Poisson Arrivals Research Articles

Message from the editor in Chief

The year 2024 promises many exciting developments for the Global Journal of Computer Sciences: Theory and Research (GJCS). In addition, every year, the number of citations continues to rise strongly in proportion to the increase in the number of articles published. In this respect, first of all, we would like to thank all our authors who have contributed to the success of our journal. While reviewing our new issue, we saw articles from different authors from different countries: Turkey, India The multiculturalism policy followed in our globally-focused journal makes us happy with every new issue. We examine the following research areas in the current issue; “Determination of reinforcement diameters of reinforced concrete deep beams with genetic algorithms”, “Single server queueing model with Poisson arrival and gamma service time distribution”, “Sustainable innovation policy in non-walled technoparks” . We will continue to contribute to literature within the scope of Computer Science with our strong team. Thanks to all our authors the contributors to this issue. Hope to see you in our new issue. Assoc. Prof. Dr. Ezgi Pelin YILDIZ Global Journal of Computer Sciences: Theory and Research (GJCS) CHEF EDITOR

Single server queueing model with Poisson arrival and gamma service time distribution

Queueing theory is pivotal in understanding waiting lines and their pervasive role in everyday life. Its profound application spans across diverse sectors, including computer programming networks, healthcare, transportation, and so on. Researchers have applied many statistical distributions in analyzing queueing data. This article focuses on the analysis of a single server queueing system with Poisson input and service times distributed according to a two-parameter gamma distribution. The study elaborates on the construction and estimation of key queueing properties such as steady state equations, average number of customers in the system, average queue length, expected size of the non-empty queue, distribution of waiting time, average waiting time in the system, and average waiting time in the queue of the M/Gamma/1 queueing model using the two-parameter gamma distribution. Keywords: Gamma distribution; Poisson; service; single server; queuing model.

Dynamic load balancing in energy packet networks

Energy Packet Networks (EPNs) model the interaction between renewable sources generating energy following a random process and communication devices that consume energy. This network is formed by cells and, in each cell, there is a queue that handles energy packets and another queue that handles data packets. We assume Poisson arrivals of energy packets and of data packets to all the cells and exponential service times. We consider an EPN model with a dynamic load balancing where a cell without data packets can poll other cells to migrate jobs. This migration can only take place when there is enough energy in both interacting cells, in which case a batch of data packets is transferred and the required energy is consumed (i.e. it disappears). We consider that data packet also consume energy to be routed to the next station. Our main result shows that the steady-state distribution of jobs in the queues admits a product form solution provided that a stable solution of a fixed point equation exists. We prove sufficient conditions for irreducibility. Under these conditions and when the fixed point equation has a solution, the Markov chain is ergodic. We also provide sufficient conditions for the existence of a solution of the fixed point equation. We then focus on layered networks and we study the polling rates that must be set to achieve a fair load balancing, i.e., such that, in the same layer, the load of the queues handling data packets is the same. Our numerical experiments illustrate that dynamic load balancing satisfies several interesting properties such as performance improvement or fair load balancing.

On the time-dependent behavior of preemptive single-server queueing systems with Poisson arrivals

On the time-dependent behavior of preemptive single-server queueing systems with Poisson arrivals

Whittle index approach to multiserver scheduling with impatient customers and DHR service times

We consider the optimal scheduling problem in a multiserver queue with impatient customers belonging to multiple classes. We assume that each customer has a random abandonment time, after which the customer leaves the system if its service has not been completed before that. In addition, we assume that the scheduler is not able to anticipate the expiration of the abandonment times but only knows their distributions and how long each customer has been in the system. Many papers consider this scheduling problem under Poisson arrivals and linear holding costs assuming further that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. To tackle this tricky problem, we apply the Whittle index approach. Unlike the earlier papers, which were restricted to exponential service times, we allow the service time distributions for which the hazard rate is decreasing. The Whittle index approach is applied to the discrete-time multiserver queueing problem with discounted costs. As our main theoretical result, we prove that the related relaxed optimization problem is indexable and derive the corresponding Whittle index explicitly. Based on this discrete-time result, we develop a reasonable heuristic for the original continuous-time multiserver scheduling problem. The performance of the resulting policy is evaluated in the M/G/M setup by numerical simulations, which demonstrate that it, indeed, gives better performance than the other policies included in the comparison.

