Stability of Parallel Server Systems

References

• Asmussen S (2003) Applied Probability and Queues (Springer Verlag, Berlin).Google Scholar
• Atar R, Keslassy I, Mendelson G (2019) Subdiffusive load balancing in time-varying queueing systems. Oper. Res. 67(6):1678–1698.Link, Google Scholar
• Baccelli F, Brémaud P (2002) Elements of Queueing Theory, 2nd ed. (Springer, Berlin).Google Scholar
• Bramson M (1994) Instability of FIFO queueing networks. Ann. Appl. Probabilities 4(2):414–431.Crossref, Google Scholar
• Bramson M (2008) Stability of Queueing Networks (Springer, Berlin).Google Scholar
• Bramson M (2011) Stability of join the shortest queue networks. Ann. Appl. Probabilities 21(4):1568–1625.Crossref, Google Scholar
• Bramson M, Lu Y, Prabhakar B (2010) Randomized load balancing with general service time distributions. Performance Evaluation Rev. 38(1):275–286.Crossref, Google Scholar
• Brandt A (1985) On stationary waiting times and limiting behavior of queues with many servers I: The general G/G/m/∞ case. Elektronische Informationsverarbeitung und Kybernetik 21:47–64.Google Scholar
• Brémaud P (1981) Point Processes and Queues: Martingale Dynamics (Springer, New York).Crossref, Google Scholar
• Brémaud P (1999) Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues (Springer, New York).Crossref, Google Scholar
• Brightwell G, Luczak M (2012) The supermarket model with arrival rate tending to one. Preprint, submitted January 26, 2012, https://arxiv.org/abs/1201.5523.Google Scholar
• Dai JG (1995) On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models. Ann. Appl. Probabilities 5(1):49–77.Crossref, Google Scholar
• Daley DJ (1987) Certain optimality properties of the first-come first-served discipline for G/G/s queues. Stochastic Processing Appl. 25:301–308.Crossref, Google Scholar
• Decreusefond L, Moyal P (2012) Stochastic Modeling and Analysis of Telecom Networks (ISTE Wiley).Crossref, Google Scholar
• Dester PS, Fricker C, Tibi D (2017) Stationary analysis of the shortest queue problem. Queueing Systems 87:211–243.Google Scholar
• Eschenfeldt P, Gamarnik D (2015) Join the shortest queue with many servers. The heavy traffic asymptotics. Math. Oper. Res. 43(3):867–886.Google Scholar
• Eschenfeldt P, Gamarnik D (2016) Supermarket queueing system in the heavy traffic regime. Short queue dynamics. Preprint, submitted January 17, 2017, https://arxiv.org/abs/1610.03522.Google Scholar
• Fayolle G, Malyshev VA, Menshikov M (1995) Topics in the Constructive Theory of Countable Markov Chains (Cambridge University Press).Crossref, Google Scholar
• Flatto L, Mc Kean HP (1977) Two queues in parallel. Comm. Pure Appl. Math. 15:255–263.Crossref, Google Scholar
• Foschini GJ, Salz J (1978) A basic routing problem and diffusion. IEEE Trans. Comm. 26:320–327.Crossref, Google Scholar
• Foss S (1981) Comparison of service disciplines in multichannel service systems. Sibirskii Matematicheskii Zhurnal 22(1):190–197.Google Scholar
• Foss S, Chernova N (1998) On the stability of a partially accessible multi-station queue with state-dependent routing. Queueing Systems 29(1):55–73.Crossref, Google Scholar
• Graham C (2000) Chaoticity on path space for a queueing network with selection of the shortest queue among several. J. Appl. Probabilities 37:198–211.Crossref, Google Scholar
• Graham C (2005) Functional central theorems for a large network in which customers join the shortest among several queues or a queueing network with selection of the shortest of several queues. Probabilities Theory Related Fields 131:97–120.Crossref, Google Scholar
• Haight FA (1958) Two queues in parallel. Biometrika 45:401–410.Crossref, Google Scholar
• Khalil HK (2002) Nonlinear Systems (Prentice Hall, Upper Saddle River, NJ).Google Scholar
• Kiefer J, Wolfowitz J (1955) On the theory of queues with many servers. Trans. Amer. Math. Soc. 78:1–18.Crossref, Google Scholar
