scholarly journals Queues with Delays in Two-State Strategies and Lévy Input

2008 ◽  
Vol 45 (2) ◽  
pp. 314-332 ◽  
Author(s):  
R. Bekker ◽  
O. J. Boxma ◽  
O. Kella
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Original Function ◽  
Steady State Distribution ◽  
State Distribution ◽  
Negative Drift

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

scholarly journals Queues with Delays in Two-State Strategies and Lévy Input

2008 ◽  
Vol 45 (02) ◽  
pp. 314-332
Author(s):  
R. Bekker ◽  
O. J. Boxma ◽  
O. Kella
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Original Function ◽  
Steady State Distribution ◽  
State Distribution ◽  
Negative Drift

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

Useful martingales for stochastic storage processes with Lévy input

1992 ◽  
Vol 29 (2) ◽  
pp. 396-403 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Lévy Process ◽  
Finite Interval ◽  
Levy Process ◽  
Secondary Process ◽  
Stochastic Integration ◽  
Steady State Distribution ◽  
State Distribution ◽  
Storage Network ◽  
Storage Processes

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

Useful martingales for stochastic storage processes with Lévy input

1992 ◽  
Vol 29 (02) ◽  
pp. 396-403 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Lévy Process ◽  
Finite Interval ◽  
Levy Process ◽  
Secondary Process ◽  
Stochastic Integration ◽  
Steady State Distribution ◽  
State Distribution ◽  
Storage Network ◽  
Storage Processes

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

scholarly journals Property Claim Services by Compound Poisson Process And Inhomogeneous Levy Process

2018 ◽  
Vol 6 (1) ◽  
pp. 32
Author(s):  
Muhammed A. S. Murad
Keyword(s):  
Poisson Process ◽  
Lévy Process ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Arrival Process ◽  
Compound Poisson ◽  
Process Factor ◽  
Before And After ◽  
Exponential Lévy Process ◽  
Loss Index

In this paper, stochastic compound Poisson process is employed to value the catastrophic insurance options and model the claim arrival process for catastrophic events, which were written in the loss period , during which the catastrophe took place. Here, a time compound process gives the underlying loss index before and after  whose losses are revaluated by inhomogeneous exponential Levy process factor. For this paper, an exponential Levy process is used to evaluate the well-known European call option in order to price Property Claim Services catastrophe insurance based on catastrophe index.

A continuous review (s, S) inventory system in a random environment

1978 ◽  
Vol 15 (03) ◽  
pp. 654-659 ◽  
Author(s):  
Richard M. Feldman
Keyword(s):  
Poisson Process ◽  
External Environment ◽  
Compound Poisson Process ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Compound Poisson ◽  
Continuous Review ◽  
Environmental Process ◽  
State Of The Environment

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

ORTHOGONAL DECOMPOSITIONS FOR LÉVY PROCESSES WITH AN APPLICATION TO THE GAMMA, PASCAL, AND MEIXNER PROCESSES

2003 ◽  
Vol 06 (01) ◽  
pp. 73-102 ◽  
Author(s):  
EUGENE LYTVYNOV
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Lévy Process ◽  
Lévy Processes ◽  
Random Variable ◽  
Levy Processes ◽  
Levy Process ◽  
Nuclear Space ◽  
Multiple Stochastic Integrals ◽  
Orthogonal Decompositions

It is well known that between all processes with independent increments, essentially only the Brownian motion and the Poisson process possess the chaotic representation property (CRP). Thus, a natural question appears: What is an appropriate analog of the CRP in the case of a general Lévy process. At least three approaches are possible here. The first one, due to Itô, uses the CRP of the Brownian motion and the Poisson process, as well as the representation of a Lévy process through those processes. The second approach, due to Nualart and Schoutens, consists of representing any square-integrable random variable as a sum of multiple stochastic integrals constructed with respect to a family of orthogonalized centered power jumps processes. The third approach, never applied before to the Lévy processes, uses the idea of orthogonalization of polynomials with respect to a probability measure defined on the dual of a nuclear space. The main aims of this paper are to develop the three approaches in the case of a general (ℝ-valued) Lévy process on a Riemannian manifold and (what is more important) to understand a relationship between these approaches. We apply the obtained results to the gamma, Pascal, and Meixner processes, in which case the analysis related to the orthogonalized polynomials becomes essentially simpler and richer than in the general case.

A continuous review (s, S) inventory system in a random environment

1978 ◽  
Vol 15 (3) ◽  
pp. 654-659 ◽  
Author(s):  
Richard M. Feldman
Keyword(s):  
Poisson Process ◽  
External Environment ◽  
Compound Poisson Process ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Compound Poisson ◽  
Continuous Review ◽  
Environmental Process ◽  
State Of The Environment

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

Diffusion approximations for queues with server vacations

1990 ◽  
Vol 22 (3) ◽  
pp. 706-729 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Heavy Traffic ◽  
Service Discipline ◽  
Single Server ◽  
Traffic Intensity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Reflecting Brownian Motion ◽  
Heavy Traffic Limit

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

Diffusion approximations for queues with server vacations

1990 ◽  
Vol 22 (03) ◽  
pp. 706-729 ◽  
Author(s):  
Offer Kella ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
Heavy Traffic ◽  
Service Discipline ◽  
Single Server ◽  
Traffic Intensity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Reflecting Brownian Motion ◽  
Heavy Traffic Limit

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

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