On the Proof of Takacs Lemma In the Stochastic Service System M/G/I

Abstract:

      L. Takacs' Lemma (ef [1] PP47-48) is a very important tool for the investigation of the transient behavior of the M/G/1 System. But the original proof contains a gap when γ(s, w) is expanded into Lagrange series by means of Lagrange theorem as given in the appendix of [1]. This is due to the fact that φ(s)=∫₀^∞e^-sxdH(x), the Laplace transform of the service time distribution , may not be analytic at s=0. An example of such a distrbution function H(x) is gives and a specific Lagrange theorem is proved By using this specific theorem, the gap in the original proof of Takacs′ Lemms is eliminated.