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FIRST PASSAGE OF A MARKOV ADDITIVE PROCESS AND GENERALIZED JORDAN CHAINS
 APPLIED PROBABILITY TRUST (15 SEPTEMBER 2010)
, 2010
"... In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan ..."
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Cited by 4 (2 self)
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In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique, which can be used to derive various further identities.
ASYMPTOTIC BLOCKING PROBABILITIES IN LOSS NETWORKS WITH SUBEXPONENTIAL DEMANDS
, 2008
"... The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes int ..."
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Cited by 3 (0 self)
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The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of blocking probabilities.
Markovmodulated Brownian motion with two reflecting barriers
 Journal of Applied Pprobability
"... We consider a Markovmodulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explaining its simplicity. Moreover, this argume ..."
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Cited by 3 (1 self)
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We consider a Markovmodulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explaining its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent exponentially distributed epoch. Our second contribution concerns transient behavior of the reflected system. We identify the joint law of the processes t,X(t), J(t) at inverse local times.
FIRST PASSAGE OF A MARKOV ADDITIVE PROCESS AND
, 2010
"... In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan ..."
Abstract
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In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique, which can be used to derive various further identities.
Loss Rate Asymptotics
"... Abstract We consider a Lévy process S t which is reflected at 0 and K > 0. The reflected process V K t is given as a solution to a Skorokhod problem, which implies a representation V , where it was expressed in terms of the characteristic triplet of S t and π K , and asymptotics of the loss rate ..."
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Abstract We consider a Lévy process S t which is reflected at 0 and K > 0. The reflected process V K t is given as a solution to a Skorokhod problem, which implies a representation V , where it was expressed in terms of the characteristic triplet of S t and π K , and asymptotics of the loss rate as K → ∞ was derived in the case of negative drift and light tails. Asymptotics for positive drift is straightforward by reversing the role of the barriers 0 and K and using a conservation law. We use the expression for the loss rate from [1] to derive asymptotics in the case of negative drift and heavy tails, as well as in the case of zero drift. In the zero drift case, functional limit theorems (with a Brownian or stable limit) play an important role and are based on continuity properties of the loss rate.
ASYMPTOTIC BLOCKING PROBABILITIES IN LOSS SYSTEMS WITH SUBEX PONENTIAL DEMANDS
 APPLIED PROBABILITY TRUST
, 2006
"... The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes int ..."
Abstract
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The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings, computing the well known Erlang formula for blocking probability in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulas and simulation results even for relatively small resource capacities and relatively large values of blocking probabilities.
unknown title
"... ABSTRACT. This survey addresses the class of queues with Lévy input, which covers the classical M/G/1 queue and reflected Brownian motion as special cases. First the stationary behavior is treated, with special attention to the case of the input process having onesided jumps (i.e., spectrally one ..."
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ABSTRACT. This survey addresses the class of queues with Lévy input, which covers the classical M/G/1 queue and reflected Brownian motion as special cases. First the stationary behavior is treated, with special attention to the case of the input process having onesided jumps (i.e., spectrally onesided Lévy processes). Then various transient metrics are focused on (such as the transient distribution, the busy period, and the workload correlation function). Distinguishing between lighttailed and heavytailed input, we give an account of results on the tail of the workload distribution; in addition we present themain asymptotic results for the various transient quantities. We then extend our basic model to various more advanced queueing systems: queues with a finite buffer, queues in which the current buffer level affects the characteristics of the Lévy input (‘feedback’), and polling type of models. The last part of the survey considers networks of queues: starting with the tandem queue, we subsequently describe the stationary behavior of a general class of Lévydriven queueing networks. At the methodological level, a variety of techniques has been used, such as transformbased techniques, martingales, rateconservation arguments, changeofmeasure, importance sampling, and large deviations.