For a discrete modified $GI/GI/c/\infty$ queue, $1\le c<\infty$, where the service times of all customers served during any busy period are independent random variables with not necessarily identical distribution functions, the joint distribution of the busy period, the subsequent idle period and the number of customers served during the busy period is derived. The formulae presented are in a convenient form for practical use. The paper is a continuation of [5], where the $M/GI/c/\infty$ discrete modified queue has been studied.