Arrival Epochs Research Articles

Inequalities between time and customer averages for HNB(W)UE arrival processes

Abstract We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.

Refining the OJ 287 2022 impact flare arrival epoch

ABSTRACT The bright blazar OJ 287 routinely parades high brightness bremsstrahlung flares, which are explained as being a result of a secondary supermassive black hole (SMBH) impacting the accretion disc of a more massive primary SMBH in a binary system. The accretion disc is not rigid but rather bends in a calculable way due to the tidal influence of the secondary. Next, we refer to this phenomenon as a variable disc level. We begin by showing that these flares occur at times predicted by a simple analytical formula, based on general relativity inspired modified Kepler equation, which explains impact flares since 1888. The 2022 impact flare, namely flare number 26, is rather peculiar as it breaks the typical pattern of two impact flares per 12-yr cycle. This is the third bremsstrahlung flare of the current cycle that follows the already observed 2015 and 2019 impact flares from OJ 287. It turns out that the arrival epoch of flare number 26 is sensitive to the level of primary SMBH’s accretion disc relative to its mean level in our model. We incorporate these tidally induced changes in the level of the accretion disc to infer that the thermal flare should have occurred during 2022 July–August, when it was not possible to observe it from the Earth. Thereafter, we explore possible observational evidence for certain pre-flare activity by employing spectral and polarimetric data from our campaigns in 2004/05 and 2021/22. We point out theoretical and observational implications of two observed mini-flares during 2022 January–February.

Overview of Earth-Moon Transfer Trajectory Modeling and Design

The Moon is the only celestial body that human beings have visited. The design of the Earth-Moon transfer orbits is a critical issue in lunar exploration missions. In the 21st century, new lunar missions including the construction of the lunar space station, the permanent lunar base, and the Earth-Moon transportation network have been proposed, requiring low-cost, expansive launch windows and a fixed arrival epoch for any launch date within the launch window. The low-energy and low-thrust transfers are promising strategies to satisfy the demands. This review provides a detailed landscape of Earth-Moon transfer trajectory design processes, from the traditional patched conic to the state-of-the-art low-energy and low-thrust methods. Essential mechanisms of the various utilized dynamic models and the characteristics of the different design methods are discussed in hopes of helping readers grasp the basic overview of the current Earth-Moon transfer orbit design methods and a deep academic background is unnecessary for the context understanding.

Performability analysis of a MMAP[2]/PH[2]/S model with PH retrial times

This work focuses over the performability analysis of a multi-server retrial queueing model with phase-type inter-retrial times in cellular networks. It is considered that the pattern of the new call arrival and handoff call arrival follows marked Markovian arrival process. The service times of both types of calls are phase-type (PH) distributed with different parameters, and inter-failure & inter-repair times of channels are exponentially distributed. For the prioritization of handoff calls, G channels are kept in reserve for handoff calls. When all the available channels, say S , are busy at the arrival epoch of a handoff call, the handoff call will be dropped. Whereas a new call will be blocked and will have an option to join the orbit of infinite capacity or leave the system without getting the connection, if at least S − G channels are busy. A new call in the orbit, termed as retrial call, retries to get the connection after a random interval which follows PH distribution. This model is analyzed as a level-dependent-quasi-birth-death process by applying an efficient method. Through numerical illustrations, the behavior of performance measures depending on the various relevant intensities is discussed.

Evolutionary Optimization of Multirendezvous Impulsive Trajectories

This paper investigates the use of evolutionary algorithms for the optimization of time-constrained impulsive multirendezvous missions. The aim is to find the minimum- Δ V trajectory that allows a chaser spacecraft to perform, in a prescribed mission time, a complete tour of a set of targets, such as space debris or artificial satellites, which move on the same orbital plane at slightly different altitudes. For this purpose, a two-level design approach is pursued. First, an outer-level combinatorial problem is defined, dealing with the simultaneous optimization of the sequence of targets and the rendezvous epochs. The suggested approach is first tested by assuming that all transfer legs last exactly the same amount of time; then, the time domain is discretized over a finer grid, allowing a more appropriate sizing of the time window allocated for each leg. The outer-level problem is solved by an in-house genetic algorithm, which features an effective permutation-preserving solution encoding. A simple, but fairly accurate, heuristic, based on a suboptimal four-impulse analytic solution of the single-target rendezvous problem, is used when solving the combinatorial problem for a fast guess at the transfer cost, given the departure and arrival epochs. The outer-level problem solution is used to define an inner-level NLP problem, concerning the optimization of each body-to-body transfer leg. In this phase, the encounter times are further refined. The inner-level problem is tackled through an in-house multipopulation self-adaptive differential evolution algorithm. Numerical results for case studies including up to 20 targets with different time grids are presented.

