Two Dimension Marginal Distributions of Crossing Time and Renewal Numbers Related to Two-Stage Erlang Processes

Abstract

The two dimensional marginal transform, probability density and cumulative probability distribution functions for the random variables T_ξN (time taken by servers during vacations), ξ_N (number of vacations taken by servers) and N_η (number of customers or units arriving in the system) are derived by taking combinations of these random variables. One random variable is controlled at one time to determine the effect of the other two random variables simultaneously.