Reflections on queue modelling from the last 50 years

Special Issue Paper


Queueing theory continues to be one of the most researched areas of operational research, and has generated numerous review papers over the years. The phrase ‘queue modelling’ is used in the title to indicate a more practical emphasis. This paper uses work taken predominantly from the last 50 years of pages of the Operational Research Quarterly and the Journal of the Operational Research Society to offer a commentary on attempts of operational researchers to tackle real queueing problems, and on research foci past and future. A new discipline of ‘queue modelling’ is proposed, drawing upon the combined strengths of analytic and simulation approaches with the responsibility to derive meaningful insights for managers.


queueing theory queueing models simulation 


  1. Alfa AS (1979). A numerical method for evaluating delay to a customer in a time-inhomogeneous, single server queue with batch arrivals. J Opl Res Soc 30: 665–667.CrossRefGoogle Scholar
  2. Ansell PS, Glazebrook KD and Kirkbride C (2003). Generalised 'join the shortest queue' policies for the dynamic routing of jobs to multi-class queues. J Opl Res Soc 54: 379–389.CrossRefGoogle Scholar
  3. Ashton R, Hague L, Brandreth M, Worthington D and Cropper S (2005). A simulation-based study of a NHS walk-in centre. J Opl Res Soc 56: 153–161.CrossRefGoogle Scholar
  4. Atkinson JB (1995). The general two-server queueing loss system: Discrete-time analysis and numerical approximation of continuous-time systems. J Opl Res Soc 46: 386–397.CrossRefGoogle Scholar
  5. Bailey NTJ (1952). A study of queues and appointment systems in hospital out-patient departments, with special reference to waiting-times. J Roy Stat Soc Ser B 14: 185–199.Google Scholar
  6. Bailey NTJ (1957). Operational research in hospital planning and design. Opl Res Q 8: 149–157.CrossRefGoogle Scholar
  7. Bhat UN (1969). Sixty years of queueing theory. Mngt Sci 15: B280–B294.CrossRefGoogle Scholar
  8. Bhat UN, Shalaby M and Fischer MJ (1979). Approximation techniques in the solution of queueing problems. Nav Res Log Q 26: 311–326.CrossRefGoogle Scholar
  9. Bitran GR and Dasu S (1992). A review of open queueing network models of manufacturing systems. Queueing Syst Theory Appl 12: 95–134.CrossRefGoogle Scholar
  10. Brahimi M and Worthington DJ (1991). Queueing models for out-patient appointment systems – a case study. J Opl Res Soc 42: 733–746.Google Scholar
  11. Brockmeyer E, Halstrom HL and Jensen A (1948). The Life and Works of A K Erlang . J Jorgenson & Co.: Copenhagen.Google Scholar
  12. Bruneel H and Kim BG (1993). Discrete-time Models for Communication Systems Including ATM . Kluwer Academic: Boston.CrossRefGoogle Scholar
  13. Bunday BD (1996). An Introduction to Queueing Theory . Arnold: London.Google Scholar
  14. Chassioti E (2005). Queueing models for call centres. PhD Thesis, Management Science, Lancaster University.Google Scholar
  15. Chassioti E and Worthington DJ (2004). A new model for call centre queue management. J Opl Res Soc 55: 1352–1357.CrossRefGoogle Scholar
  16. Cromie MV and Chaudhry ML (1976). Analytically explicit results for the queueing system M/Mx/c with charts and tables for certain measures of efficiency. Opl Res Q 27: 733–745.CrossRefGoogle Scholar
  17. Ding L and Glazebrook KD (2005). A static allocation model for the outsourcing of warranty repairs. J Opns Res Soc 56: 825–835.CrossRefGoogle Scholar