Poissonian occupation times of spectrally negative Lévy processes with applications

In this paper, we introduce the concept of Poissonian occupation times below level 0 of a spectrally negative Lévy process. In this case, occupation time is accumulated only when the process is observed to be negative at arrival epochs of an independent Poisson process. Our results extend some well known continuously observed quantities involving occupation times of spectrally negative Lévy processes. As an application, we establish a link between Poissonian occupation times and insurance risk models with Parisian implementation delays.

Estimation of the expected value of the arrival epochs given a bounded period of observation for the homogeneous and non-homogeneous Poisson processes

In this paper, we derive analytical expressions of the expected value of the arrival epochs given a bounded period of observation for the homogeneous Poisson process (HPP) and for the general case of a non-homogeneous Poisson process (NHPP). These expressions are exact, but their applicability require some computational adaptations. We illustrate the results obtained through the proposed expression by comparing them with Monte Carlo sampling. The results show that these expressions outperform Monte Carlo in precision and computational effort.

The queue revisited

We present analytic expressions (in terms of roots of the underlying characteristic equation) for the steady-state distributions of the number of customers for the finite-state queueing model with partial-batch rejection policy. We obtain the system-length distributions at a service-completion epoch by applying the imbedded Markov chain technique. Using the roots of the related characteristic equation, the method leads to giving a unified approach for solving both finite- and infinite-buffer systems. We find relationships between system-length distributions at departure, random, and arrival epochs using discrete renewal theory and conditioning on the system states. Based on these relationships, we obtain various performance measures and provide numerical results for the same. We also perform computational analysis and compare our results with respect to the solution obtained by solving a linear system of equations in terms of running times.

Two-way Communication Orbit Queues with Server Vacation

We consider a single server retrial model with two streams of calls namely, incoming and outgoing calls. Each stream consists of multiple classes of calls. As part of the internal work load, presence of outgoing calls are always assumed in the system. Arrival of incoming calls obey the Poisson law. Upon seeing a busy server at its arrival epoch, an incoming call will be directed to an orbit according to the class it belongs to and tries to get an idle sever in gap of exponential amount of time with class dependent mean. Similar kind of attempt is being made by the outgoing calls also to reach an idle server. Once the sever becomes idle, if neither an incoming nor an outgoing call turns up for an exponential amount of time, the server goes for vacation and the vacation time is assumed to be exponential. Within each stream, service times of multiple classes of calls are assumed to be independent exponential with class dependent means. Matrix Analytic Method and regenerative approach are used to derive the explicit form of the steady state probabilities. Many performance measures are computed to analyse the system performance.

A Stochastic Queueing - Inventory System with Renewal Demands and Positive Lead Time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

A stochastic queueing - inventory system with renewal demands and positive lead time

This article analyses a stochastic inventory system with a service facility. This is an extended work of Yadavalli et al. (2008) by including the positive lead time. The customer arrives according to a renewal process and demanded item is delivered to the customer after performing an exponentially distributed service time. An (s, S) type ordering policy is adopted with exponentially distributed lead times. The stationary probability distribution for number of customers in the system and inventory level at arrival epoch and at arbitrary time point are derived. Some system performance measures in the steady state are computed and using these system performance measures the long-run expected cost rate is calculated. Since the long run expected cost rate is highly complex, the mixed integer distributed ant colony optimisation is used to obtain the optimal values. A sensitivity analysis to illustrate the effects of parameters and cost on the optimal values is also carried out in this work. [Received: 13 December 2018; Accepted: 10 October 2019]

An unreliable discrete-time retrial queue with probabilistic preemptive priority, balking customers and replacements of repair times

This paper deals with a discrete-time $Geo/G/1$ retrial queueing system with probabilistic preemptive priority and balking customers, in which the server is subject to starting failures and replacements in the repair times may occur with some probability. If the server is found busy at an arrival epoch, the newly arriving customer either interrupts the customer in service to begin its own service with probability $p$ or enters the orbit with probability $1-p$. When an arriving customer (external or repeated) finds the server free, he must turn on the server. If the server is activated successfully, the customer receives service immediately. Otherwise, the server undergoes a repair process. If an external arrival finds that the server is under repair, he decides either to join the orbit with probability $q$ or leaves the system completely (balking) with probability $1-q$. Applying the supplementary variable method and the generating function technique, we analyze the Markov chain underlying the considered queueing model and derive the stationary distributions under different system states, the generating functions for the number of customers in the orbit and in the system, as well as some crucial performance measures in steady state. Especially, some corresponding results under special cases are directly obtained by setting appropriate parameter values. Further, some numerical examples are provided to examine the effect of various system parameters on queueing characteristics. Finally, an operating cost function is formulated to discuss numerically a cost optimization problem.

