# IJPAM: Volume 64, No. 2 (2010)

**ON THE M/M/1 QUEUEING MODEL WITH**

COMPULSORY SERVER VACATIONS

COMPULSORY SERVER VACATIONS

College of Mathematics and Systems Science

Xinjiang University

Urumqi, 830046, P.R. CHINA

e-mails: genigupur@yahoo.cn, geni@xju.edu.cn

**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
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 .

**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