Investigation of operation of intermediate storages applying probability density functions satisfying linear differential equation

Éva Orbán-Mihálykó,Csaba Mihálykó

doi:10.3311/pp.ch.2012-2.05

Éva Orbán-Mihálykó, Csaba Mihálykó

Open Access

https://doi.org/10.3311/pp.ch.2012-2.05

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Abstract

In this paper operation of a batch/continuous processing system connected by an intermediate storage is considered. The filling process is supposed to be random and the output process is deterministic, consequently the process in the storage is a stochastic process. We investigate the problem of determination of necessary initial amount of material to avoid emptying of the storage. We define a function which is able to handle together the probability and expected time to shortage. We set up an integral equation for it, and in a special case of density functions of inter-arrival times we transform to an integro-differential equation. We provide analytical formula for the solution. We compare the analytical solution to the results arising from Monte-Carlo simulations. In some cases we use only the form of the analytical solution but coefficients are determined by parameter fitting. Finally we use the computed functions to determine the necessary initial amount of material to a given reliability level.

Highlights

Intermediate storage is often used in chemical industry, pharmaceutical factories, food logistic or in case of environmental investments
In this paper we investigate a stochastic model, when the filling process is a batch process and the withdrawing process is a continuous one
We introduce an auxiliary function which is able to handle both the probability and the time to shortage of material in a unified way and we set up an integral equation for it

Summary

Introduction

Intermediate storage is often used in chemical industry, pharmaceutical factories, food logistic or in case of environmental investments. We investigate the case when the density function of the inter-arrival times satisfies a linear differential equation with constant coefficients subject to some general initial conditions. This set of density functions is dense in the set of continuous density functions [8] and is investigated in risk models as. Lated results) (11) the root ρi is si, roots are denoted by ρi, and the multiplicity of the solution of the linear differential equa- This value x is the necessary initial amount of material to the reliability level 1 − α.

We note that if condition

That means