Sojourn times in fluid queues with independent and dependent input and output processes

Markov Fluid Queues (MFQs) are the continuous counterparts of quasi birth–death processes, where infinitesimally small jobs (fluid drops) are arriving and are being served according to rates modulated by a continuous time Markov chain. The fluid drops are served according to the First-Come–First-Served (FCFS) discipline. The queue length process of MFQs can be analyzed by efficient numerical methods developed for Markovian fluid models. In this paper, however, we are focusing on the sojourn time distribution of the fluid drops. In the first part of the paper we derive the phase-type representation of the sojourn time when the input and output processes of the queue are dependent. In the second part we investigate the case when the input and output processes are independent. Based on the age process analysis of the fluid drops, we provide smaller phase-type representations for the sojourn time than the one for dependent input and output processes

Sample path large deviations for multiclass feedforward queueing networks in critical loading

We consider multiclass feedforward queueing networks with first in first out and priority service disciplines at the nodes, and class dependent deterministic routing between nodes. The random behavior of the network is constructed from cumulative arrival and service time processes which are assumed to satisfy an appropriate sample path large deviation principle. We establish logarithmic asymptotics of large deviations for waiting time, idle time, queue length, departure and sojourn-time processes in critical loading. This transfers similar results from Puhalskii about single class queueing networks with feedback to multiclass feedforward queueing networks, and complements diffusion approximation results from Peterson. An example with renewal inter arrival and service time processes yields the rate function of a reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org