A Stochastic Model for Two Servers sharing a common tool magazine and its analysis

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Mathematical Modeling


There are many situations in which the sharing, of all or part of a set of tools by two or more servers, is economically attractive or even necessary. For example, two (or more) automated machines (robots) may share a part of a common toolmagazine; or an expensive, but infrequently used apparatus may be shared between two operating rooms of a hospital or two units of a chemical plant; or two computer communications systems may share common data bases and so on. In this paper, we limit our attention to the case of two machines. This case already incorporates all the difficulties, which are only compounded in considering more than two machines. The holding times for the tools for each of the machines are modelled separately as independent continuous parameter Markov chains with m1 and m2 states respectively. We mainly concentrate on obtaining a configuration that minimizes the total proportion of idle time of both the machines subject to the constraint that at the most K tools may be duplicated. The objective function of the optimization problem under study will be known only in an implicit, but computable form depending on the steady-state vectors of the Markov chains describing the operation of the two machines for various choices of the decision variables (to duplicate a tool or not). We propose several heuristic algorithms to find reasonably good solutions. Explicit optimal policies are obtained in some special cases, such as when both the machines have identical holding times. Several interesting numerical examples are presented and interpreted. A FORTRAN code is developed for each of the algorithms proposed.





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ISSN: 0270-0255


This research was supported by the National Science Foundation under Grant No. ECS-8205404 and by the Air Force O&e of Scientific Research under Grant No. AFOSR 77-3236.


© 1985 Published by Elsevier B.V.