Renewal processes arising in the study of multiplex systems

Abstract

The object of this report is to discuss the improvement of reliability of systems when redundancy is introduced in the form of so-called 'multiplexing' so that a given task is performed not by one suitably chosen set of components (referred to as a 'lane') but by a number of separate lanes operating independently in parallel. Under the assumption that the failure of individual lanes can be described by a Poissonian Process it is shown that the Renewal Process representing the failure mechanism of a system composed of m lanes exhibits, for comparatively small time intervals, a dramatic improvement of reliability (from a small probability p for one lane to a much smaller probability pm for the system) whilst for the large time intervals the improvement is less than proportional to m. Thus the multiplexing appears to achieve its maximum advantage only when, after a comparatively short period of operation, all the lanes are inspected and brought to their initial state by repair or replacement; without these precautions multiplexing is still useful but the law of diminishing returns operates then: by increasing the number of lanes, less and less is added to the asymptotic reliability of the system. The discussion of the general case of m lanes is followed by a more detailed analysis of duplex and triplex systems and, in the closing section, a modified method of multiplexing (the 'majority vote') is described.