Stability analysis of parallel server systems under longest queue first

• Published in: Mathematical methods of operations research (Heidelberg, Germany) Vol. 74; no. 2; pp. 257 - 279
• Main Authors: Baharian, Golshid, Tezcan, Tolga
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2011; Springer; Springer Nature B.V
• Subjects: Business and Management; Calculus of Variations and Optimal Control; Optimization; Customer services; Drift; Graphs; Markov analysis; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Operations research; Operations Research/Decision Theory; Original Article; Queues; Queuing; Servers; Simulation; Stability; Studies; Trees; Stability; Longest queue first; Parallel server systems; Fluid model
• ISSN: 1432-2994; 1432-5217
• DOI: 10.1007/s00186-011-0362-5

We consider the stability of parallel server systems under the longest queue first (LQF) rule. We show that when the underlying graph of a parallel server system is a tree, the standard nominal traffic condition is sufficient for the stability of that system under LQF when interarrival and service times have general distributions. Then we consider a special parallel server system, which is known as the X-model, whose underlying graph is not a tree. We provide additional “drift” conditions for the stability and transience of these queueing systems with exponential interarrival and service times. Drift conditions depend in general on the stationary distribution of an induced Markov chain that is derived from the underlying queueing system. We illustrate our results with examples and simulation experiments. We also demonstrate that the stability of the LQF depends on the tie-breaking rule used and that it can be unstable even under arbitrary low loads.