Queueing Models and Service Management

Characterizing the Output Process of Two-Stage Flow Lines with Unreliable Parallel Machines and Finite Intermediate Buffer

B. Madhu Rao — 2025-09-03

This paper analyzes two-stage flow lines where raw material is processed sequentially in two stages to produce finished units. There are multiple machines at each stage that randomly break down and require repair. There is a limited amount of storage space (buffer)
between the two stages. When the buffer is full, some or all machines at the first stage may be blocked (i.e., forced to idle due to the inability to unload a finished unit), and when the buffer is empty, some or all machines at the second stage may starve (i.e., forced to idle due to a lack of jobs for processing). The state changes in the system can be described by a continuous-time Markov chain when processing times, times between machine failures, and repair times are exponentially distributed. The study focuses on the variability and autocorrelation structure of the output stream of finished products from stage two. Efficient algorithms are developed to compute steady-state system characteristics using matrix analytical methods. The paper presents detailed numerical results highlighting the qualitative features of system behavior for a wide range of parameter values. Our key finding is that the output process of the system approximates a Poisson process for buffer size as small as one, and the numbers of machines at the two stages as small as two.

Study of an Early Arrival System on GeoX/G/1 Queue with Single Vacation

Aysha Parveen — 2025-09-03

This paper investigates the GeoX/G/1 queue with single vacation in an early arrival system. The time between arrival batches follows a geometric distribution, while the service and vacation durations are expressed as integral multiples of a slot duration and can follow arbitrary distributions. This study focuses on analyzing the system length distributions at different time epochs, the waiting time distribution for a random customer within a batch, and performing a cost analysis using the theory of difference equations and the supplementary variable technique. This method has an advantage over traditional queueing analysis techniques, as it eliminates the need to compute the transition probability matrix for the embedded system length process. We evaluate key performance metrics and demonstrate the computational process through numerical examples.

A Note on Computing Approach Toward Two-tier Service Models

Hsing Paul Luh — 2025-09-03

This paper presents a new algorithm for computing the performance measure of a two-tier service queueing model. In such a system, one service provider offers service with unlimited waiting space and the other offers a finite waiting space. Due to the two queue feature, the system is formulated as a state dependent quasi-birth-and-death (QBD) process. The customer choice behavior and the observable queues make the QBD process to have a large number of boundary states. Such a structure motivates us to develop a more efficient algorithm than the classical rate matrix iteration algorithms. With the special structure of the infinitesimal generator matrix for the two-tier service system, we propose a more efficient and innovative K-matrix based algorithm for computing the stationary distribution. As the buffer size increases, the improved accuracy and computational efficiency of the K-matrix method become significant compared with the classical Geometric-Matrix method. We demonstrate the advantages of the new algorithm with numerical examples.