On trunks with negative exponential holding times serving a renewal process

A group of N trunks serves calls arriving in a renewal process, and lost calls are cleared. The number, N(k), of trunks found busy by the kth arriving customer is studied as a Markov process imbedded in a (usually) non-Markov process N(t), the number of trunks busy at t. Results of C. Palm and F. Pollaczek on the distribution of N(k) are generalized, and a study is made of bounds for, and approximations to, the probability of loss. The probability of loss is studied as a functional of the interarrival distribution function, and certain extremal properties are proven. Formulas for the mean of N(k) and for the covariance function are given, together with equilibrium curves for the probobility of loss, for the mean and variance of N(k), and for the first four values of the covariance function. Some applications to switch counting are discussed.