Pseudo Steady-State Period in Non-Stationary Infinite-Server Queue with State Dependent Arrival Intensity

Abstract

An infinite-server queueing model with state-dependent arrival process and exponential distribution of service time is analyzed. It is assumed that the difference between the value of the arrival rate and total service rate becomes positive starting from a certain value of the number of customers in the system. In this paper, time until reaching this value by the number of customers in the system is called the pseudo steady-state period (PSSP). Distribution of duration of PSSP, its raw moments and its simple approximation under a certain scaling of the number of customers in the system are analyzed. Novelty of the considered problem consists of an arbitrary dependence of the rate of customer arrival on the current number of customers in the system and analysis of time until reaching from below a certain level by the number of customers in the system. The relevant existing papers focus on the analysis of time interval since exceeding a certain level until the number of customers goes down to this level (congestion period). Our main contribution consists of the derivation of a simple approximation of the considered time distribution by the exponential distribution. Numerical examples are presented, which confirm good quality of the proposed approximation.

Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate

“…• If function µ(x) is monotonically increasing and µ(∞) > λ, the steady state always exists; • If function µ(x) is monotonically decreasing, there is no stationary regime in the system (however, there can be a pseudo steady-state period [20]); •…”

“…The approach of this paper continues our study in [20], where the methodology of the asymptotic diffusion analysis for the state-dependent infinite-server queuing system is proposed. The proposed asymptotic method lets us solve System (1) by the approximation of the process under study by a diffusion process under some limit condition.…”

“…System (1) is a system of infinite number of differential equations, so its direct solving is quite hard. We propose the method of the asymptotic diffusion analysis [20] under the limit condition of a long delay σ → 0 for System (1) solving.…”

“…In our research, a retrial queue with multiple access is considered the motivation of this feature is based on the practical problems described above. We use an asymptotic diffusion method from the class of asymptotic analysis methods of queuing theory [20,21]. We introduce a modification of the approach for the retrial queue under consideration.…”

“…If a(x) > 0, the system is in a non-stationary mode; • If equation a(x) = 0 has one root and a(∞) < 0, the system is in a steady state. It is easy to show that µ(∞) > λ is a stability condition; • If equation a(x) = 0 has two roots, the system is a non-stationary mode, but there is a pseudo steady-state period [20].…”