Nonstationary Arrival Processes Research Articles

Precise large deviations in a bidimensional risk model with arbitrary dependence between claim-size vectors and waiting times

Consider a bidimensional risk model in which an insurer simultaneously confronts two types of claims sharing a common non-stationary arrival process, and the claim-sizes {X→k;k≥1} form a sequence of i.i.d. random vectors with nonnegative components being dependent on each other. Supposing that the univariate marginal distributions of the claim-size vectors have dominatedly varying tails, precise large deviations for the aggregate amount of claims are obtained, by allowing that the claim-size vectors and claim inter-arrival (waiting) times are arbitrarily dependent.

Applying Deep Reinforcement Learning to Dynamic Staff Scheduling in a Service System with Nonstationary Arrival Process; a Case Study of Grocery Store

Applying Deep Reinforcement Learning to Dynamic Staff Scheduling in a Service System with Nonstationary Arrival Process; a Case Study of Grocery Store

Asymptotic ruin probabilities for a bidimensional risk model with heavy-tailed claims and non-stationary arrivals

This article studies a bidimensional risk model, in which an insurer simultaneously confronts two kinds of claims sharing a common non-stationary arrival process. Assuming that the arrival process satisfies a large deviation principle and the claim-size distributions are heavy tailed, an asymptotic formula for the corresponding ruin probability of this bidimensional risk model is obtained.

STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims

In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.

Modelling and simulating non-stationary arrival processes to facilitate analysis

This paper introduces a method to model and simulate non-stationary, non-renewal arrival processes that depends only on the analyst setting intuitive and easily controllable parameters. Thus, it is suitable for assessing the impact of non-stationary, non-exponential, and non-independent arrivals on simulated performance when they are suspected. A specific implementation of the method is also described and provided for download.

Transforming Renewal Processes for Simulation of Nonstationary Arrival Processes

Simulation models of real-life systems often assume stationary (homogeneous) Poisson arrivals. Therefore, when nonstationary arrival processes are required, it is natural to assume Poisson arrivals with a time-varying arrival rate. For many systems, however, this provides an inaccurate representation of the arrival process that is either more or less variable than Poisson. In this paper we extend techniques that transform a stationary Poisson arrival process into a nonstationary Poisson arrival process (NSPP) by transforming a stationary renewal process into a nonstationary, non-Poisson (NSNP) arrival process. We show that the desired arrival rate is achieved and that when the renewal base process is either more or less variable than Poisson, then the NSNP process is also more or less variable, respectively, than an NSPP. We also propose techniques for specifying the renewal base process when presented properties of, or data from, an arrival process and illustrate them by modeling real arrival data.

Convex stability and asymptotic convex ordering for non-stationary arrival processes

The notion of convex stability for a sequence of non-negative random variables is discussed in the context of several applications.

Local predictive resource reservation for handoff in multimedia wireless IP networks

This paper presents two new methods that use local information alone to predict the resource demands of and determine resource reservation levels for future handoff calls in multimedia wireless IP networks. The proposed methods model the instantaneous resource demand directly. This differs from most existing methods that derive the demands from modeling the factors that impact the demands. As a result, the proposed methods allow new and handoff calls to: (1) follow non-Poisson and/or nonstationary arrival processes; (2) have arbitrary per-call resource demands; and (3) have arbitrarily distributed call and channel holding times. The first method is based on the Wiener prediction theory and the second method is based on time series analysis. Our simulations show that they perform well even for non-Poisson and nonstationary handoff call arrivals, arbitrary per-call bandwidth demands, and nonexponentially distributed call and channel holding times. They generate closely comparable performance with an existing local method and an existing collaborative method that uses information about mobiles in neighboring cells, under assumptions for which these other methods are optimized. The proposed methods are much simpler to implement than most other existing methods with fewer capabilities.

Stationary-Process Approximations for the Nonstationary Erlang Loss Model

In this paper we consider the Mt/G/s/0 model, which has s servers in parallel, no extra waiting space, and i.i.d. service times that are independent of a nonhomogeneous Poisson arrival process. Arrivals finding all servers busy are blocked (lost). We consider approximations for the average blocking probabilities over subintervals (e.g., an hour when the expected service time is five minutes) obtained by replacing the nonstationary arrival process over that subinterval by a stationary arrival process. The stationary-Poisson approximation, using a Poisson (M) process with the average rate, tends to significantly underestimate the blocking probability. We obtain much better approximations by using a non-Poisson stationary (G) arrival process with higher stochastic variability to capture the effect of the time-varying deterministic arrival rate. In particular, we propose a specific approximation based on the heavy-traffic peakedness formula, which is easy to apply with either known arrival-rate functions or data from system measurements. We compare these approximations to exact numerical results for the Mt/M/s/0 model with linear arrival rate.

A Closure Approximation for the Nonstationary M/M/s Queue

Typical queues do not have constant arrival rates. This paper discusses effective computational methods for dealing with queues having nonstationary arrival processes. It presents a computationally undemanding approximate method for finding the time dependent mean and standard deviation of the number of customers in an s server queueing system with time-varying arrival and service rates. Results are exhibited for various 1 and 3 server queues with constant service rates and sinusoidal components in the arrival rate.