyseismic crisis, well known and extensively studied at an international level. In detail we consider the seismicity in the time interval starting on 1.1.2000 and ending on 31.12.2023. We revise the statistics of the earthquake occurrence, focusing at possible precursory time changes of the b-parameter of the Gutenberg and Richter (G&R b−value) distribution and at the time distribution of the total seismic moment inside any swarm. To estimate the G&R b−value we use a Monte Carlo method instead of the ordinary Least Squares or Maximum likelihood methods, to easily measure the uncertainty on the b−value taking into account uncertainties on the magnitude estimates. Results show that G&R a−value and b−value calculated for cumulative and discrete distributions of Mw, the moment-magnitude, and Md, the so-called duration-magnitude, are the same inside the uncertainties; a−value and b−value for Md are significantly different from the same parameters estimated for Mw, being b−value for Mw close to the value of 1.0 and b−value for Md close to 0.8. The “bounded” G&R distribution fits the data yielding a−value and b−value close to those for the unbounded distribution. The mean annual rate of exceedance, calculated for the entire catalogue, results to be 0.033±0.015 (years−1) corresponding to a return period of 30±14 years for Mw=4.5. The time dependence of G&R b-parameter show a b−value time pattern characterized by variations slightly outside 1-σ uncertainty bar, tending to the value of 1 approaching present. As evidenced by several past studies, earthquakes in CFC occur in space-time clusters, with a mean time duration of 1 day. We selected the swarms with a selection algorithm, based on the joint estimation of inter-arrival times and inter-event distance for the consecutive event couples. The plot of the event number in each cluster vs the time of occurrence, clearly show that in CFC the number of cluster occurrences and the event number in each cluster accelerates during time starting from 2010. The time-pattern of the total seismic moment in each cluster shows that, contrarily to the event number, there is no evident striking increase of the total swarm moment as a function of time.We also consider distribution of the Energy Space Density of the CFC earthquakes, ESD. This quantity shows a clearly visible enlargement of the fractured rock volumes in the last 15 months, toward West, at a depth around 3000 m below sea level. The most fractured zone coincides with the greatest contrasts in seismic attenuation.The present study confirms that the current unrest phase is still ongoing, with an enlargement of the rock fracturing zone which extends southward and westward as compared with that measured 22 months before.

Stochastic Resource Allocation Problems: Minmax and Maxmin Solutions

Problem definition: Min-max or max-min objective criteria naturally arise in many resource allocation applications, especially when the attendant goal involves distribution of a limited resource across multiple subsystems, locations, or activities. Each has a different consumption rate of the resource. In this setting, the decision maker is interested in optimizing the system performance, which can be measured in terms of the smallest or largest order statistic across subsystems. Examples include the allocation of health resources to maximize the first runout time or to minimize the maximum delay. Methodology/results: In the literature, there are no unified theoretical and computational results on the optimal allocations for these problems. Thus, in this paper, we develop key theoretical properties of the attendant stochastic resource allocation problems, providing a technical condition for the objective functions to have favorable properties, upon which we derive novel optimal allocation rules with clear interpretation. Managerial implications: The proportional allocation rule can be interpreted as a robust solution when the full distribution of the interarrival time is unknown. It generally has poor performance, especially when the resource consumption rates vary across locations, which happens regularly in practice. Our allocation rule is exact and should be used for the allocation of scarce resources—that is, when the number of resources per location is limited, which is often encountered in healthcare settings.

Stability of a Queue Fed by Scheduled Traffic at Critical Loading

Performance of a queueing system with scheduled arrivals A scheduled arrival sequence is one in which customers are scheduled to arrive at constant interarrival times, but each customer’s actual arrival time is perturbed from her scheduled arrival time by a random perturbation. In “Stability of a Queue Fed by Scheduled Traffic at Critical Loading”, V.F. Araman and P.W. Glynn consider a single server queue with deterministic service times in which customers arrive following a scheduled arrival process. Unlike a queue fed by renewal traffic, this queue is shown to be stable even when the utilization is equal to one. It is also shown that for finite mean perturbations, a necessary and sufficient condition for stability is when the positive part of the perturbation has bounded support, with no requirement on the negative part of the perturbation. Perhaps surprisingly, this criterion is not reversible, in the sense that such a queue can be stable for a scheduled traffic process in forward time, but unstable for the time-reversal of the same traffic process.

