Machine interference with general repair and running times

Abstract

Machine maintenance and repair operations, alternatively called machine interference, may be effectively represented as finite source queueing processes. This paper presents a general solution to such processes where both the repair times and running (operating) times are modelled as Erlang distributions (the (Eh/Em/1)/K model). Such a general solution is unavailable elsewhere.The solution is made possible by two devices. Firstly the finite source process is replaced by an equivalent two stage cyclic queueing process where the two stages are the repair stage and the operating stage. Secondly a one-dimensional state definition is used for each stage giving a total two-dimensional state definition for the whole process. This two-dimensional state permits the straightforward visualization and drawing of the rate diagram and the consequent extraction of the steady state balance equations from this rate diagram. A numerical solution procedure for the balance equations is also outlined as is an extension to the multiple repair facility case (the (Eh/Em/c)/K model).Notation: The notation used in this paper for finite source queueing models is (-/-/e)/K where the first slot refers to the probability distribution for the machine operating times, the second slot refers to the probability distribution for the repair times, e = 1, 2, ... refers to the number of repair facilities and K = 1, 2, ... refers to the number of machines allocated to the c repair facilities.

Queues in Reliability

“…[16] and [17]). Many other features have been introduced to extend the applicability of the machine repairmen problem including, for example, vacation models where the servers can take a vacation of random duration when the system is empty (see e.g.…”