IJPAM: Volume 64, No. 2 (2010)

ON THE M/M/1 QUEUEING MODEL WITH
COMPULSORY SERVER VACATIONS

Geni Gupur
College of Mathematics and Systems Science
Xinjiang University
Urumqi, 830046, P.R. CHINA
e-mails: [email protected], [email protected]


Abstract.By studying the spectrum of the operator corresponding to the M/M/1 queueing model with compulsory server vacations we discuss asymptotic behavior of the time-dependent solution of the model and asymptotic behavior of the time-dependent queueing length. First of all, through studying the resolvent set of the adjoint operator of the operator we obtain that all points on the imaginary axis except for zero belong to the resolvent set of the operator. Next, we prove that zero is an eigenvalue of the adjoint operator. In addition, we consider eigenvalue of the operator and get the explicit result when $\mathcal{M}=1.$ From the above results we deduce that the time-dependent solution of the model strongly converges to its steady-state solution and the time-dependent queueing length converges to a positive number when $\mathcal{M}=1$.

Received: August 2, 2010

AMS Subject Classification: 47A10, 47D03, 60K25

Key Words and Phrases: M/M/1 queueing model with compulsory server vacations, resolvent set, eigenvalue

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 64
Issue: 2