# Cybernetic Queueing and Storage Systems

• A. Ghosal
Part of the Lecture Notes in Operations Research and Mathematical Systems book series (LNE, volume 23)

## Abstract

Any system comprises a set of elements A = (a1,..., an) which are subject to some specified behaviour pattern; for example, a!s may be stochastic following any particular d.f. or deterministic or there may be a set of inter-relationships among a!s etc. All these behaviour patterns can be denoted by the set R-which takes into account any auto-regulatory process, rules, d.f.’s, inter-relations, etc. Thus we denote a system S by {A, R{ (see Klir and Valach, 1965). From this angle let us review a simple queueing or storage system <Z, X,Y,k> or <Z, u, k>; this system is subject to an input process and subject to some rules gives out an output process — in some cases the input and release processes are fed into the system which, subject to the rules of the system, govern the level of the dam or an inventory system {Z{. The simple model (1.1) or (1.2) can be regarded as the regulatory process of the system. Thus if Zt (the level of dam or waiting time) is regarded as an output process, then we can represent the process as follows [A = (X,Y), R = (rules of operation, capacity k, etc.)]:
$$S:\{ A,B\} \to Z$$
(3.1)
Thus we can have a set of systems S, S,..., such that
$${S_{i}}:\{ {A_{i}},{R_{i}}\} \to {Z_{i}}$$
(3.2)
Again Z. may be a vector instead of being a single valued variable, Zi = (Zii, Zi2,..., Zip). Thus in a queueing process we may write Z = (w,Q) where w is the waiting time process and Q is the queue size process.