Taking the multilingual convenience service hotline for a practical example, we study the numerical method for calculating the customer's sojourn time distribution function in M/G_N/1 queue with customized service. Firstly, we give the Laplace-Stieltjes (LS) transform of the customer's sojourn time by using the embedded Markov chain technique and Pollaczek-Khintchine formula. Secondly, according to the specific type of customized service time distribution function, we give the rational form of the LS transform that mentioned above. By solving the zeros with negative real parts of the denominator of the rational function, namely, the so-called characteristic roots, we finally give the customer's sojourn time probability distribution function by using the method of partial fraction and residue theory.