Tandem Queueing Systems Maximum Throughput Problem

TL;DR

This paper investigates the maximum throughput of tandem queueing systems modeled as Quasi-Birth-Death processes, analyzing how system parameters affect stability and providing insights into system behavior with multiple servers and buffer capacities.

Contribution

The paper models tandem queueing systems as QBD processes to analyze maximum throughput, including effects of buffer sizes, server order, and extending to general arrival processes.

Findings

Maximum throughput formulas become complex with more servers

Maximum throughput converges as server count increases

Reversing server order does not affect maximum throughput in homogeneous systems

Abstract

In this paper we consider the problem of maximum throughput for tandem queueing system. We modeled this system as a Quasi-Birth-Death process. In order to do this we named level the number of customers waiting in the first buffer (including the customer in service) and we called phase the state of the remining servers. Using this model we studied the problem of maximum throughput of the system: the maximum arrival rate that a given system could support before becoming saturated, or unstable. We considered different particular cases of such systems, which were obtained by modifying the capacity of the intermediary buffers, the arrival rate and the service rates. The results of the simulations are presented in our paper and can be summed up in the following conclusions: 1. The analytic formula for the maximum throughput of the system tends to become rather complicated when the number of servers increase 2. The maximum throughput of the system converges as the number of servers increases 3. The homogeneous case reveals an interesting characteristic: if we reverse the order of the servers, maximum thruoughput of the system remains unchanged The QBD process used for the case of Poisson arrivals can be extended to model more general arrival processes.