Independent Poisson Processes Research Articles

Spectrally negative Lévy risk model under mixed ratcheting-periodic dividend strategies

In this article, we consider the mixed ratcheting-periodic dividend strategies for spectrally negative Lévy risk model, in which dividend payments can both be made continuously without falling and discretely at the jump times of an independent Poisson process. The expected net present value of dividends paid up to ruin and the Laplace transform of the ruin time are obtained by using Lévy fluctuation theory. All the results are expressed in terms of scale functions. Finally, numerical results for Brownian motion with drift are given.

Capacity management for a leasing system with different equipment and batch demands

This work explores the admission and capacity allocation for a leasing system with two types of equipment and three kinds of batch demands: elementary specified, premium specified, and unspecified demands. The demands arrive following mutually independent Poisson processes, and the rental duration of equipment follows a negative exponential distribution. The lessor can satisfy partially the specified demands with the required type of equipment and satisfy partially the unspecified demands with any type of equipment. The objective is to maximize the expected discounted revenue. We formulate this problem as a Markov decision process, prove the anti‐multimodularity of the value function, and characterize the structure of the optimal policy. We show that the optimal policy has a simple structure and is characterized by state‐dependent rationing and priority thresholds. Moreover, a solution algorithm is proposed to calculate the optimal policy. We study the impacts of the system state on the optimal action and find that the optimal action has limited sensitivity to the system state. Numerical studies are conducted to compare the performance of the optimal policy with that of two heuristic methods and to derive some managerial insights by analysis. We further discuss batch acceptance.

Sequential common rate decrease detection, isolation, and estimation in multiple Poisson processes

ABSTRACT We consider the detection of a common change when M independent Poisson processes are monitored simultaneously where only a portion of the processes have rate decrease after the change time. In contrast to the detection of rate increase (Wu and Wu (2022)), the individual CUSUM and Shiryayev–Roberts (S–R) processes need to be calculated continuously in parallel. A combined CUSUM-SR procedure is proposed by using the sum of S–R processes as the detection process for a common change and after the detection, the M individual CUSUM processes are used to isolate the changed panels with FDR control. The medians of change time estimates from the individual CUSUM or S–R processes from the isolated changed panels are then used to estimate the common change time. Coal mining disaster data in USA are used for illustration.

Closed-Form Characterization of the MGF of AoI in Energy Harvesting Status Update Systems

This paper considers a real-time status update system in which an energy harvesting (EH)-powered transmitter node observes some physical process, and sends its sensed measurements in the form of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">status updates</i> to a destination node. The status update and harvested energy packets are assumed to arrive at the transmitter according to independent Poisson processes, and the service time of each status update is assumed to be exponentially distributed. We quantify the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">freshness</i> of status updates when they reach the destination using the concept of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Age of Information (AoI)</i> . Unlike most of the existing analyses of AoI focusing on the evaluation of its average value when the transmitter is not subject to energy constraints, our analysis is focused on understanding the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">distributional properties</i> of AoI through the characterization of its moment generating function (MGF). In particular, we use the stochastic hybrid systems (SHS) framework to derive closed-form expressions of the MGF of AoI under several queueing disciplines at the transmitter, including non-preemptive and preemptive in service/waiting strategies. Using these MGF results, we further obtain closed-form expressions for the first and second moments of AoI in each queueing discipline. We demonstrate the generality of this analysis by recovering several existing results for the corresponding system with no energy constraints as special cases of the new results. Our numerical results verify the analytical findings, and demonstrate the necessity of incorporating the higher moments of AoI in the implementation/optimization of real-time status update systems rather than just relying on its average value.

Sequential common change detection, isolation, and estimation in multiple poisson processes

In this article, motivated by detecting the occurrence of an epidemic when the arrival rates of patients increase in a portion of M panels or detecting the deterioration of a system composed of M independent components that causes an increase in failure rates in a portion of components, we consider the detection of a common change when M independent Poisson processes are monitored simultaneously where only a portion of the processes have rate increases after the change time. M individual cumulative sum (CUSUM) processes and Shiryaev-Roberts (S-R) processes are calculated recursively in parallel at each pooled arrival time. A systematic procedure is proposed by using the sum of M S-R processes as the detection process for a common change. After the detection, the M individual CUSUM processes are used to isolate the changed panels with false discovery rate (FDR) control and then the medians of the change time estimates from each individual CUSUM process or S-R process based on the isolated panels are used to estimate the common change time. The model can be generalized to different prechange rates, jittered change time, and unknown postchange rates.

