Abstract
This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.
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References
Asmussen S (2003) Applied probability and queues, 2nd edn. Springer
Bonald T, Borst S, Hegde N, Proutière A (2004) Wireless data performance in multi-cell scenarios. In: Proc. ACM Sigmetrics/Performance 2004, pp 378–388
Bonald T, Massoulié L, Proutière A, Virtamo J (2006) A queueing analysis of max-min fairness, proportional fairness and balanced fairness. Queueing Syst 53(1–2):65–84
Cohen J, Boxma OJ (1983) Boundary value problems in queueing system analysis. North-Holland, Amsterdam
Dai JG (1995) On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann Appl Probab 5(1):49–77
de Veciana G, Lee T-J, Konstantopoulos T (2001) Stability and performance analysis of networks supporting elastic services. IEEE/ACM Trans Netw 9(1):2–14
Fayolle G, Iasnogorodski R (1979) Two coupled processors: the reduction to a Riemann–Hilbert problem. Z Wahrsch Verw Gebiete 47(3):325–351
Fayolle G, Malyshev VA, Menshikov MV (1995) Topics in the constructive theory of countable Markov chains. Cambridge University Press
Jonckheere M, Borst SC (2006) Stability of multi-class queueing systems with state-dependent service rates. In: Proc. Valuetools’06
Kallenberg O (2002) Foundations of modern probability, 2nd edn. Springer
Kamae T, Krengel U, O’Brien GL (1977) Stochastic inequalities on partially ordered spaces. Ann Probab 5(6):899–912
Liu X, Chong E, Shroff N (2003) A framework for opportunistic scheduling in wireless networks. Comp Netw 41:451–474
Massey WA (1987) Stochastic orderings for Markov processes on partially ordered spaces. Math Oper Res 12(2):350–367
Meyn SP (1995) Transcience of multiclass queueing networks with via fluid limit models. Ann Appl Probab 5(4):946–957
Meyn SP, Tweedie RL (1993) Markov chains and stochastic stability. Springer.
Neuts MF (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv Appl Probab 10:185–212
Rao RR, Ephremides A (1988) On the stability of interacting queues in a multiple-access system. IEEE Trans Inf Theory 34(5):918–930
Robert P (2003) Stochastic networks and queues. Springer
Rogers LCG, Williams D (1994) Diffusions, Markov processes, and martingales, vol I, 2nd edn. Wiley
Szpankowski W (1988) Stability conditions for multidimensional queueing systems with computer applications. Oper Res 36(6):944–957
Szpankowski W (1994) Stability conditions for some distributed systems: buffered random access systems. Adv Appl Probab 26(2):498–515
Tweedie RL (1982) Operator-geometric stationary distributions for Markov chains, with application to queueing models. Adv Appl Probab 14:368–391
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Borst, S., Jonckheere, M. & Leskelä, L. Stability of Parallel Queueing Systems with Coupled Service Rates. Discrete Event Dyn Syst 18, 447–472 (2008). https://doi.org/10.1007/s10626-007-0021-4
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DOI: https://doi.org/10.1007/s10626-007-0021-4