Comparison of steady-state methods computing Markov modulated fluid models

2008 ◽  
Author(s):  
Wassim Abbessi ◽  
Hédi Nabli
Keyword(s):  
Steady State ◽  
Fluid Models ◽  
Markov Modulated

scholarly journals Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

2021 ◽  
Vol 53 (1) ◽  
pp. 1-29
Author(s):  
Ari Arapostathis ◽  
Guodong Pang ◽  
Yi Zheng
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Regime Switching ◽  
Convergence Rates ◽  
Death Process ◽  
Ergodic Properties ◽  
State Diffusion ◽  
Markov Modulated ◽  
Steady State Diffusion ◽  
Birth Death

AbstractWe study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied.

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (02) ◽  
pp. 461-472
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

scholarly journals Perturbation analysis of Markov modulated fluid models

2017 ◽  
Vol 33 (4) ◽  
pp. 473-494 ◽  
Author(s):  
Sarah Dendievel ◽  
Guy Latouche
Keyword(s):  
Perturbation Analysis ◽  
Fluid Models ◽  
Markov Modulated

scholarly journals Geometrical Optics Approach to Markov-Modulated Fluid Models

2005 ◽  
Vol 114 (1) ◽  
pp. 45-93 ◽  
Author(s):  
Diego Dominici ◽  
Charles Knessl
Keyword(s):  
Geometrical Optics ◽  
Fluid Models ◽  
Markov Modulated

scholarly journals On Exceedance Times for Some Processes with Dependent Increments

2014 ◽  
Vol 51 (01) ◽  
pp. 136-151 ◽  
Author(s):  
Søren Asmussen ◽  
Sergey Foss
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Risk Process ◽  
Fluid Models ◽  
Insurance Risk ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Time Position ◽  
Heavy Tailed ◽  
Markov Modulated

Let {Z n } n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = sup n≥0 Z n be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Z τ at this time, position Z τ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (2) ◽  
pp. 461-472 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

scholarly journals On Exceedance Times for Some Processes with Dependent Increments

2014 ◽  
Vol 51 (1) ◽  
pp. 136-151
Author(s):  
Søren Asmussen ◽  
Sergey Foss
Keyword(s):  
Random Walk ◽  
Limit Theorem ◽  
Risk Process ◽  
Fluid Models ◽  
Insurance Risk ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Time Position ◽  
Heavy Tailed ◽  
Markov Modulated

Let {Zn}n≥0 be a random walk with a negative drift and independent and identically distributed increments with heavy-tailed distribution, and let M = supn≥0Zn be its supremum. Asmussen and Klüppelberg (1996) considered the behavior of the random walk given that M > x for large x, and obtained a limit theorem, as x → ∞, for the distribution of the quadruple that includes the time τ = τ(x) to exceed level x, position Zτ at this time, position Zτ-1 at the prior time, and the trajectory up to it (similar results were obtained for the Cramér-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of τ. The class of models includes Markov-modulated models as particular cases. We also study fluid models, the Björk-Grandell risk process, give examples where the order of τ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli and Schmidt (1999), Foss and Zachary (2002), and Foss, Konstantopoulos and Zachary (2007).

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