Poisson Arrival Process Research Articles

Multi server queuing model with dynamic power shifting and performance efficiency factor

The need for high performance models in the queuing systems; availability in the fields of computing and communication systems and logistics management poses extensive challenges in design and development of the appropriate modeling. In this work, we propose a robust probabilistic model to mitigate these problems by appropriately choosing a multi-server queuing system with dynamic service facility. The arrival substances, such as packets or jobs or customers or consumers, follow a Poisson arrival process and these arrivals enter a queuing system according to FIFO discipline. The system designed in the system is armed with a fixed number of service stations, in which some servers are capable of allocating additional service capacity by adjusting dynamically. When a customer arrives at the system, if a free server is available, it is immediately served by one of the free processors; if no server is free, that is all are busy, the arrived job is accommodated in the queue and waits for service in the system. In this paper, we proposed a stochastic model to handle peak loads efficiently, boosting service capacity during burst arrivals. Numerical results presented in this paper are generated by the spectral expansion method to demonstrate the model’s performance, offering insights into its efficiency and accuracy. Furthermore, we derived some important special cases of speedup factor, which provide the mathematical estimation of system performance in terms of computation and communication times.

Local stability of traffic equilibria in an isotropic network

For a static economic model of auto traffic in an isotropic zone, this paper classifies possible equilibria into three types, by whether traffic is hypercongested and by the relative slopes of “supply” and “demand” curves. We then conduct a local stability analysis of each type when density, demand and the unit travel time (inverse speed) evolve gradually and simultaneously according to dynamical systems of differential equations. Some hypercongested equilibria may be stable when demand adjusts quickly enough to congestion. Other hypercongested equilibria, which have counterintuitive comparative statics, are never stable. Non-hypercongested equilibria can be unstable under special circumstances. A discrete event simulation with a dynamic Poisson arrival process supports the results of the formal analysis.

Electricity consumption simulation using random coefficient periodic autoregressive model

The concept of a continuous-time conditional linear random process involves a random kernel being integrated stochastically by a process with independent increments, which is often called a “generative process”. In cases where the “generative process” is Poisson, the resulting model represents an investigated signal as a sum of numerous stochastically dependent random impulses each of which occurs according to some inhomogeneous Poisson arrival process. This model can be applied to represent various processes related to energy consumption, such as electricity loads of electrical power systems, gas and water consumption, and other energy resources, while also considering the signals’ cyclostationarity, which is usually caused by the rhythmic nature of consumer behaviour. A member of the discretetime conditional linear cyclostationary random processes class is the random coefficient periodic autoregressive (RCPAR) model, which is appropriate for use in energy informatics, including estimation, forecasting, and computer simulation purposes. The primary objective of the paper is to establish a procedure for simulating the hourly electricity consumption of small and medium-sized enterprises using the RCPAR model, which has periodic parameters and creates cyclostationary properties while also accounting for the investigated process conditional heteroscedasticity. The statistical estimation step of the proposed procedure uses the general methodology for estimating the parameters of the RCPAR model and the methods of statistical analysis of cyclostationary signals. This step is used to identify the simulation characteristics. The simulation step is based on the methods of cyclostationary white noise generation and its transformation by a digital linear filter with random parameters. The last ones are obtained using the Gaussian random vectors computer simulation methods, taking into account the cyclostaionarity property.

