Asymptotic behavior of the time-dependent solution of an M/G/1 queueing model
1 College of Mathematics and Systems Science, Xinjiang University, Urumqi, 830046, P.R. China
2 School of Mathematical Sciences, Xinjiang Normal University, Urumqi, 830054, P.R. China
Boundary Value Problems 2013, 2013:17 doi:10.1186/1687-2770-2013-17Published: 11 February 2013
We study the spectrum on the imaginary axis of the underlying operator which corresponds to the M/G/1 queueing model with exceptional service time for the first customer in each busy period that was described by infinitely many partial differential equations with integral boundary conditions and obtain that all points on the imaginary axis except 0 belong to the resolvent set of the operator and 0 is an eigenvalue of the operator and its adjoint operator. Thus, by combining these results with our previous results, we deduce that the time-dependent solution of the model converges strongly to its steady-state solution. Moreover, we show that our result on convergence is optimal.
MSC: 47A10, 47D99.