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Table Of Integrals Series And Products

Tables of integrals

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

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An Introduction To The Theory Of Point Processes

Calderas are collapse structures related to the emptying of magmatic reservoirs, often associated with large eruptions from long-lived magmatic systems. Understanding how magma is transferred from a magma reservoir to the surface before eruptions is a major challenge. Here we exploit the historical, archaeological and geological record of Campi Flegrei caldera to estimate the surface deformation preceding the Monte Nuovo eruption and investigate the shallow magma transfer. Our data suggest a progressive magma accumulation from ~1251 to 1536 in a 4.6 ± 0.9 km deep source below the caldera centre, and its transfer, between 1536 and 1538, to a 3.8 ± 0.6 km deep magmatic source ~4 km NW of the caldera centre, below Monte Nuovo; this peripheral source fed the eruption through a shallower source, 0.4 ± 0.3 km deep. This is the first reconstruction of pre-eruptive magma transfer at Campi Flegrei and corroborates the existence of a stationary oblate source, below the caldera centre, that has been feeding lateral eruptions for the last ~5 ka. Our results suggest: 1) repeated emplacement of magma through intrusions below the caldera centre; 2) occasional lateral transfer of magma feeding non-central eruptions within the caldera. Comparison with historical unrest at calderas worldwide suggests that this behavior is common.

Magma transfer at Campi Flegrei caldera (Italy) before the 1538 AD eruption

Abstract In modeling of port dynamics it seems reasonable to assume that the ships arrive on a somewhat scheduled basis and that there is a constant lay period during which, in a uniform way, each vessel can arrive at the port. In the present paper, we study the counting process N ( t ) which represents the number of scheduled vessels arriving during the time interval (0, t ], $$t&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Specifically, we provide the explicit expressions of the probability generating function, the probability distribution and the expected value of N ( t ). In some cases of interest, we also obtain the probability law of the stationary counting process representing the number of arrivals in a time interval of length t when the initial time is an arbitrarily chosen instant. This leads to various results concerning the autocorrelations of the random variables $$X_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , $$i\in \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> , which give the actual interarrival time between the $$(i-1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -th and the i -th vessel arrival. Finally, we provide an application to a stochastic model for the queueing behavior at the port, given by a queueing system characterized by stationary interarrival times $$X_i$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , exponential service times and an infinite number of servers. In this case, some results on the average number of customers and on the probability of an empty queue are disclosed. 

/pdf/vessels-arrival-process-and-its-application-to-the-ship-m-3fbelqlc.pdf

Vessels Arrival Process and its Application to the SHIP/M/
 
 
 
 $$\infty$$
 
 ∞
 
 Queue

https://link.springer.com/content/pdf/10.1007/s11009-023-10003-8.pdf

Vessels Arrival Process and its Application to the SHIP/M/∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty$$\en

Barbara Martinucci

Papers

Vessels Arrival Process and its Application to the SHIP/M/ $$\infty$$ ∞ Queue