Pseudo conservation for partially fluid, partially lossy queueing systems

  • Veeraruna Kavitha
  • Jayakrishnan Nair
  • Raman Kumar Sinha
S.I.: Queueing Theory and Network Applications


We consider a queueing system with heterogeneous customers. One class of customers is eager; these customers are impatient and leave the system if service does not commence immediately upon arrival. Customers of the second class are tolerant; these customers have larger service requirements and can wait for service. In this paper, we establish pseudo-conservation laws relating the performance of the eager class (measured in terms of the long run fraction of customers blocked) and the tolerant class (measured in terms of the steady state performance, e.g., sojourn time, number in the system, workload) under a certain partial fluid limit. This fluid limit involves scaling the arrival rate as well as the service rate of the eager class proportionately to infinity, such that the offered load corresponding to the eager class remains constant. The workload of the tolerant class remains unscaled. Interestingly, our pseudo-conservation laws hold for a broad class of admission control and scheduling policies. This means that under the aforementioned fluid limit, the performance of the tolerant class depends only on the blocking probability of the eager class, and not on the specific admission control policy that produced that blocking probability. Our psuedo-conservation laws also characterize the achievable region for our system, which captures the space of feasible tradeoffs between the performance experienced by the two classes. We also provide two families of complete scheduling policies, which span the achievable region over their parameter space. Finally, we show that our pseudo-conservation laws also apply in several scenarios where eager customers have a limited waiting area and exhibit balking and/or reneging behaviour.


Heterogeneous queues Multiclass queues Loss systems Fluid limits Pseudo-conservation Achievable region Complete policies Reneging Balking 


  1. Ancker, C. J, Jr., & Gafarian, A. V. (1963). Some queuing problems with balking and reneging-II. Operations Research, 11(6), 928–937.CrossRefGoogle Scholar
  2. Baxendale, Peter H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. The Annals of Applied Probability, 15(1B), 700–738.CrossRefGoogle Scholar
  3. Bertsimas, D., Paschalidis, I., & Tistsiklis, J. N. (1994). Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance. The Annals of Applied Probability, 4, 43–75.CrossRefGoogle Scholar
  4. Coffman, E. G., & Mitrani, I. (1979). A characterization of waiting time performance realizable by single server queues. Operations Research, 28, 810–821.CrossRefGoogle Scholar
  5. de Haan, Roland, Boucherie, Richard J., & van Ommeren, Jan-Kees. (2009). A polling model with an autonomous server. Queueing Systems, 62(3), 279–308.CrossRefGoogle Scholar
  6. Federgruen, A., & Groenevelt, H. (1988). M/G/c queueing systems with multiple agent classes: Characterization and control of achievable performance under nonpre-emptive priority rules. Management Science, 9, 1121–1138.CrossRefGoogle Scholar
  7. Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1). Hoboken: Wiley.Google Scholar
  8. Feller, W. (1972). An introduction to probability theory and its applications (Vol. 2). Hoboken: Wiley.Google Scholar
  9. Harchol-Balter, M. (2013). Performance modeling and design of computer systems: queueing theory in action. Cambridge: Cambridge University Press.Google Scholar
  10. Hoel, P. G., Port, S. C., & Stone, C. J. (1986). Introduction to stochastic processes. Long Grove: Waveland Press.Google Scholar
  11. Kavitha, V., & Sinha, R. (2017). Achievable region with impatient customers. In Proceedings of Valuetools.Google Scholar
  12. Kavitha, V., & Sinha, R. (2017b). Queuing with heterogeneous users: Block probability and Sojourn times. ArXiv preprint
  13. Kleinrock, L. (1964). A delay dependent queue discipline. Naval Research Logistics Quarterly, 11, 329–341.CrossRefGoogle Scholar
  14. Kleinrock, L. (1965). A conservation law for wide class of queue disciplines. Naval Research Logistics Quarterly, 12, 118–192.CrossRefGoogle Scholar
  15. Li, B., Li, L., Li, B., Sivalingam, K. M., & Xi-Ren, C. (2004). Call admission control for voice/data integrated cellular networks: Performance analysis and comparative study. IEEE Journal on Selected Areas in Communications, 22(4), 706–718.CrossRefGoogle Scholar
  16. Mahabhashyam, Sai  Rajesh, & Gautam, Natarajan. (2005). On queues with Markov modulated service rates. Queueing Systems Theory Applications, 51(1–2), 89–113.CrossRefGoogle Scholar
  17. Meyn, S. P., & Tweedie, R. L. (1993). Markov chains and stochastic stability, communications and control engineering series. London: Springer.CrossRefGoogle Scholar
  18. Mitrani, I., & Hine, J. (1977). Complete parametrized families of job scheduling strategies. Acta Informatica, 8, 61–73.Google Scholar
  19. Sesia, S., Baker, M., & Toufik, I. (2011). LTE-the UMTS long term evolution: From theory to practice. Hoboken: Wiley.CrossRefGoogle Scholar
  20. Shaked, M., & Shanthikumar, J. G. (2007). Stochastic orders. Berlin: Springer.CrossRefGoogle Scholar
  21. Shanthikumar, J. G., & Yao, D. D. (1992). Multiclass queueing systems: Polymatroidal structure and optimal scheduling control. Operations Research, 40(3–supplement–2), S293–S299.CrossRefGoogle Scholar
  22. Sleptchenko, A., van Harten, A., & van der Heijden, M. C. (2003). An exact analysis of the multi-class M/M/k priority queue with partial blocking (pp. 527–548). Milton Park: Taylor & Francis.Google Scholar
  23. Takács, L. (1962). An Introduction to queueing theory.Google Scholar
  24. Tang, S., & Li, W. (2004). A channel allocation model with preemptive priority for integrated voice/data mobile networks. In Proceedings of the 1st international conference on quality of service in heterogeneous wired/wireless networks.Google Scholar
  25. White, Harrison, & Christie, Lee S. (1958). Queuing with pre-emptive priorities or with breakdown. Operations Research, 6(1), 79–95.CrossRefGoogle Scholar
  26. Whitt, Ward. (1999). Improving service by informing customers about anticipated delays. Management Science, 45(2), 192–207.CrossRefGoogle Scholar
  27. Zhang, Yan, Soong, Boon-Hee, & Ma, Miao. (2006). A dynamic channel assignment scheme for voice/data integration in GPRS networks. Elsevier Computer Communications, 29, 1163–1163.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IEORIndian Institute of TechnologyMumbaiIndia
  2. 2.EEIndian Institute of Technology BombayMumbaiIndia

Personalised recommendations