Transient behaviour of time-varying tandem queueing networks

Most of the large-scale service systems in real life are subject to time-varying conditions, such as arrival rates, service rates, and other factors that can affect system performance. These systems can be adequately modelled using time-varying queueing systems, where one or several parameters change over time. Analysis of transient behaviour in such time-varying queueing systems is more challenging than their steady-state analysis. This study deals with the transient analysis of Markovian queues connected in tandem, where both the service and arrival processes at each station depend on time. To begin with, we derive the transient distributional relationship between the average workload and the customer’s waiting time in a single-server non-Markovian queue with time-varying arrival and service rates. We then generalise the transient laws for single-server queues to a k-station tandem network. Furthermore, we develop an algorithm for analysing transient performance measures in a k-station tandem queueing network and conduct a numerical study based on the algorithm. Numerical study supports the effectiveness of the algorithm, and the results provide insights into the transient behaviour of tandem networks, specifically in bottleneck scenarios. The study reveals that the location of the bottleneck station in a line has a significant impact on average workload in the stations.