scholarly journals HEAVY TRAFFIC LIMITS VIA BROWNIAN EMBEDDINGS

2006 ◽  
Vol 20 (4) ◽  
pp. 595-598 ◽  
Author(s):  
Erol A. Peköz ◽  
Jose Blanchet
Keyword(s):  
Brownian Motion ◽  
Heavy Traffic ◽  
Continuous Mapping ◽  
Queue Size ◽  
Reflected Brownian Motion ◽  
Probabilistic Argument ◽  
Brownian Motion With Drift ◽  
Queuing Models ◽  
Topological Argument ◽  
Critical Model

For theGI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

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.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (04) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

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.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

scholarly journals Parisian Time of Reflected Brownian Motion with Drift on Rays and Its Application in Banking

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 127
Author(s):  
Angelos Dassios ◽  
Junyi Zhang
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Liquidity Risk ◽  
Finite Collection ◽  
Recursive Method ◽  
Simulation Algorithm ◽  
Reflected Brownian Motion ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
First Time

In this paper, we study the Parisian time of a reflected Brownian motion with drift on a finite collection of rays. We derive the Laplace transform of the Parisian time using a recursive method, and provide an exact simulation algorithm to sample from the distribution of the Parisian time. The paper was motivated by the settlement delay in the real-time gross settlement (RTGS) system. Both the central bank and the participating banks in the system are concerned about the liquidity risk, and are interested in the first time that the duration of settlement delay exceeds a predefined limit. We reduce this problem to the calculation of the Parisian time. The Parisian time is also crucial in the pricing of Parisian type options; to this end, we will compare our results to the existing literature.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (4) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

scholarly journals Parisian Time of Reflected Brownian Motion With Drift on Rays

2020 ◽  
Author(s):  
Angelos Dassios ◽  
Junyi Zhang
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Liquidity Risk ◽  
Finite Collection ◽  
Recursive Method ◽  
Simulation Algorithm ◽  
Reflected Brownian Motion ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
First Time

In this paper, we study the Parisian time of a reflected Brownian motion with drift on a finite collection of rays. We derive the Laplace transform of the Parisian time using a recursive method, and provide an exact simulation algorithm to sample from the distribution of the Parisian time. The paper is motivated by the settlement delay in the real-time gross settlement (RTGS) system. Both the central bank and the participating banks in the system are concerned about the liquidity risk, and are interested in the first time that the duration of settlement delay exceeds a predefined limit, we reduce this problem to the calculation of the Parisian time. The Parisian time is also crucial in the pricing of Parisian type options; to this end, we will compare our results with the existing literature.

scholarly journals Level sets and drift estimation for reflected Brownian motion with drift

2020 ◽  
Author(s):  
Alejandro Cholaquidis ◽  
Ricardo Fraiman ◽  
Ernesto Mordecki ◽  
Cecilia Papalardo
Keyword(s):  
Brownian Motion ◽  
Level Sets ◽  
Reflected Brownian Motion ◽  
Brownian Motion With Drift ◽  
Drift Estimation

Storage-Limited Queues in Heavy Traffic

1991 ◽  
Vol 5 (4) ◽  
pp. 499-522 ◽  
Author(s):  
E. G. Coffman ◽  
A. A. Pukhalskii ◽  
M. I. Reiman
Keyword(s):  
Brownian Motion ◽  
Arrival Time ◽  
Storage Capacity ◽  
Queueing System ◽  
Heavy Traffic ◽  
Capacity Constraints ◽  
Reflected Brownian Motion ◽  
Heavy Traffic Limit ◽  
System 1

This paper models primary computer storage in the context of a general (GI/GI/l) queueing system. Queued items are described by sizes, or storage requirements, as well as by arrival and service times; the sum of the sizes of the items in the system is the occupied storage. Capacity constraints are represented by two different protocols for determining whether an arriving item is admitted to the system: (1) an item is accepted if and only if at its arrival time the currently occupied storage does not exceed a given constantC> 0, and (2) an item is accepted if and only if at its arrival time the occupied storage is at mostC, and the occupied storage plus the item's size is at mostC(l + ε) for some given ε > 0. We prove for both systems that in heavy traffic the occupied storage, suitably normalized, converges weakly to reflected Brownian motion with boundaries at 0 and atC. A distinctive feature of the proof is the characterization of reflected Brownian motion as a limit of unrestricted penalized processes.These results make more plausible an earlier conjecture of the authors, i.e., that one obtains the same heavy traffic limit when the admission rule is: accept an item if and only if at its arrival time the occupied storage plus the item's size is no greater thanC.

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