scholarly journals Exact overflow asymptotics for queues with many Gaussian inputs

2003 ◽  
Vol 40 (03) ◽  
pp. 704-720 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Time Horizon ◽  
Exact Results ◽  
Service Capacity ◽  
Stationary Increments ◽  
Overflow Probability ◽  
Finite Time Horizon ◽  
Transient Probability

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

scholarly journals Exact overflow asymptotics for queues with many Gaussian inputs

2003 ◽  
Vol 40 (3) ◽  
pp. 704-720 ◽  
Author(s):  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Gaussian Processes ◽  
Continuous Time ◽  
Time Horizon ◽  
Exact Results ◽  
Service Capacity ◽  
Stationary Increments ◽  
Overflow Probability ◽  
Finite Time Horizon ◽  
Transient Probability

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.

On the supremum distribution of integrated stationary Gaussian processes with negative linear drift

1999 ◽  
Vol 31 (01) ◽  
pp. 135-157 ◽  
Author(s):  
Jinwoo Choe ◽  
Ness B. Shroff
Keyword(s):  
Gaussian Processes ◽  
Extreme Value Theory ◽  
Continuous Time ◽  
Value Theory ◽  
Numerical Examples ◽  
Stationary Increments ◽  
Negative Drift ◽  
Stationary Gaussian Processes ◽  
Continuous Time Processes ◽  
Asymptotic Upper Bound

In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. Our bound is valid for both discrete- and continuous-time processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance.

scholarly journals On the Monitoring Error of the Supremum of a Normal Jump Diffusion Process

2011 ◽  
Vol 48 (4) ◽  
pp. 1021-1034 ◽  
Author(s):  
Ao Chen ◽  
Liming Feng ◽  
Renming Song
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Riemann Zeta Function ◽  
Time Horizon ◽  
Jump Diffusion ◽  
Mathematical Finance ◽  
The Riemann Zeta Function ◽  
Jump Diffusion Process ◽  
Riemann Zeta ◽  
Finite Time Horizon

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the form a0 / N1/2 + a1 / N3/2 + · · · + b1 / N + b2 / N2 + b4 / N4 + · · ·, where N is the number of monitoring intervals. We obtain explicit expressions for the coefficients {a0, a1, …, b1, b2, …}. In particular, a0 is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

scholarly journals On the Monitoring Error of the Supremum of a Normal Jump Diffusion Process

2011 ◽  
Vol 48 (04) ◽  
pp. 1021-1034 ◽  
Author(s):  
Ao Chen ◽  
Liming Feng ◽  
Renming Song
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Riemann Zeta Function ◽  
Time Horizon ◽  
Jump Diffusion ◽  
Mathematical Finance ◽  
The Riemann Zeta Function ◽  
Jump Diffusion Process ◽  
Riemann Zeta ◽  
Finite Time Horizon

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the forma0/N1/2+a1/N3/2+ · · · +b1/N+b2/N2+b4/N4+ · · ·, whereNis the number of monitoring intervals. We obtain explicit expressions for the coefficients {a0,a1, …,b1,b2, …}. In particular,a0is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.

On the supremum distribution of integrated stationary Gaussian processes with negative linear drift

1999 ◽  
Vol 31 (1) ◽  
pp. 135-157 ◽  
Author(s):  
Jinwoo Choe ◽  
Ness B. Shroff
Keyword(s):  
Gaussian Processes ◽  
Extreme Value Theory ◽  
Continuous Time ◽  
Value Theory ◽  
Numerical Examples ◽  
Stationary Increments ◽  
Negative Drift ◽  
Stationary Gaussian Processes ◽  
Continuous Time Processes ◽  
Asymptotic Upper Bound

In this paper we study the supremum distribution of a class of Gaussian processes having stationary increments and negative drift using key results from Extreme Value Theory. We focus on deriving an asymptotic upper bound to the tail of the supremum distribution of such processes. Our bound is valid for both discrete- and continuous-time processes. We discuss the importance of the bound, its applicability to queueing problems, and show numerical examples to illustrate its performance.

scholarly journals A Constrained Markovian Diffusion Model for Controlling the Pollution Accumulation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1466
Author(s):  
Beatris Adriana Escobedo-Trujillo ◽  
José Daniel López-Barrientos ◽  
Javier Garrido-Meléndez
Keyword(s):  
Dynamic Programming ◽  
Dirichlet Problem ◽  
Stochastic Control ◽  
Finite Time ◽  
Time Horizon ◽  
Closed Loop ◽  
Programming Techniques ◽  
Pollution Accumulation ◽  
Finite Time Horizon ◽  
The Cost

This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered here are supposed to be unbounded, and the obtained strategies are of non-stationary closed-loop type. The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing three algorithms for the purpose of replicating the results. There, the reader can find a result on the interchangeability of limits in a Dirichlet problem.

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