Cybernetic Queueing and Storage Systems

Abstract

Any system comprises a set of elements A = (a₁,..., a_n) 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 Z_t (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, Z_i = (Z_ii, Z_i2,..., Z_ip). 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.