On the expected length of an orderly path

Lutz et al. (Ann Oper Res 289:463–472, 2020) derived an integral expression for the expected length of an orderly path. We show here that the integral expression can be reduced to a closed form expression in terms of the Gauss hypergeometric function, a generalised hypergeometric function and the Appell hypergeometric function of the first kind. We check the correctness of the closed form expression numerically as well as provide a Mathematica code. Finally, we show that the closed form expression is more efficient than known integral expressions.