, Volume 13, Issue 2, pp 359–412 | Cite as

Optimal design and control of queues

  • Lotfi Tadj
  • Gautam Choudhury


We have divided this review into two parts. The first part is concerned with the optimal design of queueing systems and the second part deals with the optimal control of queueing systems. The second part, which has the lion’s share of the review since it has received the most attention, focuses mainly on the modelling aspects of the problem and describes the different kinds of threshold (control) policy models available in the literature. To limit the scope of this survey, we decided to limit ourselves to research on papers dealing with the three policies (N, T, and D), where a cost function is designed specifically and optimal thresholds that yield minimum cost are sought.

Key Words

Queue optimal design optimal control N-policy D-policy T-policy F-policy Q-policy 

AMS subject classification

60K10 60K25 90B22 90B25 


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  1. Aalto S. (1998). Optimal Control of Batch Service Queues with Compound Poisson Arrivals and Finite Service Capacity.Mathematical Methods of Operations Research 48, 317–335.CrossRefGoogle Scholar
  2. Aalto S. (2000). Optimal Control of Batch Service Queues with Finite Service Capacity and Linear Holding Costs.Mathematical Methods of Operations Research 51, 263–285.CrossRefGoogle Scholar
  3. Agarwal R.P. and Dshalalow J.H. (2005). New Fluctuation Analysis of D-Policy Bulk Queues with Multiple Vacations.Mathematical and Computer Modelling 41, 253–269.CrossRefGoogle Scholar
  4. Alfa A.S. and Frigui I. (1996). Discrete NT-Policy Single Server Queue with Markovian Arrival Process and Phase Type Service.European Journal of Operational Research 88, 599–613.CrossRefGoogle Scholar
  5. Altman E. and Nain P. (1993). Optimal Control of an M/G/1 Queue with Repeated Vacations of the Server.IEEE Transactions on Automatic Control 38, 1766–1775.CrossRefGoogle Scholar
  6. Altman E. and Nain P. (1996). Optimality of a Threshold Policy in the M/M/1 Queue with Repeated Vacations.Mathematical Methods of Operations Research 44, 75–96.CrossRefGoogle Scholar
  7. Artalejo J.R. (1997). Analysis of an M/G/1 Queue with Constant Repeated Attempts and Server Vacations.Computers and Operations Research 24, 493–504.CrossRefGoogle Scholar
  8. Artalejo J.R (1999a). Accessible Bibliography on Retrial Queues.Mathematical and Computer Modelling 30, 1–6.CrossRefGoogle Scholar
  9. Artalejo J.R (1999b). A Classified Bibliography of Research on Retrial Queues: Progress in 1990–1999.TOP 7, 187–211.CrossRefGoogle Scholar
  10. Artalejo J.R. (2001a). The D-Policy for the M/G/1 Queue: Queue Length and Optimality.Electronic Modelling 23, 35–43.Google Scholar
  11. Artalejo J.R. (2001b). On the M/G/1 Queue with D-Policy.Applied Mathematical Modelling 25, 1055–1069.CrossRefGoogle Scholar
  12. Artalejo J.R. (2002). A Note on the Optimality of the N- and D-policies for the M/G/1 Queue.Operations Research Letters 23, 35–43.Google Scholar
  13. Artalejo J.R. and Falin G.I. (2002). Standard and Retrial Queueing Systems: A Comparative Analysis.Revista Matemática Complutense 15, 101–129.Google Scholar
  14. Artalejo J.R. and Hernández-Lerma O. (2003). Performance Analysis and Optimal Control of the Geo/Geo/c Queue.Performance Evaluation 52, 15–39.CrossRefGoogle Scholar
  15. Artalejo J.R and Lopez-Herrero M.J. (2003). On the M/M/m Queue with Removable Server, In: Srinivasan S.K. and Vijayakumar A. (ed.),Stochastic Point Processes. Narosa Publishing House, 124–144.Google Scholar
  16. Bahary E. and Kolesar P. (1972). Multilevel Bulk Service Queues.Operations Research 20, 406–420.Google Scholar
  17. Baker K.R. (1973). A Note on Operating Policies for the Queue M/M/1 with Exponential Startup.INFOR 11, 71–72.Google Scholar
  18. Balachandran K.R. (1971). Queue Length Dependent Priority Queues.Management Science 17, 463–471.Google Scholar
  19. Balachandran K.R. (1973). Control Policies for a Single Server System.Management Science 19, 1013–1018.Google Scholar
  20. Balachandran K.R. and Tijms H. (1975). On the D-policy for the M/G/1 Queue.Management Science 21, 1073–1076.Google Scholar
  21. Bell C.E. (1971). Characterization and Computation of Optimal Policies for Operating an M/G/1 Queueing System with Removable Server.Operations Research 19, 208–218.Google Scholar
  22. Bell C.E. (1980). Optimal Operation of an M/M/2 Queue with Removable Servers.Operations Research 28, 1189–1204.Google Scholar
  23. Bellman R.E. (1957).Dynamic Programming, Princeton University Press.Google Scholar
  24. Benson F. (1952). Further Notes on the Productivity of Machines Requiring Attention at Random Times.Journal of the Royal Statistic Society B 14, 200–210.Google Scholar
  25. Bhat U.N. and Rao S.S. (1972). A Statistical Technique for the Control of Traffic Intensity in the Queueing Systems M/G/1 and GI/M/1.Operations Research 20, 955–966.Google Scholar
