Strong stability in a G/M/1 queueing system. (English) Zbl 1097.60076

Benaouicha, Mustapha; Aissani, Djamil

Teor. Jmovirn. Mat. Stat. 71, 22-32 (2004) and Theory Probab. Math. Stat. 71, 25-36 (2005).

The authors study the strong stability of the stationary distribution of the imbedded Markov chain in the G/M/1 queueing system, after perturbation of the service law. Let \(\mu\) be a parameter of exponentially distributed service time; let \(\bar\tau\) be a mean time between arrivals in G/M/1 queueing system; let \(\sigma\) be a unique solution of the equation \(\sigma=F^{*} (\mu-\mu\sigma)= \int_{0}^{\infty} e^{-(\mu-\mu\sigma)x}\,dF(x)\), where \(F^{*}\) is the Laplace transform of the probability density function of the demands inter-arrival times.
The following result is proved: Suppose that in the G/M/1 system the geometric ergodicity condition \(\mu\bar\tau>1\) holds. Then for all \(\beta\in R^{+}\) such that \(1<\beta<\sigma^{-1}\) the imbedded Markov chain \(\bar X_{n}\) in G/M/1 system is strongly \(v\)-stable for the function \(v(k,x)=\beta^{k}e^{\delta x}\), where \(0<\delta=\mu-\mu/\beta<\mu\) and \(\rho=\beta F^{*}(\mu-\mu/\beta)<1\).
The authors show that under some hypotheses the characteristics of the G/G/1 queueing system can be approximated by the corresponding characteristics of the G/M/1 system. The stability inequalities with exactly computing of the constants are obtained.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60K25	Queueing theory (aspects of probability theory)
68M20	Performance evaluation, queueing, and scheduling in the context of computer systems
90B22	Queues and service in operations research

Keywords:

stationary distribution; imbedded Markov chain; perturbation of the service law