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First passage of a Markov additive process and generalized Jordan chains
Get PDFIn 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
Reflecting thoughts
No full textThe purpose of this note is to provide an equivalent definition and an alternative proof of uniqueness of the one-dimensional reflection map which is more a direct derivation that structurally leads to the form of the map when it exists, does not involve integration (neither in the definition nor in the proof) and for which no assumptions on the driving process is needed. Also, it is argued that, with the proposed definition, the reflection map exists provided that the driving process is lower semicontinuous from the right, but is not necessarily right continuous and does not necessarily have left limits. These ideas are then easily extended to the multidimensional case.Reflection mapping Regulator Local time Skorohod problem
Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts
Get PDFIn this paper we study a reflected Markov-modulated Brownian motion with a two sided
reflection in which the drift, diffusion coefficient and the two boundaries are (jointly)
modulated by a finite state space irreducible continuous time Markov chain. The goal is
to compute the stationary distribution of this Markov process, which in addition to the
complication of having a stochastic boundary can also include jumps at state change
epochs of the underlying Markov chain because of the boundary changes. We give the
general theory and then specialize to the case where the underlying Markov chain has
two states. Moreover, motivated by an application of optimal dividend strategies, we
consider the case where the lower barrier is zero and the upper barrier is subject to
control. In this case we generalized earlier results from the case of a reflected Brownian
motion to the Markov modulated cas
Superposition of renewal processes and an application to multi-server queues
No full textThe aim of this paper is to compare the waiting times of customers in multiple-server queues, where the idle times are removed, with different numbers of servers. For this purpose we develop some results regarding the vector-valued marked point process whose points are arrival epochs of the superposition of renewal processes with different continuous inter-arrival distribution and the marks are the vectors of forward recurrence times of the various renewal processes at these arrival epochs.Renewal processes Superposition Forward recurrence times Joint distribution Multi-server queue
On the area between a Lévy process with secondary jump inputs and its reflected version
Get PDFWe study the stochastic properties of the area under some function of the difference between (i) a spectrally positive Lévy process that jumps to a level whenever it hits zero, and (ii) its reflected version . Remarkably, even though the analysis of each of these areas is challenging, we succeed in attaining explicit expressions for their difference. The main result concerns the Laplace-Stieltjes transform of the integral of (a function of) the distance between and until hits zero. This result is extended in a number of directions, including the area between and and a Gaussian limit theorem. We conclude the paper with an inventory problem for which our results are particularly useful
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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Stability and Structural Properties of Stochastic Storage Networks
Get PDFWe establish stability, monotonicity, concavity and subadditivity properties for open stochastic storage networks in which the driving process has stationary increments. A principal example is a stochastic fluid network in which the external inputs are random but all internal flows are deterministic. For the general model, the multi-dimensional content process is tight under the natural stability condition. The multi-dimensional content process is also stochastically increasing when the process starts at the origin, implying convergence to a proper limit under the natural stability condition. In addition, the content process is monotone in its initial conditions. Hence, when any content process with nonzero initial conditions hits the origin, it couples with the content process starting at the origin. However, in general, a tight content process need not hit the origin
Synchronized Lévy queues
Get PDFWe 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
Another look into decomposition results
Get PDFIn this note, we identify a simple setup from which one may easily infer various decomposition results for queues with interruptions as well as càdlàg processes with certain secondary jump inputs. Special cases are processes with stationary or stationary and independent increments. In the Lévy process case, the decomposition holds not only in the limit but also at independent exponential times, due to the Wiener-Hopf decomposition. A similar statement holds regarding the GI/GI/1 setting with multiple vacation
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