Waiting-Time Distributions for Application to a Series of Service Centers

The method described in this paper is a result of analyzing job shop production processes as a network of waiting lines at machine (production) centers. The waiting time distribution for a series of N work centers, P[∑j=1j=N Xj ≦ τ], is derived as a function of the waiting-time distributions at the individual work centers, P[Xj ≦ τ], j = 1, 2, …, N, where the cumulative waiting-time probability distribution at each work center is of the form P[Xj ≦ τ] = l − Kj exp(Cjτ), with Cj and Kj constant. The resulting formula for total waiting time is amenable to desk computation and may be useful in estimating job flow time distributions in production control applications for which the first-come, first-served, steady-state model described below is applicable. Also, the method of proof may suggest an approach for convolution in other waiting-line models.