scholarly journals Synchronized Lévy queues

2020 ◽  
Vol 57 (4) ◽  
pp. 1222-1233
Author(s):  
Offer Kella ◽  
Onno Boxma
Keyword(s):  
Steady State ◽  
Lévy Process ◽  
Levy Process ◽  
Special Structure ◽  
Stieltjes Transform ◽  
Random Vectors ◽  
Parallel Queues ◽  
Buffer Content ◽  
Stieltjes Transforms

AbstractWe consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors.

scholarly journals ON A GENERIC CLASS OF LÉVY-DRIVEN VACATION MODELS

2009 ◽  
Vol 24 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
Michel Mandjes
Keyword(s):  
Steady State ◽  
Content Distribution ◽  
Levy Process ◽  
Active Period ◽  
Queuing Systems ◽  
Stieltjes Transform ◽  
Server Vacation ◽  
Vacation Models ◽  
Buffer Content ◽  
Generic Class

This article analyzes a generic class of queuing systems with server vacation. The special feature of the models considered is that the duration of the vacations explicitly depends on the buffer content evolution during the previous active period (i.e., the time elapsed since the previous vacation). During both active periods and vacations, the buffer content evolves as a Lévy process. For two specific classes of models, the Laplace–Stieltjes transform of the buffer content distribution at switching epochs between successive vacations and active periods, and in steady state, is derived.

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 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.

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 LAN property for a simple Lévy process

2014 ◽  
Vol 352 (10) ◽  
pp. 859-864 ◽  
Author(s):  
Arturo Kohatsu-Higa ◽  
Eulalia Nualart ◽  
Ngoc Khue Tran
Keyword(s):  
Lévy Process ◽  
Levy Process

scholarly journals Recent developments on the Stieltjes transform of generalized functions

1987 ◽  
Vol 10 (4) ◽  
pp. 641-670 ◽  
Author(s):  
Ram Sankar Pathak ◽  
Lokenath Debnath
Keyword(s):  
Final Section ◽  
Generalized Functions ◽  
Stieltjes Transform ◽  
Poisson Transform ◽  
Open Questions ◽  
The Laplace Transform ◽  
Recent Developments ◽  
The Subject ◽  
Abelian Theorems ◽  
Stieltjes Transforms

This paper is concerned with recent developments on the Stieltjes transform of generalized functions. Sections 1 and 2 give a very brief introduction to the subject and the Stieltjes transform of ordinary functions with an emphasis to the inversion theorems. The Stieltjes transform of generalized functions is described in section 3 with a special attention to the inversion theorems of this transform. Sections 4 and 5 deal with the adjoint and kernel methods used for the development of the Stieltjes transform of generalized functions. The real and complex inversion theorems are discussed in sections 6 and 7. The Poisson transform of generalized functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom are included in sections 8, 9 and 10. The Stieltjes transforms of different orders and the fractional order integration and further generalizations of the Stieltjes transform are discussed in sections 11 and 12. Sections 13, 14 and 15 are devoted to Abelian theorems, initial-value and final-value results. Some applications of the Stieltjes transforms are discussed in section 16. The final section deals with some open questions and unsolved problems. Many important and recent references are listed at the end.

scholarly journals On the optimal dividend problem for a spectrally negative Lévy process

2007 ◽  
Vol 17 (1) ◽  
pp. 156-180 ◽  
Author(s):  
Florin Avram ◽  
Zbigniew Palmowski ◽  
Martijn R. Pistorius
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Spectrally Negative Lévy Process ◽  
Optimal Dividend
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