Performance of Stochastic Inventory System with a Fresh Item, Returned Item, Refurbished Item, and Multi-Class Customers

This paper deals with an integrated and interconnected stochastic queuing-inventory system with a fresh item, a returned item, and a refurbished item. This system provides a multi-type service facility to an arriving multi-class customer through a dedicated channel. It sells fresh and refurbished items, buys used items from customers, refurbishes the used items for resale, and provides a repair service for defective items. The assumption of purchasing a used item from the customer and allowing them to buy a fresh item is a new idea in stochastic queuing-inventory modeling. To do so, this system has four parallel queues to receive four classes of customers and five dedicated servers to provide a multi-type service facility. Customers are classified according to the type of service they require. Each class of arrival follows an independent Poisson process. The service time of each dedicated server is assumed to be exponentially distributed and independent. This system assumes an instantaneous ordering policy for the replenishment of a fresh item. In the long run of this considered system, the joint probability distribution of the seven-dimensional stochastic process, significant system performance measures, and the optimum total cost are to be derived using the Neuts matrix geometric technique. The main objective of the system was to increase the occurrence of all kinds of customers by providing a multi-type service facility in one place. Buying a used item is unavoidable in an emerging society because it helps form a green society. Furthermore, the numerical result shows that the assumption of a system that allows a customer to sell their used item and purchase a new item will increase the number of customers approaching the system.

Analysis of a single server serving three queues m[x 1 ] /g1/1, m[x 2 ] /g2/1 , m[x 3 ] /g3/1 with priority services, working breakdown, modified bernoulli vacation

In this paper considers M[X1]/G1/1, M[X2]/G2/1 , M[X3]/G3/1 general queueing system with priority services . Three types of customers from different classes arrive at the system in different independent Poisson process. The server follows the non preemptive priority rule subject to working breakdown, and modified Bernoulli vacation with general (arbitrary) vacation periods. After completing the service, if there is no high priority customers present in the system. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained. Also the average number of customer in the priority and non priority, preemptive priority queue and the average waiting time are derived.

Flexibility can Hurt Dynamic Matching System Performance

We study the performance of stochastic matching models with general compatibility graphs. Items of different classes arrive to the system according to independent Poisson processes. Upon arrival, an item is matched with a compatible item according to the First Come First Matched discipline and both items leave the system immediately. If there are no compatible items, the new arrival joins the queue of unmatched items of the same class. Compatibilities between item classes are defined by a connected graph, where nodes represent the classes of items and the edges the compatibilities between item classes. We show that such a model may exhibit a non intuitive behavior: increasing the matching flexibility by adding new edges in the matching graph may lead to a larger average population at the steady state. This performance paradox can be viewed as an analog of the Braess paradox. We show sufficient conditions for the existence or non-existence of this paradox. This performance paradox in matching models appears when specific independent sets are in saturation, i.e., the system is close to the stability condition.

Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls

This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of N−1 (N≥2) categories occur according to N−1 independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit.

Age of Information in Multi-source Updating Systems Powered by Energy Harvesting

This paper considers a multi-source updating system in which a transmitter node powered by energy harvesting (EH) sends status updates about multiple sources of information to a destination, where the freshness of status updates is measured in terms of Age of Information (AoI). The status updates of each source and harvested energy packets are assumed to arrive at the transmitter according to independent Poisson processes, and the service time of each status update is assumed to be exponentially distributed. Further, the transmitter can harvest energy only when its server is idle. Unlike most of the existing queueing-theoretic analyses of AoI that focus on characterizing its average when the transmitter has a reliable energy source and is hence not powered by EH (referred henceforth as a non-EH transmitter), our analysis is focused on understanding the distributional properties of AoI in multi-source systems through the characterization of its moment generating function (MGF). In particular, we use the stochastic hybrid systems (SHS) framework to derive closed-form expressions of the average/MGF of AoI under several queueing disciplines at the transmitter, including non-preemptive and source-agnostic/source-aware preemptive in service strategies. The generality of our results is demonstrated by recovering several existing results as special cases.

