IJPAM: Volume 51, No. 3 (2009)

ASYMPTOTIC ANALYSIS OF THE M/G/1 QUEUEING
SYSTEM WITH ADDITIONAL OPTIONAL SERVICE
AND NO WAITING CAPACITY

Ma Yuan-Yuan$^1$, Geni Gupur$^2$
$^{1,2}$College of Mathematics and System Science
Xinjiang University
Urumqi, 830046, P.R. CHINA
$^2$e-mail: geni@xju.edu.cn, genigupur@yahoo.cn


Abstract.In this paper, we will do dynamic analysis for the M/G/1 queueing system with additional optional service and no waiting capacity by using functional analysis. First we will convert the mathematical model of the queueing system into an abstract Cauchy problem in a Banach space, next we will prove that the operator corresponding to the model generates a positive contraction $C_0$-semigroup, which is isometric for the initial value of the model. Thus we will obtain that the model has a unique positive time-dependent solution which satisfies probability condition. Third we will prove that the $C_0$-semigroup is a quasi-compact operator. From which we will deduce that the $C_0$-semigroup converges exponentially to a positive projection operator, and for special case, the time-dependent solution of the model converges strongly to the steady-state solution as time tends to infinite. Fourth we will discuss eigenvalues of the operator in the left half complex plane when the service rates are constants and then will give expression of the project operator by using the residue theorem in complex analysis. Last from the above steps we deduce that the time-dependent solution of the model converges exponentially to the steady-state solution of the model.

Received: May 2, 2007

AMS Subject Classification: 47D03, 47A10

Key Words and Phrases: $C_0$-semigroup, quasi-compact operator, eigenvalue, resolvent set

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 51
Issue: 3