Analysis of delays in medium access control devices in cognitive radio systems with correlation of time intervals in request flows

This article is about the development of a method for analyzing the average waiting time of a request in a queue with correlated inter-arrival and service times. The ON-OFF process is often used to model the primary user channel occupancy in different questions related to cognitive radio, such as resource allocation, medium access control protocols, spectrum handoff and others. It is assumed that the time intervals of the ON- and OFF- periods are not correlated. More realistic models of primary user activity require the presence of correlation not only between consequent periods, but also the presence of correlation between channel occupancy and other quantities interpreted as service time in models based on queuing systems. As a result, here is proposed a variant of constructing the probability density function of a random variable which is the difference between the service time of the i-th request and the time interval between the moments of arriving of the i-th and (i-1)-th requests with correlated inter-arrival and service times. Solution based on the use of a two-dimensional characteristic function, so in this case it is possible to more purposefully specify the components of the statistical connection of previously described sequences. These components are the cumulants of the described distribution. For a queuing system with hyperexponential distribution of inter-arrival and service intervals, an analytical expression for the average waiting time of a request in a queue is found.

Multi-objective QoS optimization in swarm robotics

The “Internet of Robotic Things” (IoRT) is a concept that connects sensors and robotic objects. One of the practical applications of IoRT is swarm robotics, where multiple robots collaborate in a shared workspace to accomplish assigned tasks that may be challenging or impossible for a single robot to conquer. Swarm robots are particularly useful in critical situations, such as post-earthquake scenarios, where they can locate survivors and provide assistance in areas inaccessible to humans. In these life-saving situations, reliable and prompt communication among swarm robots is of utmost importance. To address the need for highly dependable and low-latency communication in swarm robotics, this research introduces a novel hybrid approach called Multi-objective QoS optimization based on Support vector regression and Genetic algorithm (MQSG). The MQSG method consists of two main phases: Parameter Relationship Identification and Parameter Optimization. In the Parameter Relationship Identification phase, the relationship between network inputs (Packet inter-arrival time, Packet size, Transmission power, Distance between sender and receiver) and outputs (quality of service (QoS) parameters) is established using support vector regression. In the parameter optimization phase, a multi-objective function is created based on the obtained relationships from the Parameter Relationship Identification phase. By solving this multi-objective function, optimal values for each QoS parameter are determined, leading to enhanced network performance. Simulation results demonstrate that the MQSG method outperforms other similar algorithms in terms of transmission latency, packet delivery rate, and the number of retransmitted packets.

A Uniform Approach for Analyzing Queues with Correlated Interarrival and Service Times

A Uniform Approach for Analyzing Queues with Correlated Interarrival and Service Times

Asymptotic estimates for ruin probabilities in a bidimensional delay-claim risk model with subexponential claims

This article studies the asymptotic behaviors for some common ruin probabilities of a bidimensional delay-claim risk model, in which each claim follows a subexponential distribution and each claim-size inherits some dependence structure. Meanwhile, the asymptotic behavior for sum ruin probability of the bidimensional delay-claim risk model is investigated when the two components of each claim inter-arrival time vector are arbitrarily dependent.

Moderate deviations for multidimensional aggregate claims with arbitrary dependence between claim sizes and waiting times

Consider a multidimensional renewal risk model, in which the claim sizes X k , k ≥ 1 , form a sequence of independent and identically distributed random vectors with non negative components allowed to be dependent on each other, and the inter-arrival times form a sequence of p-extended negatively dependent random variables. By allowing arbitrary dependence between claim sizes and waiting times, we obtain the moderate deviations for the multidimensional aggregate claims with consistently varying tails.

The impact of the correlation coefficient of interarrival and service times on queueing performance: The [formula omitted] case

This paper is concerned on an M/M/1 queue with correlated interarrival and service times. In particular, we assume that the interarrival time and service time of a customer have a bivariate exponential distribution. By utilizing a Markov modulated fluid flow (MMFF) process associated with the age process of the customer in service, we obtain a number of queueing quantities in closed form. Using the solutions, the impact of the correlation coefficient of the interarrival and service times on a variety of queueing quantities is explored quantitatively and qualitatively. Specifically, we establish a monotonic relationship between the decay rates of queueing quantities and the correlation coefficient of the interarrival and service times. We also show that the decay rates are increasing in the correlation coefficient. In addition, queues with the maximum/minimum correlation coefficient are analyzed. Two examples are presented to demonstrate the importance of using the correlation coefficient in queueing performance/economic analysis.

Regenerative Analysis and Approximation of Queueing Systems with Superposed Input Processes

A single-server queueing system with n classes of customers, stationary superposed input processes, and general class-dependent service times is considered. An exponential splitting is proposed to construct classical regeneration in this (originally non-regenerative) system, provided that the component processes have heavy-tailed interarrival times. In particular, we focus on input processes with Pareto interarrival times. Moreover, an approximating GI/G/1-type system is considered, in which the independent identically distributed interarrival times follow the stationary Palm distribution corresponding to the stationary superposed input process. Finally, Monte Carlo and regenerative simulation techniques are applied to estimate and compare the stationary waiting time of a customer in the original and in the approximating systems, as well as to derive additional information on the regeneration cycles’ structure.

