Summary: We consider a type 1 counter with exponential dead time and particles arriving in a Poisson process. A bivariate continuous time Markov process based on the number of registered particles and the state of the counter is constructed. This Markov process is then used to establish a system of Kolmogorov-type differential equations for the joint probabilities of the number of registered particles and the state of the counter. These probabilities are given in terms of probability generating functions.