A sample path approach to mean busy periods for Markov-modulated queues and fluids

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.

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Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model

Astin Bulletin ◽

10.2143/ast.35.2.2003457 ◽

2005 ◽

Vol 35 (02) ◽

pp. 351-361 ◽

Cited By ~ 6

Author(s):

Andrew C.Y. Ng ◽

Hailiang Yang

Keyword(s):

Regime Switching ◽

Joint Distribution ◽

Risk Model ◽

Insurance Risk ◽

Additive Process ◽

Deficit At Ruin ◽

Markov Additive Process ◽

The Deficit At Ruin ◽

Markov Modulated ◽

Markovian Regime Switching

In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

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Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution

Journal of Applied Probability ◽

10.2307/3213864 ◽

1985 ◽

Vol 22 (3) ◽

pp. 611-618 ◽

Cited By ~ 29

Author(s):

A. G. Nobile ◽

L. M. Ricciardi ◽

L. Sacerdote

Keyword(s):

Asymptotic Behavior ◽

Steady State ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Steady State Distribution ◽

State Distribution ◽

First Passage ◽

The Mean

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time.

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Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

Advances in Applied Probability ◽

10.1017/apr.2020.2 ◽

2020 ◽

Vol 52 (2) ◽

pp. 404-432

Author(s):

Irmina Czarna ◽

Adam Kaszubowski ◽

Shu Li ◽

Zbigniew Palmowski

Keyword(s):

Insurance Risk ◽

Additive Process ◽

Markov Additive Process ◽

Surplus Process ◽

Current State ◽

Markov Modulated ◽

Exit Problems ◽

State Of The Environment ◽

Risk Processes ◽

Omega Model

AbstractIn this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

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

Journal of Applied Probability ◽

10.1017/s0021900200098119 ◽

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.

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DISTRIBUTION OF NUCLEOTIDE DIFFERENCES BETWEEN TWO RANDOMLY CHOSEN CISTRONS IN A FINITE POPULATION

Genetics ◽

10.1093/genetics/85.2.331 ◽

1977 ◽

Vol 85 (2) ◽

pp. 331-337

Author(s):

Wen-Hsiung Li

Keyword(s):

Steady State ◽

Population Size ◽

Effective Population Size ◽

Mutation Rate ◽

Finite Population ◽

Effective Population ◽

Steady State Distribution ◽

State Distribution ◽

Simple Estimate ◽

The Mean

ABSTRACT Watterson's (1975) formula for the steady-state distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population has been extended to transient states. The rate for the mean of this distribution to approach its equilibrium value is 1/2 N and independent of mutation rate, but that for the variance is dependent on mutation rate, where N denotes the effective population size. Numerical computations show that if the heterozygosity (i.e., the probability that two cistrons are different) is low, say of the order of 0.1 or less, the probability that two cistrons differ at two or more nucleotide sites is less than 10 percent of the heterozygosity, whereas this probability may be as high as 50 percent of the heterozygosity if the heterozygosity is 0.5. A simple estimate for the mean number (d) of site differences between cistrons is d = h/(1 - h) where h is the heterozygosity. At equilibrium, the probability that two cistrons differ by more than one site is equal to h 2, the square of heterozygosity.

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Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution

Journal of Applied Probability ◽

10.1017/s0021900200029363 ◽

1985 ◽

Vol 22 (03) ◽

pp. 611-618 ◽

Cited By ~ 21

Author(s):

A. G. Nobile ◽

L. M. Ricciardi ◽

L. Sacerdote

Keyword(s):

Asymptotic Behavior ◽

Steady State ◽

Diffusion Processes ◽

First Passage Time ◽

Passage Time ◽

Steady State Distribution ◽

State Distribution ◽

First Passage ◽

The Mean

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time.

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On the settling time of the congested GI/G/1 queue

Advances in Applied Probability ◽

10.1017/s000186780002320x ◽

1990 ◽

Vol 22 (04) ◽

pp. 929-956 ◽

Cited By ~ 1

Author(s):

George D. Stamoulis ◽

John N. Tsitsiklis

Keyword(s):

Fluid Model ◽

Settling Time ◽

Interarrival Time ◽

Steady State Distribution ◽

State Distribution ◽

Variation Distance ◽

Service Rates ◽

Image Position ◽

Number Of Customers ◽

First Time

We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N 0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N 0/(μ –λ); that is, we prove that where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.

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

Journal of Applied Probability ◽

10.2307/3213292 ◽

1981 ◽

Vol 18 (2) ◽

pp. 461-472 ◽

Cited By ~ 7

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.

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Markov-modulated linear fluid networks with Markov additive input

Journal of Applied Probability ◽

10.1017/s0021900200022634 ◽

2002 ◽

Vol 39 (02) ◽

pp. 413-420 ◽

Cited By ~ 3

Author(s):

Offer Kella ◽

Wolfgang Stadje

Keyword(s):

Markov Chain ◽

External Input ◽

Release Rates ◽

Constant Multiple ◽

Additive Process ◽

Markov Additive Process ◽

Fluid Networks ◽

Markov Modulated

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

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