  18. Ding L, Glazebrook KD and Kirkbride C (2008). Allocation models and heuristics for the outsourcing of repairs for a dynamic warranty population. Mngt Sci 54: 594–607.CrossRefGoogle Scholar
  19. Edmond ED and Maggs RP (1978). How useful are queue models in port investment decisions for container berths? J Opl Res Soc 29: 741–750.CrossRefGoogle Scholar
  20. Eick SG, Massey WA and Whitt W (1993). The physics of the Mt/G/∞ queue. Opns Res 41: 731–742.CrossRefGoogle Scholar
  21. Faulkner JA (1968). The use of closed queues in the deployment of coal-face machinery. Opl Res Q 19: 15–23.CrossRefGoogle Scholar
  22. Feldman Z, Mandelbaum A, Massey WA and Whitt W (2008). Staffing of time-varying queues to achieve time-stable performance. Mngt Sci 54: 324–338.CrossRefGoogle Scholar
  23. Fletcher A, Halsall D, Huxham S and Worthington D (2007). The DH accident and emergency department model: A national generic model used locally. J Opl Res Soc 58: 1554–1562.CrossRefGoogle Scholar
  24. Grassmann W (1977). Transient solutions in markovian queues: An algorithm for finding them and determining their waiting-time distributions. Eur J Opl Res 1: 396–402.CrossRefGoogle Scholar
  25. Green DH and Hartley MG (1966). The simulation of some simple control policies for a signalized intersection. Opl Res Q 17: 263–277.CrossRefGoogle Scholar
  26. Green LV, Kolesar PJ and Whitt W (2007). Coping with time-varying demand when setting staffing requirements for a service system. Product Opn Mngt 16: 13–39.Google Scholar
  27. Griffiths JD (1995). Queueing at the Suez Canal. J Opl Res Soc 46: 1299–1309.CrossRefGoogle Scholar
  28. Griffiths JD, Holland W and Williams JE (1991). Estimation of queues at the Channel Tunnel. J Opl Res Soc 42: 365–373.CrossRefGoogle Scholar
  29. Griffiths JD, Leonenko GM and Williams JE (2006). The transient solution to M/Ek/1 queue. Opns Res Lett 34: 349–354.CrossRefGoogle Scholar
  30. Griffiths JD, Price-Lloyd N, Smithies M and Williams JE (2005). Modelling the requirement for supplementary nurses in an intensive care unit. J Opl Res Soc 56: 126–133.CrossRefGoogle Scholar
  31. Groom KN (1977). Planning emergency ambulance services. Opl Res Q 28: 641–651.CrossRefGoogle Scholar
  32. Gross D and Harris CM (1985). Fundamentals of Queueing Theory . John Wiley & Sons Inc.: New York.Google Scholar
  33. Hasslinger G and Rieger ES (1996). Analysis of open discrete time queueing networks: A refined decomposition approach. J Opl Res Soc 47: 640–653.CrossRefGoogle Scholar
  34. Jackson JR (1957). Networks of waiting lines. Opns Res 5: 518–521.CrossRefGoogle Scholar
  35. Jackson JR (1963). Jobshop-like queueing systems. Mngt Sci 10: 131–142.CrossRefGoogle Scholar
  36. Jackson RRP (1954). Queueing systems with phase type service. Opl Res Q 5: 109–120.CrossRefGoogle Scholar
  37. Jackson RRP (1956). Random queueing processes with phase-type service. J Roy Stat Soc Ser B 18: 129–132.Google Scholar
  38. Jackson RRP, Welch JD and Fry J (1964). Appointment systems in hospitals and general practice: Design of an appointments system. Opl Res Q 15: 219–237.CrossRefGoogle Scholar
  39. Ke JC and Wang KH (1999). Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns. J Opl Res Soc 50: 275–282.CrossRefGoogle Scholar
  40. Kimber RM, Daly P, Barton J and Giokas C (1986). Predicting time-dependent distributions of queues and delays for road traffic at roundabouts and priority junctions. J Opl Res Soc 37: 87–97.CrossRefGoogle Scholar
  41. Koenigsberg E (1982). Twenty five years of cyclic queues and closed queue networks: A review. J Opl Res Soc 33: 605–619.CrossRefGoogle Scholar