MAESTRO-X: Distributed Orchestration of Rotary-Wing UAV-Relay Swarms

This work details a scalable framework to orchestrate a swarm of rotary-wing UAVs serving as cellular relays to facilitate beyond line-of-sight connectivity and traffic offloading for ground users. First, a Multiscale Adaptive Energyconscious Scheduling and TRajectory Optimization (MAESTRO) framework is developed for a single UAV. Aiming to minimize the time-averaged latency to serve user requests, subject to an average UAV power constraint, it is shown that the optimization problem can be cast as a semi-Markov decision process, and exhibits a multiscale structure: outer actions on radial wait velocities and terminal service positions minimize the longterm delay-power trade-off, optimized via value iteration; given these outer actions, inner actions on angular wait velocities and service trajectories minimize a short-term delay-energy cost; finally, rate adaptation is embedded along the trajectory to leverage air-to-ground channel propagation conditions. A novel hierarchical competitive swarm optimization scheme is developed in the inner optimization, to devise high-resolution trajectories via iterative pair-wise updates. Next, MAESTRO is eXtended to UAV swarms (MAESTRO-X) via scalable policy replication, enabled by a decentralized command-and-control network augmented with: (1) spread maximization to proactively position UAVs to serve future requests; (2) consensus-driven conflict resolution to orchestrate scheduling decisions based on delay-energy costs including queuing dynamics; (3) adaptive frequency reuse to improve spectrum utilization across the network; and (4) a piggybacking mechanism allowing UAVs to serve multiple ground users simultaneously. Numerical evaluations show that, for user requests of 10 Mbits, generated according to a Poisson arrival process with rate 0.2 req/min/UAV, single-agent MAESTRO offers 3.8× faster service than a high-altitude platform and 29% faster than a static UAV deployment; moreover, for a swarm of 3 UAV-relays, MAESTRO-X delivers data payloads 4.7× faster than a successive convex approximation scheme; and remarkably, a single UAV optimized via MAESTRO outclasses 3 UAVs optimized via a deep-Q network by 38%.

Scaling Limits of a Tandem Queue with Two Infinite Orbits

This paper considers a tandem queueing network with a Poisson arrival process of incoming calls, two servers, and two infinite orbits by the method of asymptotic analysis. The servers provide services for incoming calls for exponentially distributed random times. Blocked customers at each server join the orbit of that server and retry to enter the server again after an exponentially distributed time. Under the condition of low retrial rates, we prove that the joint stationary distribution of scaled numbers of calls in the orbits weakly converges to a two-variable Normal distribution.

Performance Analysis and Energy Efficiency of MU- (OFDMA & MIMO) Based Hybrid MAC Protocol of IEEE 802.11ax WLANs

Future wireless networks are characterized by densely populated stations (STAs) with limited battery capacity. These networks demand high aggregate throughput, increased energy efficiency, and high Random Access (RA) capability, i.e., the system's ability to allow new STAs to join the network quickly. WiFi-6 (also known as IEEE 802.11 ax) is the new generation WLAN standard proposed to meet these diverse system requirements. In this paper, we analyze the effect of Multi-User Multi-Input Multi-Output (MU-MIMO) & MU-Orthogonal Frequency Division Multiple Access (MU-OFDMA)-based Hybrid Access (HA) mechanism of WiFi-6 systems. We propose a fixed Point model for analyzing this HA mechanism. By solving the system model, we investigate the impact on critical system metrics such as throughput, energy efficiency, Random Access capability, and delay due to a variation in (i) the number of resource units (RUs), (ii) the channel sounding interval, and (iii) the Target Wake-Up time interval. We have studied the effect of the non-saturated traffic model driven by the Poisson and Pareto arrival processes. Further, we have modeled the buffer status-based dynamic scheduling of RUs for the MU-OFDMA-based HA mechanism. We have also analyzed the novel Multi-Traffic Identifier-based frame aggregation feature. Further, we have analyzed the hand-off mechanism for a multi-BSS (Basic Service Set) scenario using the Stochastic geometry-based mathematical tool. We have used Matlab-based simulations for our study and found that the simulation results closely match the analytical results. Our study sheds light on the trade-off that exists between different performance metrics of the 802.11ax HA mechanism.

Real-Time HAP-Assisted Vehicular Edge Computing for Rural Areas

Non-Terrestrial Networks (NTNs) are expected to be a key component of 6th generation (6G) networks to support broadband seamless Internet connectivity and expand the coverage even in rural and remote areas. In this context, High Altitude Platforms (HAPs) can act as edge servers to process computational tasks offloaded by energy-constrained terrestrial devices such as Internet of Things (IoT) sensors and ground vehicles (GVs). In this letter, we analyze the opportunity to support Vehicular Edge Computing (VEC) via HAP in a rural scenario where GVs can decide whether to process data onboard or offload them to a HAP. We characterize the system as a set of queues in which computational tasks arrive according to a Poisson arrival process. Then, we assess the optimal VEC offloading factor to maximize the probability of real-time service, given latency and computational capacity constraints.

