scholarly journals Some control problems with random intervention times

2001 ◽  
Vol 33 (2) ◽  
pp. 404-422 ◽  
Author(s):  
Hui Wang
Keyword(s):  
Poisson Process ◽  
Impulse Control ◽  
Value Functions ◽  
Control Cost ◽  
Signal Process ◽  
Arrival Times ◽  
Quadratic Deviation ◽  
Ergodic Problem ◽  
Impulse Controls ◽  
Admissible Controls

We consider the problem of optimally tracking a Brownian motion by a sequence of impulse controls, in such a way as to minimize the total expected cost that consists of a quadratic deviation cost and a proportional control cost. The main feature of our model is that the control can only be exerted at the arrival times of an exogenous uncontrolled Poisson process (signal). In other words, the set of possible intervention times are discrete, random and determined by the signal process (not by the decision maker). We discuss both the discounted problem and the ergodic problem, where explicit solutions can be found. We also derive the asymptotic behavior of the optimal control policies and the value functions as the intensity of the Poisson process goes to infinity, or roughly speaking, as the set of admissible controls goes from the discrete-time impulse control to the continuous-time bounded variation control.

A Class of Stochastic Control Problem Governed by a Poisson Process

2012 ◽  
Vol 450-451 ◽  
pp. 46-55
Author(s):  
Shao Lin Tian ◽  
Ji Chun Li ◽  
Kun Hui Liu
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Poisson Process ◽  
Objective Function ◽  
Impulse Control ◽  
Sufficient Condition ◽  
Signal Process ◽  
Impulse Control Problem ◽  
State Process ◽  
Optimal Impulse

In this paper, we examine an optimal impulse control problem of stochastic system, whose state follows a Brownian motion. Here we want to maximum the objective function. The main feature of our model is that the controlled state process includes an impulse control governed by a Poisson process. In other words, the set of possible intervention times are discrete, random and determined by the signal process. Here we not only present a theorem giving a sufficient condition on the existence of an optimal control and its corresponding objective function, but also provide an explicit solution obtained under some simplified conditions.

scholarly journals Singular perturbations for a subelliptic operator

2018 ◽  
Vol 24 (4) ◽  
pp. 1429-1451 ◽  
Author(s):  
Paola Mannucci ◽  
Claudio Marchi ◽  
Nicoletta Tchou
Keyword(s):  
Singular Perturbation ◽  
Singular Perturbation Problems ◽  
Value Functions ◽  
Infinitesimal Operator ◽  
Lipschitz Continuous ◽  
Subelliptic Operator ◽  
Whole Space ◽  
Perturbation Problems ◽  
Ergodic Problem ◽  
Logarithmic Growth

We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

INDIVIDUAL EQUILIBRIUM DYNAMIC ROUTING IN A MULTIPLE SERVER RETRIAL QUEUE

2000 ◽  
Vol 14 (1) ◽  
pp. 9-26 ◽  
Author(s):  
Anthony C. Brooms
Keyword(s):  
Nash Equilibrium ◽  
Poisson Process ◽  
Sojourn Time ◽  
Service System ◽  
Transmission Rate ◽  
Retrial Queue ◽  
Dynamic Learning ◽  
Arrival Times ◽  
The Public ◽  
Expected Sojourn Time

Customers arrive sequentially to a service system where the arrival times form a Poisson process of rate λ. The system offers a choice between a private channel and a public set of channels. The transmission rate at each of the public channels is faster than that of the private one; however, if all of the public channels are occupied, then a customer who commits itself to using one of them attempts to connect after exponential periods of time with mean μ−1. Once connection to a public channel has been made, service is completed after an exponential period of time, with mean ν−1. Each customer chooses one of the two service options, basing its decision on the number of busy channels and reapplying customers, with the aim of minimizing its own expected sojourn time. The best action for an individual customer depends on the actions taken by subsequent arriving customers. We establish the existence of a unique symmetric Nash equilibrium policy and show that its structure is characterized by a set of threshold-type strategies; we discuss the relevance of this concept in the context of a dynamic learning scenario.

A study on the arrival process of lift passengers in a multi-storey office building

2011 ◽  
Vol 33 (4) ◽  
pp. 437-449 ◽  
Author(s):  
J-M Kuusinen ◽  
J Sorsa ◽  
M-L Siikonen ◽  
H Ehtamo
Keyword(s):  
Poisson Process ◽  
Time Of Day ◽  
Office Building ◽  
Arrival Process ◽  
Batch Arrivals ◽  
Arrival Times ◽  
Piecewise Constant ◽  
Practical Applications ◽  
Video Recordings ◽  
Traffic Simulations

This article presents a study on the process of how passengers arrive at lift lobbies to travel to their destinations. Earlier studies suggest that passengers arrive at the lift lobbies individually with exponentially distributed inter-arrival times, that is, according to a Poisson process. This study was carried out in a multi-storey office building. The data was collected using a questionnaire, digital video recordings and the lift monitoring system. The results show that, in the studied building, passengers arrive in batches whose size varies with the time of day and the floor utilization. In addition, the batch arrivals follow a time-inhomogeneous Poisson process with piecewise constant arrival rates. Practical applications: This article contributes to the basic understanding of passenger behaviour, and how people move around in buildings and arrive at the lift lobbies. It is proposed that the model for the passenger arrival process should take into account that passengers do not always arrive individually but also in batches. The passenger arrival process affects the design of elevators. It will also affect the passenger generation in building traffic simulations.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (4) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

scholarly journals Impulse Control of Proportional Reinsurance with Constraints

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Hui Meng ◽  
Tak Kuen Siu
Keyword(s):  
Optimal Control ◽  
Diffusion Process ◽  
Impulse Control ◽  
Dividend Policy ◽  
Proportional Reinsurance ◽  
Value Functions ◽  
Insurance Company ◽  
Control Policies

We consider an insurance company whose surplus follows a diffusion process with proportional reinsurance and impulse dividend control. Our objective is to maximize expected discounted dividend payouts to shareholders of the company until the time of bankruptcy. To meet some essential requirements of solvency control (e.g., bankruptcy not soon), we impose some constraints on the insurance company's dividend policy. Under two types of constraints, we derive the value functions and optimal control policies of the company.

A generalized spatial-temporal model of rainfall based on a clustered point process

1995 ◽  
Vol 450 (1938) ◽  
pp. 163-175 ◽  
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Random Variable ◽  
Cell Types ◽  
Exponential Random Variable ◽  
Two Dimensional ◽  
Arrival Times ◽  
Hourly Rainfall ◽  
Temporal Model ◽  
Different Cell Types

The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the x - y plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to n with different parameters for the different cell types, so that the random variables of an arbitrary cell, e. g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (04) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

An invariance property of Poisson processes

1969 ◽  
Vol 6 (02) ◽  
pp. 453-458 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Random Variables ◽  
Poisson Processes ◽  
Invariance Property ◽  
Homogeneous Poisson Process ◽  
Arrival Times ◽  
Random Functions

In this paper we shall investigate point processes generated by random variables of the form 〈gi (Ti ]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti 〉.

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