In this paper we consider the repairmen problem from the point of view of production theory. This problem, which has a large literature,3 can be described as follows. There are m identical machines which operate independently and which normally require no human care. However, at any time a machine may break down and call for service, according to an exponential failure distribution. Furthermore, it is assumed that there are r repairmen, each capable of repairing one machine at a time. The time required for servicing a machine is again taken as a random variable with an exponential distribution. If fewer than r machines are being repaired a failed machine is immediately serviced. On the other hand, if r or more machines are being repaired, the failed machine joins a queue and waits for service. The queue discipline assumes that all repairmen are working if a machine is waiting, but it is otherwise arbitrary. If we let x(t) be the total output produced at time t, then x(t) is proportional to the number of machines operating at time t, n(t), which follows a stochastic birth and death process. Our purpose in this paper is to study the limiting behaviour of x(t) over time and, in particular, the behaviour of the expected value of x(t) as m and r become large. We here prove that as the number of machines and repairmen increases, expected output becomes, in a way to be specified below, proportional to the number of repairmen or to the number of machines, depending upon whether the ratio of repairmen to machines is below or above a certain value. It has previously been shown, by means of numerical calculations [4], that the production function relating expected output to the number of machines and to the number of repairmen exhibits smooth isoquants (with continuous marginal rates of substitution). It seemed, therefore, that the random character of the production process had introduced smoothing effects into an otherwise fixed-coefficients technology.4 But, as is proved here, for a large number of machines, the production function is of the fixed-coefficients type. In Figure 1, the isoquant relating the inputs per unit of output is seen to approach an L shape as m becomes large. The critical value (the ray OA) depends on the parameters of the stochastic process, i.e. the average failure time and the average repair time. More generally, it can be shown that the sequence of output processes, properly translated and scaled, tends to be a limiting diffusion process, the well-known OrnsteinUhlenbeck process.
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