Using infinite server queues with partial information for occupancy prediction

Motivated by demand prediction for the custodial prison population in England and Wales, this paper describes an approach to the study of service systems using infinite server queues, where the system has non-empty initial state and the elapsed time of individuals initially present is not known. By separating the population into initial content and new arrivals, we can apply several techniques either separately or jointly to those sub-populations, to enable both short-term queue length predictions and longer-term considerations such as managing congestion and analysing the impact of potential interventions. The focus in the paper is the transient behaviour of the M t / G / ∞ queue with a non-homogeneous Poisson arrival process and our analysis considers various possible simplifications, including approximation. We illustrate the approach in that domain using publicly available data in a Bayesian framework to perform model inference.

A dual risk model with additive and proportional gains: ruin probability and dividends

AbstractWe consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

Strategic behavior in queues with arrival rate uncertainty

This paper shows how to account for arrival rate uncertainty in the analysis of queueing models with strategic customer behavior. We consider a general queueing model with a Poisson arrival process whose rate is random, and realized once for the entire process. We show that the distribution of the arrival rate at arrival instants is the size-biased counterpart of the original distribution. In particular, the ASTA (arrivals see time averages) property does not hold but rather a rate-biased version of it which we define and coin by the term RASTA (Rate-biased ASTA). We show that the RASTA phenomenon plays a crucial role in the analysis of strategic behavior of customers who evaluate the consequences of the actions they take upon arrival. By studying such a system with a single server and strategic customers who decide whether to join or balk without observing the queue, we exemplify the importance of RASTA in deriving the equilibrium behavior.

Callable products for an implicit time model with dependent demands

AbstractCapacity providers such as airlines and hotels have traditionally increased revenues by practicing market segmentation and revenue management, enabling them to sell the same capacity pool to different consumers at different prices. Callable products can enhance profits and improve consumers' welfare by allowing the firm to broker capacity between consumers with different willingness to pay. A consumer who buys a callable product gives the capacity provider the right to recall capacity at a prespecified recall price. This article studies callable products in the context of the model most commonly used in industry, which handles time implicitly imposing fewer restrictions on the nature of randomness compared to the Poisson arrival process favored in academia. In the implicit time model, capacity providers set booking limits to protect capacity for future high‐fare demand. Our numerical study identifies conditions where callable products result in significant gains in profits.

On optimal stochastic jumps in multi server queue with impatient customers via stochastic control

<p style='text-indent:20px;'>In this paper, a queuing system as multi server queue, in which customers have a deadline and they request service from a random number of identical severs, is considered. Indeed there are stochastic jumps, in which the time intervals between successive jumps are independent and exponentially distributed. These jumps will be occurred due to a new arrival or situation change of servers. Therefore the queuing system can be controlled by restricting arrivals as well as rate of service for obtaining optimal stochastic jumps. Our model consists of a single queue with infinity capacity and multi server for a Poisson arrival process. This processes contains deterministic rate <inline-formula><tex-math id="M1">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and exponential service processes with <inline-formula><tex-math id="M2">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> rate. In this case relevant customers have exponential deadlines until beginning of their service. Our contribution is to extend the Ittimakin and Kao's results to queueing system with impatient customers. We also formulate the aforementioned problem with complete information as a stochastic optimal control. This optimal control law is found through dynamic programming.</p>

The price of anarchy in loss systems

AbstractAssume a multi‐server memoryless loss system. Each server is associated with a service rate and a value of service. Customers from a common Poisson arrival process are routed to the servers in an unobservable way, where the goal is to maximize the long‐run expected reward per customer (which is the service value times the probability that the customer is not blocked). We first solve this problem under two criteria: social optimization and Nash equilibrium. Our main result is that the price of anarchy, defined as the ratio between the expected gain under the two criteria, is bounded by . We also show, via examples, that this bound is tight for any number of servers.

Validity of Exponential Distribution for Modelling Inter-failure Arrival Times of Windows based Industrial Process Control Data Exchange

Using a reliability function with a known classical distribution, at an instance in time, the chances of an activity success can be predicted. Based on this curve, a proper reset period can be appointed. This would help to keep the risk under control. For this purpose, a test bed mimicking the machine to machine (M2M) communication of a Digital Oil Rig is employed. It contains an Embedded Computer Server and an ordinary computer acting as a client querying data from the server. On this testbed, as the communication protocol Open Platform Communications - Unified Architecture (OPC UA) is run. In “most occasions, the arrival processes are assumed as a Poisson Arrival Process. The Question of a concern: "Is this assumption valid in case of an Embedded Computer responding to queries?". Our results indicate that the failure arrivals are independent identically distributed (IID) and can be modeled with an Exponential Distribution by accepting a reasonable error.

