PriMera Scientific Engineering (ISSN: 2834-2550)

Review Article

Volume 5 Issue 5

Control Theory for Queuing Systems with Moving Servers and Some Open Problems

Asaf Hajiyev*

October 30, 2024

View PDF

Abstract

For queuing systems with moving servers, the control policy which means delays of a beginning service is introduced. In the capacity of efficiency index of systems is taken a customer’s average waiting time before service. Although it seems that it is a paradoxical idea to introduce delays of beginning service, it is shown that for some systems it gives a gain in a customer’s average waiting time before service. The class of queuing systems for which it is advisable to introduce delays is described. The form of an optimal function minimizing the efficiency index is found.

It is shown that if the intervals between neighbor services have exponential distribution, then the gain in a customer’s average waiting time before service equals 10% and independent of parameter of exponential distribution. For uniform distribution such gain equals 3.5% and also independent of parameter of uniform distribution. The criterion to define for which systems the gain is greater are given. Some open problems and numerical examples demonstrating theoretical results are given.

Keywords: queues with moving servers; a customer’s average waiting time; delay of beginning service; optimal function

References

  1. Asmussen S. “Applied Probab. and Queues”. Springer-Verlag (2020).
  2. Baccelli F and Bremaud P. “Elements of Queueing Theory, 2003”. Springer-Verlag, Berlin (2003).
  3. Belyaev YuK., et al. “Markov approximation of stochastic model of motion on a two-road lane”. М., МАДИ (2002): 32 (in Russian).
  4. Blank M. “Ergodic properties of a simple deterministic traffic flow model”. J. Stat. Phys 111 (2003): 903-930.
  5. Borovkov AA. “Asymptotic methods in queuing theory”. John Willey Son, NY (1984).
  6. Bozejko W and Bocewicz G. “Modelling and Performance Analysis of Cyclic Systems”. Springer, Studies in Systems, Decision and Control 241 (2020).
  7. Cox D. “Renewal theory”. Chapman and Hall (1967).
  8. Franken P., et al. “Queues and Point Processes”. Wiley, Chichester (1982).
  9. Gazis DC. “Traffic Theory”. Berlin: Springer (2002).
  10. Gnedenko BV and Kovalenko IN. “Introduction to Queuing Theory”. Birkhauser (1989).
  11. Haight F. “Mathematical Theory of Traffic Flow”. Academic Press (1968).
  12. Hajiyev AH. Mammadov TSh. “Mathematical models of moving particles and their Applic”. Lambert, Academic Publish, Germany (2013): 134.
  13. Hajiyev AH and Mammadov TSh. “Cyclic queues with delays”. RAS, Doklady, Mathem 1 (2009).
  14. Hajiyev AH and Mammadov TSh. “Mathematical models of moving particles and its application”. Theory of Probab. Appl 56.4 (2011): 1-14.
  15. Kerner BS. “Introduction to Modern Traffic Flow Theory and Control”. Berlin: Springer (2009).
  16. Khinchin AYa. “Mathematical theory of queues”. М (1963) (in Russian).
  17. Kleinrock L. “Queuing systems”. John Wiley Sons 1.2 (2008).
  18. Long Z., et al. “Dynamic Scheduling of Multiclass Many-Server Queues with Abandonment: The Generalized cµ/h Rule”. Oper. Res 68 (2020): 1218-1230.
  19. Lee H-S and Srinivasan MM. “The shuttle dispatch problem with compound Poisson arrivals: controls at two terminals”. Queuing Systems 6 (1990): 207-222.
  20. Moeschlin O and Poppinga C. “Controlling traffic lights at a bottleneck with renewal arrival processes”. Proc. Inst. Math. And Mech. Azerb. National Acad. Sci 14 (2001): 187-194.
  21. Nagatani T. “The physics of traffic jams”. Rep. Prog. Phys 65 (2002): 1311-1356.
  22. Nagel K, Wagner R and Woesler R. “Still flowing: Approaches to traffic flow and traffic jam modelling”. Oper. Research 51.5 (2003): 681-710.
  23. Newell GF. “A simplified theory of kinematic waves in highway traffic, part I: General theory”. Transp. Research Part B. Methodological (1993).
  24. Newell GF. “Applications of queuing theory”. London: Chapman and Hall (1982).
  25. Renyi A. “On two mathematical models of the traffic on a divided highway”. J. Appl. Probab 1 (1964): 311-320.
  26. Ross SM. “Average delay in queues with non-stationary Poisson arrivals”. J. Appl. Probab 15 (1978): 602-609.
  27. Saati TL. “Elements of queueing theory, with applications”. McGrow Hill Book Comp (1961).
  28. Shiryaev AN. “Probability”. Springer-Verlag, New York- Berlin-Heidelberg (2013): 580.
  29. Zhou W, Huang W and Zhang R. “A two-stage queueing network on form postponement supply chain with correlated demands”. Appl. Math. Model 38 (2014): 2734-2743.