A Discrete-Time Queueing Model in a Random Environment

• Published in: Queueing Theory and Network Applications pp. 330 - 348
• Main Authors: Nobel, Rein, Rondaij, Annette
• Format: Book Chapter
• Language: English
• Published: Cham Springer International Publishing 2019
• Series: Lecture Notes in Computer Science
• Subjects: Discrete-time queue; Generating functions; Random environment
• ISBN: 3030271803; 9783030271800
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-27181-7_20

To study the effect of burstiness in arrival streams on the congestion in queueing systems this paper presents a one-server queueing model in a random environment in discrete time. The environment can be in two states. The number of time slots between two consecutive transitions of the environment follows a geometric distribution with a transition-dependent parameter. In every slot customers arrive in batches, and the batch-size distribution depends on the environment. Each customer requires a generally distributed service time. Arriving customers are put in a queue which is served in FIFO order. Arrivals have precedence over departures and departures have precedence over a change of the environment. The generating functions of the number of customers in the queue and the individual waiting time will be derived. Numerical results will show the effect of the burstiness in the arrival stream on the waiting-time and the queue-size distribution by calculating in parallel the corresponding results for the standard discrete-time model with a mixed batch-size distribution, ceteris paribus.