Transient analysis of a bulk stream queue with arbitrarily distributed arrival intervals

Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad campus, Hyderabad 500078, India

^* Corresponding author: faridaparvezb@hyderabad.bits-pilani.ac.in

Received: 3 October 2023
Accepted: 10 May 2024

Abstract

We study the classical infinite buffer single server queueing model with renewal input of customers in batches of random size, having arbitrarily distributed arrival intervals and exponentially distributed service times. Using the technique of supplementary variable and shift operator we derive closed form expression of the time dependent system content distribution in terms of its Laplace transform. The analysis is mainly based on the root-finding technique of the non-linear characteristic equation in terms of the Laplace transform variable. Additionally, using asymptotic properties of Laplace transform, we deduce the corresponding steady-state distribution. We discuss some special cases of the model, thus providing an alternative approach in deriving the transient distribution. We further evaluate certain performance measures and present extensive numerical examples in tabular and graphical form to illustrate the applicability of our theoretical work. The effect of system parameters, interarrival time distribution and traffic intensity on the system behavior is also demonstrated.