This paper is motivated by the response-time analysis of distributed information systems, where transactions are handled by a sequence of front-end server and back-end server actions. We study sojourn times in an open queueing network with a single Processor Sharing (PS) node and an arbitrary number of M multi-server First-Come-First-Served (FCFS) nodes. Customers arrive at the PS according to a Poisson process. After departing from the PS node a customer jumps to FCFS node k with probability pk, and departs from the system with probability 1 - p, where p = Σk-1M pk (0 < p < 1). After receiving service at a FCFS node, a customer jumps back to the PS node. For this model, we focus on the mean and the variability of the sojourn time of an arbitrary customer in the system. The model is a product-form network, which immediately leads to a closed-form expression for the mean sojourn times. The variance of the sojourn times, however, does not admit an exact expression; the complexity is caused by the possibility of overtaking. To this end, we propose a new methodology for deriving closed-form approximations for the variance of sojourn times in queueing networks with feedback. Numerical results from extensive experimentation with simulations demonstrates that the approximations are highly accurate for a wide range of parameter values.
Gijsen, B. M. M., van der Mei, R. D., van den Berg, H. L., & van Wingerden, K. M. C. (2006). Sojourn-time approximations in queueing networks with feedback. Performance evaluation, 63(A219/8), 743-758. [10.1016/j.peva.2005.08.002]. https://doi.org/10.1016/j.peva.2005.08.002