Central limit theorem for a birth–growth model with poisson arrivals and random growth speed

AbstractWe consider Gaussian approximation in a variant of the classical Johnson–Mehl birth–growth model with random growth speed. Seeds appear randomly in $\mathbb{R}^d$ at random times and start growing instantaneously in all directions with a random speed. The locations, birth times, and growth speeds of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed, the time distribution, and a weight function $h\;:\;\mathbb{R}^d \times [0,\infty) \to [0,\infty)$ , we prove a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Such models have previously been considered, albeit with fixed growth speed. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.

Gerber-Shiu Metrics for a Bivariate Perturbed Risk Process

We consider a two-dimensional risk model with simultaneous Poisson arrivals of claims. Each claim of the first input process is at least as large as the corresponding claim of the second input process. In addition, the two net cumulative claim processes share a common Brownian motion component. For this model we determine the Gerber–Shiu metrics, covering the probability of ruin of each of the two reserve processes before an exponentially distributed time along with the ruin times and the undershoots and overshoots at ruin.

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.

The kind of silence: managing a reputation for voluntary disclosure in financial markets

We create a continuous-time setting in which to investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies) in the period between two consecutive mandatory disclosure dates: one with full and transparent disclosure termed candid, the other, termed sparing, under which values only above a dynamic threshold are disclosed. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and the optimal times for management to switch between the two. The model presented develops a number of insights, based on a very simple ordinary differential equation characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black–Scholes model and Poisson arrival of signals of firm value. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about). Significantly, with constant pay-for-performance parameters, at most one switching can occur.

Open networks of infinite server queues with non-homogeneous multivariate batch Poisson arrivals

In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size with a multinomial distribution.

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.

MATHEMATICAL MODELLING IN WAITING QUEUE IN SHOPPING MALL

This project is an attempt to analyze the use of queuing theory in a local shopping mall. Also considers the mall as a multi server queuing system following Poisson arrival and Exponential service based system. A comparison of the results using MATLAB pertaining to double server and multiple server queuing system is also provided.

Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

Analytical Study of Two Serial Channels with Priority and Reneging

Most of the studies of queuing theory, which are useful in our daily life has been investigated by many researchers. The present research is the study of pre-emptive priority queuing system consisting two serial channels in stochastic environment. The impatient behavior of customer’s will be discussed with exponential service distribution and Poisson arrivals. Higher priority customers have pre-emptive priority over the low priority customers. The G.F. technique is used to derive the performance measures of high & low priority queues and assuming FCFS discipline in busy schedule of higher priority class. Also evaluate queue behavior graphically and discussed a special case at the end which shows utilization of channels.

Performance Analysis of Modified SRPT in Multiple-Processor Multitask Scheduling

In this paper we study the multiple-processor multitask scheduling problem in both deterministic and stochastic models, where each job have several tasks and is complete only when all its tasks are finished. We consider and analyze Modified Shortest Remaining Processing Time (M-SRPT) scheduling algorithm, a simple modification of SRPT, which always schedules jobs according to SRPT whenever possible, while satisfying the non-preemptive constraint. The M-SRPT algorithm is proved to achieve a competitive ratio of θ(log α + β) for minimizing response time, where α denotes the ratio between maximum job workload and minimum job workload, β represents the ratio between maximum non-preemptive task workload and minimum job workload. In addition, the competitive ratio achieved is shown to be optimal (up to a constant factor), when there are constant number of machines. We further consider the problem under Poisson arrival and general workload distribution (i. e., M/GI/N system), and show that M-SRPT achieves asymptotic optimal mean response time when the traffic intensity ρ approaches 1, if job size distribution has finite support. Beyond finite job workload, the asymptotic optimality of M-SRPT also holds for infinite job size distributions with certain probabilistic assumptions, for example, M/M/N system with finite task workload.

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.

