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.

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.

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.

Kinetic analysis of mechanism of intestinal Na+-dependent sugar transport

1985 ◽  
Vol 248 (5) ◽  
pp. C498-C509 ◽  
Author(s):  
D. Restrepo ◽  
G. A. Kimmich
Keyword(s):  
Experimental Data ◽  
Steady State ◽  
Sugar Transport ◽  
Enzyme Kinetic ◽  
Steady State Distribution ◽  
State Distribution ◽  
Least Squares Fit ◽  
Coupling Stoichiometry ◽  
Rate Limiting ◽  
Kinetics Of

Zero-trans kinetics of Na+-sugar cotransport were investigated. Sugar influx was measured at various sodium and sugar concentrations in K+-loaded cells treated with rotenone and valinomycin. Sugar influx follows Michaelis-Menten kinetics as a function of sugar concentration but not as a function of Na+ concentration. Nine models with 1:1 or 2:1 sodium:sugar stoichiometry were considered. The flux equations for these models were solved assuming steady-state distribution of carrier forms and that translocation across the membrane is rate limiting. Classical enzyme kinetic methods and a least-squares fit of flux equations to the experimental data were used to assess the fit of the different models. Four models can be discarded on this basis. Of the remaining models, we discard two on the basis of the trans sodium dependence and the coupling stoichiometry [G. A. Kimmich and J. Randles, Am. J. Physiol. 247 (Cell Physiol. 16): C74-C82, 1984]. The remaining models are terter ordered mechanisms with sodium debinding first at the trans side. If transfer across the membrane is rate limiting, the binding order can be determined to be sodium:sugar:sodium.

PROCESSOR SHARING G-QUEUES WITH INERT CUSTOMERS AND CATASTROPHES: A MODEL FOR SERVER AGING AND REJUVENATION

2017 ◽  
Vol 31 (4) ◽  
pp. 420-435 ◽  
Author(s):  
J.-M. Fourneau ◽  
Y. Ait El Majhoub
Keyword(s):  
Steady State ◽  
Product Form ◽  
Processor Sharing ◽  
Service Capacity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Signal Arrival ◽  
Open Networks ◽  
Networks Of Queues

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

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