• Kingman JFC (1961) Two similar queues in parallel. Ann. Math. Statist. 32(4):1314–1323.Crossref, Google Scholar
• Kulkarni VG (2017) Modeling and Analysis of Stochastic Systems (Chapman and Hall/CRC).Google Scholar
• Liberzon D (2003) Switching in Systems and Control (Birkäuser).Crossref, Google Scholar
• Loynes RM (1962) The stability of queues with non-independent interarrivals and service times. Proc. Cambridge Philos. Soc. 58:497–520.Crossref, Google Scholar
• Lu SH, Kumar PR (1991) Distributed scheduling based on due dates and buffer priorities. IEEE Trans. Automated Control 36(12):1406–1416.Crossref, Google Scholar
• Lu Y, Xie Q, Kliot G, Geller A, Larus JR, Greenberg A (2011) Join-idle-queue: A novel load balancing algorithm for dynamically scalable web services. Performance Evaluation 68(11):1056–1071.Crossref, Google Scholar
• Luczak MJ, McDiarmid C (2006) On the maximum queue length in the supermarket model. Ann. Probabilities 34(2):493–527.Crossref, Google Scholar
• Luczak MJ, McDiarmid C (2007) Asymptotic distributions and chaos for the supermarket model. Electronic J. Probabilities 12:75–99.Google Scholar
• Marshall AW, Olkin I (1979) Inequalities: Theory of Majorization and Its Applications (Academic Press, New York).Google Scholar
• Mitzenmacher M (1996) The power of two choices in randomized load balancing. PhD thesis, University of California, Berkeley.Google Scholar
• Moyal P (2017a) On the stability of non-monotonic systems of parallel queues. Discrete Event Dynamic Systems 27(1):85–107.Crossref, Google Scholar
• Moyal P (2017b) A pathwise comparison of parallel queues. Discrete Event Dynamic Systems 27(3):573–584.Crossref, Google Scholar
• Moyal P, Perry O (2017) On the instability of matching queues. Ann. Appl. Probabilities 27(6):3385–3434.Crossref, Google Scholar
• Pang G, Talreja R, Whitt W (2007) Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probability Survey 4:193–267.Crossref, Google Scholar
• Perry O, Whitt W (2016) Chattering and congestion collapse in an overload switching control. Stochastic Systems 6(1):132–210.Link, Google Scholar
• Reiman MI (1984) Some diffusion approximations with state space collapse. Modelling and Performance Evaluation Methodology (Springer, Berlin), 207–240.Crossref, Google Scholar
• Robert P (2003) Stochastic Networks and Queues (Springer-Verlag).Crossref, Google Scholar
• Rybko AN, Stolyar AL (1992) Ergodicity of stochastic processes describing the operations of open queueing networks. Problemy Peredachi Informatsii 28:3–26 (in Russian).Google Scholar
• Scheller-Wolf A (2003) Necessary and sufficient conditions for delay moments in FIFO multiserver queues with and application comparing s slow servers with one fast one. Oper. Res. 51(5):748–758.Link, Google Scholar
• Turner SR (1998) The effect of increasing routing choice on resource pooling. Probabilities Engrg. Inform. Sci. 12(1):109–124.Crossref, Google Scholar
• Tweedie RL (1981) Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markoc processes. J. Appl. Probabilities 18(1):122–130.Crossref, Google Scholar
• Vvedenskaya ND, Dobrushin RLV, Karpelevich FI (1996) Queueing system with selection of the shortest of two queues: An asymptotic approach. Problemy Peredachi Informatsii 32(1):20–34.Google Scholar
• Weber RW (1978) On the optimal assignment of customers to parallel servers. J. Appl. Probabilities 15(2):406–413.Crossref, Google Scholar
• Whitt W (1985) Deciding which queue to join: some counterexamples. Oper. Res. 34(1):55–62.Link, Google Scholar
• Whitt W (2002) Stochastic Process Limits (Springer, New York).Crossref, Google Scholar
• Winston W (1977) Optimality of the shortest line discipline. J. Appl. Probabilities 14(1):181–189.Crossref, Google Scholar
• Xu J, Hajek B (2013) The supermarket game. Stochastic Systems 3(2):405–441.Link, Google Scholar