Waiting time probabilities in the M/G/1 + M queue

We consider an M/G/1 queueing system where the customers may leave the queue if their services do not commence before an exponentially distributed random time. The (conditional) offered waiting time distribution is approximated by a gamma distribution via matching the first and second moments of the actual waiting time. A simulation study is conducted to assess the accuracy of the approximation and it reveals that the approximation performs satisfactorily under general conditions on service time distributions.