The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement*

Erlang's classical model for telephone traffic in a loss system is considered: N trunks, calls arriving in a Poisson process and negative exponential holding times; calls which cannot be served at once are dismissed without retrials. Let N (t) be the number of trunks in use at t. An explicit formula for the covariance R (·) of N (·) in terms of the characteristic values of the transition matrix of the Markov process N (·) is obtained. Also, R (·) is expressed purely in terms of constants and the “recovery” function, i.e. the transition probability Pr {N(t) = N|N(0) = N}; R (·) is accurately approximated byR(0)e^r₁ᵗwith r1the largest negative characteristic value, itself well approximated (underestimated) by − E{N (·)}/R(0). Exact and approximate formulas for sampling error in traffic measurement are deduced from these results.