Call center data modeling: a queueing science approach based on Markovian arrival process

ABSTRACT In this paper we analyze the well-known ‘Anonymous bank’ call center dataset from a queueing science viewpoint. For this purpose, fitted distributions for both the inter-arrival and service times as well as for customers patiences are integrated in a simulator to infer quantities of interest related to call centers managerial decisions as waiting times, abandonment rates and queue lengths. In particular, it is shown how a type of Markov renewal process, the Markovian arrival process (MAP), is able to capture some of the characterizing properties of arrivals in a modern call center as overdispersion and positive correlation between arrival counts. The work provides a new inference approach for the MAP based on the count process descriptors and presents new properties concerning the dependence structure of the cumulated number of arrivals in a MAP.

Transient analysis of a bulk stream queue with arbitrarily distributed arrival intervals

We study the classical infinite buffer single server queueing model with renewal input of customers in batches of random size, having arbitrarily distributed arrival intervals and exponentially distributed service times. Using the technique of supplementary variable and shift operator we derive closed form expression of the time dependent system content distribution in terms of its Laplace transform. The analysis is mainly based on the root-finding technique of the non-linear characteristic equation in terms of the Laplace transform variable. Additionally, using asymptotic properties of Laplace transform, we deduce the corresponding steady-state distribution. We discuss some special cases of the model, thus providing an alternative approach in deriving the transient distribution. We further evaluate certain performance measures and present extensive numerical examples in tabular and graphical form to illustrate the applicability of our theoretical work. The effect of system parameters, interarrival time distribution and traffic intensity on the system behavior is also demonstrated.

Adaptive Flow Timeout Management in Software-Defined Optical Networks

Current trends in network traffic management rely on the efficient control of individual flows. Software-defined networking popularized this notion. Per-flow management is perfectly viable in standard IP networks, in which packet processing is in the electric domain. However, optical networks provide more restrictions and constraints making per-flow traffic management difficult. One of the most important challenges is to reduce the concurrent number of flows present in the flow tables to make the switching process quicker. In this paper, we propose a mechanism to manage flow timeout values that uses idle timeout and hard timeout parameters. To calculate the appropriate values of the parameters, the mechanism analyzes the packet inter-arrival times. The algorithm also takes into account the current occupancy of the flow table.

Parameter Estimation and Testing for the Doubly Geometric Process with Lognormal Distribution: Application to Bladder Cancer Patients’ Data

The geometric process (GP) has been widely utilized as a stochastic monotone model in the fields of probability and statistics. However, its practical application is often limited by certain assumptions. To address this, [Wu (2018). Doubly geometric process and applications. Journal of the Operational Research Society, 69(1), 66–67] introduced the doubly geometric process (DGP) as an extension of the GP model, relaxing some of its assumptions. Due to its ability to overcome the limitations of the GP model, the DGP has gained significant popularity in recent times. This study focuses on the parameter estimation problem for the DGP when the distribution of the first interarrival time follows a lognormal distribution with parameters [Formula: see text] and [Formula: see text]. We employ the maximum likelihood method to obtain estimates for both the model parameters and the distribution parameters. Additionally, we investigate the asymptotic joint distribution and statistical properties such as asymptotic unbiasedness and consistency of the estimators. Furthermore, we propose a novel test procedure to distinguish between the GP and DGP models. To assess the performance of the estimators and the proposed test procedure, we conduct a simulation study involving various sample sizes and parameter values. Finally, we present an application of the developed methods in fitting data from bladder cancer patients.

Queues with correlated inter-arrival and service times

Queues with correlated inter-arrival and service times

The Robust Classical MTBF Test

For the reliability demonstration of a repairable system, testers expediently assume that the rate of occurrence of failures (ROCOF) can be modeled by a homogenous Poisson process (HPP), also called an ordinary renewal process (ORP), and therefore the times between failure, or interarrival times, are exponentially distributed with a constant mean time between failures (MTBF). This often-used test is colloquially called the “classical MTBF test” because it has been widely used by both government and industry for the past 50 years. The test is simple; operate a system for a period and the sample estimate for MTBF is the operating time divided by the number of failures. A confidence interval is applied to the sample estimate to determine the test result. This paper considers the validity of the test result if the repairable system has an increasing or decreasing MTBF, that is, a nonhomogeneous Poisson process (NHPP). Specifically, the change in consumer’s risk is analyzed if the repairable system has a decreasing MTBF (increasing ROCOF). A simple risk mitigation strategy consisting of a burn-in period equal to the threshold MTBF is also considered and found to be effective. The classical MTBF test is shown to be a robust test and a burn-in period prior to testing makes the classical MTBF test even more robust.