An inventory system with replacement and refurbishment of failed items

We consider a continuous review inventory system with two types of inventory, namely type-I and type-II. Type-I and type-II inventory, respectively, contain fresh and refurbished items. Two streams of customers, called type-A and type-B customers arrive to the system in which type-A customer demands for an item either from type-I or type-II inventory according to a Bernoulli trial and type-B customer occurs to replace the failed item with new one from type-I inventory. The arrival time points of type-A and type-B customers follow independent Poisson Processes. Failed items are stored in a place, called pool, of infinite size for refurbishment. After refurbishment, these items are kept in type-II inventory. The refurbishment of pooled item is stopped once type-II inventory reaches its maximum level S 2 and is again commenced only when type-II inventory level falls to s 2. The refurbishment time follows exponential distribution. Type-I inventory is replenished according to a (s, S) policy with exponential lead time. The joint pdf of number of items in the pool, type-II inventory level, status of refurbishment and type-I inventory level is obtained in the steady state. Various system performance measures are derived and total expected cost rate is calculated. Results are illustrated numerically.

Stochastic non-bipartite matching models and order-independent loss queues

The problem of appropriately matching items subject to compatibility constraints arises in a number of important applications. While most of the literature on matching theory focuses on a static setting with a fixed number of items, several recent works incorporated time by considering a stochastic model in which items of different classes arrive according to independent Poisson processes and assignment constraints are described by an undirected non-bipartite graph on the classes. In this paper, we prove that the Markov process associated with this model has the same transition diagram as in a product-form queueing model called an order-independent loss queue. This allows us to adapt existing results on order-independent (loss) queues to stochastic matching models and, in particular, to derive closed-form expressions for several performance metrics, like the waiting probability or the mean matching time, that can be implemented using dynamic programming. Both these formulas and the numerical results that they allow us to derive are used to gain insight into the impact of parameters on performance. In particular, we characterize performance in a so-called heavy-traffic regime in which the number of items of a subset of the classes goes to infinity while items of other classes become scarce.

Exact analysis of production lines with Coxian-2-distributed processing times and parallel machines

This study considers production lines where each station has parallel machines with 2-phase Coxian processing times. The setting in this paper is designed specifically for scrutiny of the replenishment of raw materials and finished goods inventories with intermediate buffers in-between stations. Each buffer has a limit of capacity. Raw material supply and demand for finished goods are generated according to independent stationary Poisson processes. Coxian-2 processing times can be utilized to model failure-prone machines with exponential service times, times to failure, and repair times. The second phase of Coxian-2 can also be considered as a rework operation visited with a predefined probability. We model the line as a continuous-time Markov chain and propose recursive algorithms to generate the transition rate matrix. Although the general recursive form is specific to 3-station 4-buffer lines, routines for calculating the number of states and generating the states work for any M−station (M + 1)-buffer systems. The developed model allows obtaining steady-state distribution and performance metrics such as throughput, the average number of items in the buffers, and average system cost consisting of production, holding, and shortage costs. Furthermore, we enrich our study with numerical experiments and analyze the impacts of buffer capacities, processing rates of the machines and the number of parallel machines on the system performance. Moreover, the exact analysis provided in this paper can also be used as the decomposition block for the performance analysis of longer lines.

A triple [formula omitted]-thresholds base-stock policy subject to uncertainty environment, returns and order cancellations

This paper studies a continuous-review (S,s,ℓ) inventory model that extends the familiar (S,s)-policy by including returns and order cancellations. The demand and return processes follow independent compound Poisson processes. We assume a random shelf life, a random lead time and early losses. Our objective is to find a replenishment level S, a reorder level s, and a threshold cancellation level ℓ that minimizes the total expected cost per time unit under lost sales. The cost structure includes variable and fixed costs for an order, a holding cost, a penalty cost, a handling cost for returns, a relocation cost, and costs for end of life and early loss. Via Markovian framework, we derive the steady-state probabilities for the inventory level, and construct closed-form expressions for the average cost functions. Using a numerical study, we investigate how the best S,s and ℓ, are impacted by the system’s parameters, such as returns, demands (rate, mean and variability), and order cancellations. We illustrate the impact of the return flow on the best policy and the cost, and show that ignoring returns may yield a substantial loss. Consequently, we compare our model to a relatively simple one with no returns but with less demand, and specify when the latter model can be used as a good approximation. Finally, we identify situations where an earlier-cancellation policy is more cost effective. Our framework is completed by providing two efficient search algorithms for numerically finding the optimal parameters. We show that both algorithms significantly reduce the computational effort compared to the basic algorithm.

Optimal timing of investments modeled as perpetual American options in a Levy market

Using a real option approach, this paper models an arbitrary real life investment, which typically has a long maturity date, as a perpetual American call option in a Levy market. Expressions for the moments, characteristic function and infinitesimal generator of the associated jump-diffusion Levy process, defined by two independent compound Poisson processes and two correlated standard Brownian motions, are derived and these fundamental results are employed to determine the optimal time for investment. An application of the results to a Build Operate and Transfer investment is furnished.