Performance and sensitivity analysis of an M/G/1 queue with retrial customers due to server vacation

This paper treats an M/G/1 queue with retrial customers due to server vacation, which can be used to model a hospital service system. When an arriving customer finds the server on vacation at his arrival epoch, he either enters the retrial group with probability p or leaves the system with probability 1-p. Otherwise, the new arriving customer begins his service immediately if the server is idle upon his arrival or joins the original queue with infinite capacity under FCFS if the sever is busy at his arrival epoch. First, we develop the steady-state analysis by the supplementary variable method and the generating function approach, and give some important performance measures. Moreover, we obtain the Laplace-Stieltjes transform of the sojourn time in the system of an arbitrary customer and prove that the Little’s law still holds in our model. Finally, sensitivity analysis and optimal joining policy are given for illustrative purpose.

Analysis of a batch-service queue with variable service capacity, correlated customer types and generally distributed class-dependent service times

Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server’s capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain.We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance.

Multimodularity in the Stochastic Appointment Scheduling Problem with Discrete Arrival Epochs

We address the problem of designing appointment scheduling strategies in a stochastic environment accounting for patient no-shows, nonpunctuality, general stochastic service times, and unscheduled emergency walk-ins. A good appointment schedule seeks to help outpatient clinics to utilize their resources efficiently while containing patients’ waiting times. The task of identifying an optimal schedule is modeled as a nonlinear integer program, where the objective function is the outcome of stochastic analysis in transient state. We maintain the discrete nature of the appointment scheduling problem by considering arrival epochs with discrete supports. By looking at discrete-time snapshots of the random evolution of a single-server queueing model, we characterize probabilistically the system’s workload over time as a function of an appointment schedule, and we derive recursive expressions for the performance measures of interest. Subsequently, we unfold discrete convexity properties of the optimization problem. We prove that under general conditions the objective function is supermodular and componentwise convex. Under assumptions on patient punctuality, we prove that the optimal scheduling strategy minimizes a multimodular function, a property which guarantees that a locally optimal schedule is also globally optimal. The size of the local neighborhood, however, grows exponentially with the dimension of the problem. To the best of our knowledge, this study is the first to develop and implement an algorithm for minimizing locally a multimodular function over nonnegative integer vectors via submodular set-function minimization over ring families, a task that can be performed in polynomial time. This paper was accepted by David Simchi-Levi, operations management.

Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes

We develop CIATA, a combined inversion-and-thinning approach for modeling a nonstationary non-Poisson process (NNPP), where the target arrival process is described by a given rate function and its associated mean-value function together with a given asymptotic variance-to-mean (dispersion) ratio. CIATA is based on the following: (i) a piecewise-constant majorizing rate function that closely approximates the given rate function from above; (ii) the associated piecewise-linear majorizing mean-value function; and (iii) an equilibrium renewal process (ERP) whose noninitial interrenewal times have mean 1 and variance equal to the given dispersion ratio. Transforming the ERP by the inverse of the majorizing mean-value function yields a majorizing NNPP whose arrival epochs are then thinned to deliver an NNPP having the specified properties. CIATA-Ph is a simulation algorithm that implements this approach based on an ERP whose noninitial interrenewal times have a phase-type distribution. Supporting theorems establish that CIATA-Ph can generate an NNPP having the desired mean-value function and asymptotic dispersion ratio. Extensive simulation experiments substantiated the effectiveness of CIATA-Ph with various rate functions and dispersion ratios. In all cases, we found approximate convergence of the dispersion ratio to its asymptotic value beyond a relatively short warm-up period. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0828 .

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 “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect 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 order statistics of its service distribution. 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.

Equilibrium and pricing analysis for an unreliable retrial queue with limited idle period and single vacation

This paper treats an M/M/1 retrial queue with an unreliable server, where the server is subject to random breakdowns and repairs when he is serving a customer. Whenever the server becomes idle, he will spare a limited period to wait possible new arrivals or repeated customers if any. If no customer arrives during this period, the server will take a single vacation. Under a given reward-cost structure, arriving customers have to decide to balk or to enter the orbit if they find the server unavailable at their arrival epochs. Firstly, we study some important performance measures. Secondly, we derive the equilibrium joining strategies for the customers, respectively, in noncooperative and cooperative cases. Thirdly, we study the pricing problem that the social planner faces to eliminate the gap between the individual optimal and socially optimal strategies. Finally, we present some numerical examples to illustrate the effect of some system parameters on optimal joining probabilities and Maximum of social welfare.

Poissonian potential measures for Lévy risk models

This paper studies the potential (or resolvent) measures of spectrally negative Lévy processes killed on exiting (bounded or unbounded) intervals, when the underlying process is observed at the arrival epochs of an independent Poisson process. Explicit representations of these so-called Poissonian potential measures are established in terms of newly defined Poissonian scale functions. Moreover, Poissonian exit measures are explicitly solved by finding a direct relation with Poissonian potential measures. Our results generalize Albrecher et al. (2016) in which Poissonian exit identities are solved. As an application of Poissonian potential measures, we extend the Gerber–Shiu analysis in Baurdoux et al. (2016) to a (more general) Parisian risk model subject to Poissonian observations.

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