Queing Theory, a Tool for Polio Eradication in Nigeria

To curb the spread of contagious diseases and the recent polio outbreak in Nigeria, health departments must set up and operate clinics to dispense medications or vaccines. Residents arrive according to an external (not necessarily Poisson) Arrival process to the clinic. When a resident arrives, he goes to the first workstation, based on his or her information, the resident moves from one workstation to another in the clinic. The queuing network is decomposed by estimating the performance of each workstation using a combination of exact and approximate models. A key contribution of this research is to introduce approximations for workstations with batch arrivals and multiple parallel servers, for workstations with batch service processes and multiple parallel servers, and for self service workstations. We validated the models for likely scenarios using data collected from one of the states vaccination clinics in the country during the vaccination exercises.

A Change-Detection-Based Thompson Sampling Framework for Non-Stationary Bandits

We consider a non-stationary two-armed bandit framework and propose a change-detection-based Thompson sampling (TS) algorithm, named TS with change-detection (TS-CD), to keep track of the dynamic environment. The non-stationarity is modeled using a Poisson arrival process, which changes the mean of the rewards on each arrival. The proposed strategy compares the empirical mean of the recent rewards of an arm with the estimate of the mean of the rewards from its history. It detects a change when the empirical mean deviates from the mean estimate by a value larger than a threshold. Then, we characterize the lower bound on the duration of the time-window for which the bandit framework must remain stationary for TS-CD to successfully detect a change when it occurs. Consequently, our results highlight an upper bound on the parameter for the Poisson arrival process, for which the TS-CD achieves asymptotic regret optimality with high probability. Finally, we validate the efficacy of TS-CD by testing it for edge-control of radio access technique (RAT)-selection in a wireless network. Our results show that TS-CD not only outperforms the classical max-power RAT selection strategy but also other actively adaptive and passively adaptive bandit algorithms that are designed for non-stationary environments.

Performance Approximation for Time-Dependent Queues with Generally Distributed Abandonments

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

Detection of an appropriate pharmaceutical company to get a suitable vaccine against COVID-19 with minimum cost under the quality control process.

Some international pharmaceutical companies have succeeded in producing vaccines against COVID‐19. Countries all over the world have aimed to obtain these vaccines with minimum cost. We consider a set of K‐independent Markovian waiting lists. Each list contains a set of countries, where each one of them has an exponential service time and a Poisson arrival process. These companies differ in some characteristics such as the vaccine production cost and the speed of the required quantity delivery. We present a new detection model that helps in providing an appropriate decision to choose a suitable company. Moreover, the concept of balking and the retention of reneged countries is taken into consideration under the quality control process of each waiting list. Under steady state, we face an interesting and difficult discrete stochastic optimization problem. Its solution gives an optimal distribution of the searching effort, which is bounded by a known probability distribution. A simulation study has been derived to get the minimum value of the paid cost random values. The highest service rate, the total expected profit of each queuing system, and the optimum performance measures, which depend on this cost, have been obtained to show the effectiveness of this model.

Hawkes processes in insurance: Risk model, application to empirical data and optimal investment

In this paper we study a risk model with claim arrivals based on general compound Hawkes processes and show that it is suitable to model empirical insurance data. We review a law of large numbers and functional central limit theorem for this model and derive a pure diffusion approximation which allows analytical calculation of finite-time and infinite-time ruin probabilities. We use the approximation to study the influence of replacing the classical Poisson arrival process by a general compound Hawkes process on optimal investment strategies for an insurer in an incomplete market by applying results from asset–liability management.

Strategic behavior in queues with arrival rate uncertainty

We consider a general queueing model with a Poisson arrival process whose rate is random, and realized once for the entire process. We show that the distribution of the arrival rate at arrival instants is the size-biased counterpart of the original distribution. In particular, the ASTA (arrivals see time averages) property does not hold but rather a rate-biased version of it that we define and coin by the term RASTA (Rate-biased ASTA). We show that the RASTA phenomenon plays a crucial role in the analysis of strategic behavior of customers who evaluate the consequences of the actions they take upon arrival. We study such a system with a single server and strategic customers who decide whether to join or balk without observing the queue.