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.

Ergodic control of diffusions with random intervention times

AbstractWe study an ergodic singular control problem with constraint of a regular one-dimensional linear diffusion. The constraint allows the agent to control the diffusion only at the jump times of an independent Poisson process. Under relatively weak assumptions, we characterize the optimal solution as an impulse-type control policy, where it is optimal to exert the exact amount of control needed to push the process to a unique threshold. Moreover, we discuss the connection of the present problem to ergodic singular control problems, and illustrate the results with different well-known cost and diffusion structures.

On a double barrier hybrid dividend strategy in a compound Poisson risk model with stochastic income

The paper features a hybrid dividend payment strategy on an insurance surplus with stochastic income. The hybrid dividend strategy works as follows: A delay-clock starts whenever the surplus up-shoots, due to a premium arrival, to a level between barriers ‘k’ and ‘ $$ \ell $$ ’, such that $$ 0 \leqslant u \leqslant k \leqslant \ell < \infty $$ . Further, the clock restarts along with each premium or claim arrival, provided the surplus still lies between ‘k’ and ‘ $$ \ell $$ ’. However, when the surplus exits to a level outside the range between two barriers, the clock terminates. The insurer pays a dividend of an amount exceeding the level ‘k’ to shareholders on two scenarios. In one situation, the insurer pays dividend if delay-clock alarms before a premium or claim arrival. In the other situation, the insurer pays dividend instantaneously if surplus exceeds the second barrier ‘ $$ \ell $$ ’. The aggregate premiums and claims are both following mutually independent compound Poisson processes, both having exponential amount sizes which are further independent to corresponding inter-arrival times. A scheme for determining analytic expressions for piecewise Volterra integral equations, with boundary conditions, which are satisfied by the expected discounted dividend paid before ruin and the Gerber–Shiu function are discussed in detail when the delay-clock also follows an exponential distribution. We discuss a numerical example to show the tractability of the scheme and to brief the impact of hybrid dividend strategy over the classical De Finetti barrier strategy for paying dividends.

The Nonlinear Time Sequence Analysis in the Alpine-Himalayan Earthquake Zone

The characteristics of the earthquake activity in the Eurasian earthquake zone, which is the second largest earthquake zone in the world, was investigated by researchers. The earthquake activity of the Eurasian earthquake zone was analysed in various disciplines, such as earth dynamics, rock mechanics, geology and tectonics. The emergence of fractal theory provided a new direction in exploring the characteristics of the earthquake activity in the Eurasian earthquake zone. This study processed the data on the earthquake activity in the Eurasian earthquake zone by self-similarity method and scaled invariant feature test and used the rescaled range analysis method to analyse the nonlinear time series fractal characteristics of the earthquake activity in the Eurasian earthquake zone. Results show that the time series of earthquake activity in the study area is not an independent Poisson process, which exhibits the characteristics of scale invariance and long-range correlation. Approximately 80% of the H values of the earthquake activity iteratively increase and decrease for moderate earthquakes, which is mainly concentrated during the increasing stage. The time difference of the H value between the two-neighbouring earthquake shows that the H value fluctuates in the active earthquake region and is stationary in the relatively stable region. Strong earthquakes will likely occur in the next few years because the H value fluctuates.

Two queues with time-limited polling and workload-dependent service speeds

In this paper, we study a single-server polling model with two queues. Customers arrive at the queues according to two independent Poisson processes. The server spends random amounts of time in each queue, regardless of the amounts of work present at the queues. The service speed is not constant; it is assumed that the server works at speed at queue i when its current workload equals xi , i = 1, 2. We first focus on the case that all visit times are constant. In the two-queue case, we then compute the Laplace–Stieltjes transform (LST) of the steady-state joint workload distribution. Using a different method in which we exploit the independence of the workloads at visit endings, we compute the joint LST of workloads in the case of an arbitrary number of queues with constant visit times. Next, we consider a two-queue polling model with constant visit times at the first queue and general visit times at the second, and we derive the marginal workload distribution at the first queue. We also investigate the case of a two-queue polling model with exponentially distributed visit times. We determine the steady-state marginal workload distributions, and we formulate a two-dimensional Volterra integral equation for the LST of the steady-state joint workload distribution. We finally show that this equation can be solved by a fixed-point iteration.