Instability of LRTF multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the longest remaining time first (LRTF) service discipline. It is a reentrant line with two servers and four customer classes, Poisson arrivals and deterministic service times. Our instability proof can be generalized to a class of light-tailed stochastic model primitives, including independent, identically distributed exponential service times. A similar example of an unstable deterministic, strictly subcritical LRTF network is also given. Our arguments work for any service discipline such that there are no departures from any network station during its busy period.

Demand estimation with infrequent purchases and small market sizes

We propose a demand estimation method that allows for a large number of zero‐ sale observations, rich unobserved heterogeneity, and endogenous prices. We do so by modeling small market sizes through Poisson arrivals. Each of these arriving consumers solves a standard discrete choice problem. We present a Bayesian IV estimation approach that addresses sampling error in product shares and scales well to rich data environments. The data requirements are traditional market‐level data as well as a measure of market sizes or consumer arrivals. After presenting simulation studies, we demonstrate the method in an empirical application of air travel demand.

Internet of Drones: Improving Multipath TCP over WiFi with Federated Multi-Armed Bandits for Limitless Connectivity

We consider multipath TCP (MPTCP) flows over the data networking dynamics of IEEE 802.11ay for drone surveillance of areas using high-definition video streaming. Mobility-induced handoffs are critical in IEEE 802.11ay (because of the smaller coverage of mmWaves), which adversely affects the performance of such data streaming flows. As a result of the enhanced 802.11ay network events and features (triggered by beamforming, channel bonding, MIMO, mobility-induced handoffs, channel sharing, retransmissions, etc.), the time taken for packets to travel end-to-end in 802.11ay are inherently time-varying. Several fundamental assumptions inherent in stochastic TCP models, including Poisson arrivals of packets, Gaussian process, and parameter certainty, are challenged by the improved data traffic dynamics over IEEE 802.11ay networks. The MPTCP model’s state estimation differs largely from the actual network values. We develop a new data-driven stochastic framework to address current deficiencies of MPTCP models and design a foundational architecture for intelligent multipath scheduling (at the transport layer) considering lower layer (hybrid) beamforming. At the heart of our cross-layer architecture is an intelligent learning agent for actuating and interfacing, which learns from experience optimal packet cloning, scheduling, aggregation, and beamforming using successful features of multi-armed bandits and federated learning. We demonstrate that the proposed framework can estimate and optimize jointly (explore–exploit) and is more practicable for designing the next generation of low-delay and robust MPTCP models.

A Markov Decision Model for Managing Display-Advertising Campaigns

Problem definition: Managers in ad agencies are responsible for delivering digital ads to viewers on behalf of advertisers, subject to the terms specified in the ad campaigns. They need to develop bidding policies to obtain viewers on an ad exchange and allocate them to the campaigns to maximize the agency’s profits, subject to the goals of the ad campaigns. Academic/practical relevance: Determining a rigorous solution methodology is complicated by uncertainties in the arrival rates of viewers and campaigns, as well as uncertainty in the outcomes of bids on the ad exchange. In practice, ad hoc strategies are often deployed. Our methodology jointly determines optimal bidding and viewer-allocation strategies and obtains insights about the characteristics of the optimal policies. Methodology: New ad campaigns and viewers are treated as Poisson arrivals, and the resulting model is a Markov decision process, where the state of the system is the number of undelivered impressions in queue for each campaign type in each period. We develop solution methods for bid optimization and viewer allocation and perform a sensitivity analysis with respect to the key problem parameters. Results: We solve for the optimal dynamic, state-dependent bidding and allocation policies as a function of the number of ad impressions in queue, for both the finite horizon and steady-state cases. We show that the resulting optimization problems are strictly concave in the decision variables and develop and evaluate a heuristic method that can be applied to large problems. Managerial implications: Numerical analysis of our heuristic solution shows that its errors are generally small and that the optimal dynamic, state-dependent bidding policies obtained by our model are significantly better than optimal static policies. Our proposed approach is managerially attractive because it is easy to implement in practice. We identify the capacity of the impression queue as an important managerial control lever and show that it can be more effective than using higher bids to reduce delay penalties. We quantify potential operational benefits from the consolidation of ad campaigns, as well as merging ad exchanges. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.1142 .