Pseudo steady-state period in non-stationary infinite-server queue with state dependent arrival intensity

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

Transient Analysis of Large-scale Stochastic Service Systems

The transient analysis of large-scale systems is often difficult even when the systems belong to the simplest M/M/n type of queues. To address analytical difficulties, previous studies have been conducted under various asymptotic regimes by suitably accelerating parameters, thereby establishing some useful mathematical frameworks and giving insights into important characteristics and intuitions. However, some studies show significant limitations when used to approximate real service systems: (i) they are more relevant to steady-state analysis; (ii) they emphasize proofs of convergence results rather than numerical methods to obtain system performance; and (iii) they provide only one set of limit processes regardless of actual system size. Attempting to overcome the drawbacks of previous studies, this dissertation studies the transient analysis of large-scale service systems with time-dependent parameters. The research goal is to develop a methodology that provides accurate approximations based on a technique called uniform acceleration, utilizing the theory of strong approximations. We first investigate and discuss the possible inaccuracy of limit processes obtained from employing the technique. As a solution, we propose adjusted fluid and diffusion limits that are specifically designed to approximate large, finite-sized systems. We find that the adjusted limits significantly improve the quality of approximations and hold asymptotic exactness as well. Several numerical results provide evidence of the effectiveness of the adjusted limits. We study both a call center which is a canonical example of large-scale service systems and an emerging peer-based Internet multimedia service network known as P2P. Based on our findings, we introduce a possible extension to systems which show non-Markovian behavior that is unaddressed by the uniform acceleration technique. We incorporate the denseness of phase-type distributions into the derivation of limit processes. The proposed method offers great potential to accurately approximate performance measures of non-Markovian systems with less computational burden

Analysis of buffer allocations in time-dependent and stochastic flow lines

This thesis reviews and classifies the literature on the Buffer Allocation Problem under steady-state conditions and on performance evaluation approaches for queueing systems with time-dependent parameters. Subsequently, new performance evaluation approaches are developed. Finally, a local search algorithm for the derivation of time-dependent buffer allocations is proposed. The algorithm is based on numerically observed monotonicity properties of the system performance in the time-dependent buffer allocations. Numerical examples illustrate that time-dependent buffer allocations represent an adequate way of minimizing the average WIP in the flow line while achieving a desired service level