  42. Koole G (2008). Special issue on call center management. Mngt Sci 54: 237.CrossRefGoogle Scholar
  43. Lawrie NL (1980). An application of queueing theory to a teletraffic problem. J Opl Res Soc 31: 975–981.CrossRefGoogle Scholar
  44. Lee AM and Longton PA (1959). Queueing processes associated with airline passenger check-in. Opl Res Q 10: 56–71.CrossRefGoogle Scholar
  45. Lee HS, Bouhchouch A, Dallery Y and Frein Y (1998). Performance evaluation of open queueing networks with arbitrary configuration and finite buffers. Ann Opns Res 79: 181–206.CrossRefGoogle Scholar
  46. Massey WA and Whitt W (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Syst 13: 183–250.CrossRefGoogle Scholar
  47. Mayhew LD (1987). Resource inputs and performance outputs in social security offices. J Opl Res Soc 38: 913–928.CrossRefGoogle Scholar
  48. Neuts MF (1975). Computational uses of the method of phases in the theory of queues. Comput Math Appl 1: 151–166.CrossRefGoogle Scholar
  49. Neuts MF (1984). Matrix-analytic methods in queueing theory. Eur J Opl Res 15: 2–12.CrossRefGoogle Scholar
  50. Neuts MF (1995). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach . John Hopkins University Press: Baltimore.Google Scholar
  51. Parkan C (1987). Simulation of a fast-food operation where dissatisfied customers renege. J Opl Res Soc 38: 137–148.CrossRefGoogle Scholar
  52. Perros HG (1994). Queueing Networks with Blocking: Exact and Approximate Solutions . Oxford University Press: New York.Google Scholar
  53. Preater J (2001). A bibliography of queues in health and medicine . Keele University: UK.Google Scholar
  54. Royston G, Halsall J, Halsall D and Braithwaite C (2003). Operational research for informed innovation: NHS Direct as a case study in the design, implementation and evaluation of a new public service. J Opl Res Soc 54: 1022–1028.CrossRefGoogle Scholar
  55. Saaty TL (1966). Seven more years of queues. A lament and a bibliography. Nav Res Log Q 13: 447–476.CrossRefGoogle Scholar
  56. Samanta SK, Gupta UC and Sharma RK (2007). Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations. J Opl Res Soc 58: 368–377.CrossRefGoogle Scholar
  57. Sharma OP (1990). Markovian Queues . Ellis Horwood: Chichester.Google Scholar
  58. Taylor J and Jackson RRP (1954). An application of the birth and death process to the provision of spare machines. Opl Res Q 5: 95–108.CrossRefGoogle Scholar
  59. van Ackere A and Ninios P (1993). Simulation and queueing theory applied to a single-server queue with advertising and balking. J Opl Res Soc 44: 407–414.CrossRefGoogle Scholar
  60. Wall AD and Worthington DJ (1994). Using discrete distributions to approximate general service time distributions in queueing models. J Opl Res Soc 45: 1398–1404.CrossRefGoogle Scholar
  61. Whitt W (1983). The main paper: The queueing network analyzer. Bell Syst Tech J 92: 2779–2815.CrossRefGoogle Scholar
  62. Whitt W (2006). Fluid models for multiserver queues with abandonments. Opns Res 54: 37–54.CrossRefGoogle Scholar
  63. Williams TM (1980). Nonpreemptive multi-server priority queues. J Opl Res Soc 31: 1105–1107.CrossRefGoogle Scholar
  64. Worthington DJ (1987). Queueing models for hospital waiting lists. J Opl Res Soc 38: 413–422.CrossRefGoogle Scholar
  65. Worthington D and Wall A (1999). Using the discrete time modelling approach to evaluate the time-dependent behaviour of queueing systems. J Opl Res Soc 50: 777–788.CrossRefGoogle Scholar

Copyright information

© Palgrave Macmillan 2009

Authors and Affiliations

  1. 1.Lancaster UniversityLancasterUK

Personalised recommendations