Queuing With Fixed and Variable Channels

Abstract

The process treated in this paper is one in which the number of service channels is limited and the queue is unlimited. Arrivals and services are under steady-state conditions and exponentially distributed. Arrivals form in a single queue under strict queue discipline. The number of manned channels increases from a fixed minimum number, σ, when the queue reaches a given length N. When the maximum number of channels S, (S > σ) are operating, no further increases are possible and the queue is unbounded. Channels are cancelled when the number of units in the queue drops to v(O ≦ v ≦ N − 2) and a service is completed. Seven model equations describing the process are solved, giving explicit expressions for the state probabilities. The usual measures of effectiveness are developed, together with a new measure appropriate to this model, i.e., the mean number of channel starts per unit time. Sample computations are made on the measures of effectiveness, indicating their sensitivity to the parameters and their application to various problems.