  26. Blackburn J.D. (1972). Optimal Control of a Single Server Queue with Balking and Reneging.Management Science 19, 297–313.Google Scholar
  27. Borthakur A., Medhi J. and Gohain R. (1987). Poisson Input Queueing System with Startup Time and under Control Operating Policy.Computers and Operations Research 14, 33–40.CrossRefGoogle Scholar
  28. Boxma O.J. (1976). Note on a Control Problem of Balachandran and Tijms.Management Science 22, 916–917.Google Scholar
  29. Brigham G. (1955). On a Congestion Problem in an Aircraft Factory.Operations Research 3, 412–428.Google Scholar
  30. Brill P.H. and Hlynka M. (2004). An Exponential Queue with Competition for Service.European Journal of Operational Research 126, 587–602.CrossRefGoogle Scholar
  31. Brosh I. (1970). The Policy Space Structure of Markovian Systems with two Types of Service.Management Science 16, 607–621.Google Scholar
  32. Chae K.C. and Lee H.W. (1995). MX/G/1 Vacation Model with N-Policy: Heuristic Interpretation of mean Waiting Time.Journal of Operational Research Society 46, 258–264.CrossRefGoogle Scholar
  33. Choudhury G. (1997). On a Two Server Poisson Input Queue under a Control Operating Policy with a General Startup Time.IAPQR-Transactions 22, 115–126.Google Scholar
  34. Choudhury G. (2005). An M/G/1 Queueing System with Two Phase Service under D-Policy.International Journal of Information and Management Sciences (to appear).Google Scholar
  35. Choudhury G. and Borthakur A. (2000). The Stochastic Decomposition Results of Batch Arrival Poisson Queue with a Grand Vacation Process.Sankhya 62, 448–462.Google Scholar
  36. Choudhury G. and Madan K.C. (2005). A Two Stage Batch Arrival Queueing System with a Modified Bernoulli Schedule Vacation under N-Policy.Mathematical and Computer Modelling 42, 71–85.CrossRefGoogle Scholar
  37. Choudhury G. and Paul M. (2004). A Batch Arrival Queue with an Additional Service Channel underN-Policy.Applied Mathematics and Computation 156, 115–130.CrossRefGoogle Scholar
  38. Choudhury G. and Paul M. (2005). A Batch Arrival Queue with Second Optional Service Channel underN-Policy.Stochastic Analysis and Applications (to appear).Google Scholar
  39. Cooper R.B. (1990).Introduction to the Theory of Queues (3rd edition). CEE Press Books.Google Scholar
  40. Crabill T.B. (1972). Optimal Control of a Service Facility with Variable Exponential Service Times and Constant Arrival Rate.Management Science 18, 560–566.Google Scholar
  41. Crabill T.B. (1974). Optimal Heuristic Control of a Stochastic Service System with Variable Service Rates and Fixed Switch-Over Costs. Technical Report, The New York Rand Institute.Google Scholar
  42. Crabill T.B., Gross D. and Magazine M.J. (1977). A Classified Bibliography of Research on Optimal Design and Control of Queues.Operations Research 25, 219–232.Google Scholar
  43. Deb R. (1976). Optimal Control of Batch Service Queues with Switching Costs.Advances in Applied Probability 8, 177–194.CrossRefGoogle Scholar
  44. Deb R. (1984). Optimal Control of Bulk Queues with Compound Poisson Arrivals and Batch Service.Opsearch 21, 227–245.Google Scholar
  45. Deb R.K. and Serfozo R.F. (1973). Optimal Control of Batch Service Queues.Advances in Applied Probability 5, 340–361.CrossRefGoogle Scholar
  46. Doganata Y.N. (1990). NT-Policy for M/G/1 Queue with Starter, In: E. Arikan (ed.),Communication, Control, and Signal Processing. Elsevier.Google Scholar
  47. Deng Y., Braun W.J. and Zhao Y.Q. (1999). M/M/1 Queueing System with Delayed Controlled Vacation.OR Transactions 3, 17–30.Google Scholar
  48. Doshi B.T. (1986). Queueing Systems with Vacations — A survey.Queueing Systems 1, 29–66.CrossRefGoogle Scholar
  49. Doshi B.T. (1990). Single-Sever Queues with Vacations, In: H. Takagi, (ed),Stochastic Analysis of Computer and Communication Systems. North-Holland, 217–265.Google Scholar
  50. Dshalalow J.H. (1996). On applications on excess level process to (N,D) Policy bulk Queueing System.Journal of Applied Mathematics and Stochastic Analysis 9, 551–562 (correction inJournal of Applied Mathematics and Stochastic Analysis 10 (1997), 207–208).CrossRefGoogle Scholar
  51. Dshalalow J.H. (1998). Queueing Processes in Bulk Systems under D-Policy.Journal of Applied Probability 35, 976–989.CrossRefGoogle Scholar
  52. Dshalalow J.H. (2001). A Note on D-Policy Bulk Queueing System.Journal of Applied Probability 38, 280–283.CrossRefGoogle Scholar
  53. Dudin A. (1981). Optimal Assignment of the Rate for Service of Customers in a Multilinear Two-Rate Service System.Automation and Remote Control 11, 74–82.Google Scholar
  54. Dudin A. (1991a). About Controlled M/Ĝ/1 Type Queues with Unreliable Server and the Loss of Time at Switching. Proceedings of the Seventh Belarus Winter Workshop in Queueing Theory, 59–61.Google Scholar
  55. Dudin A. (1991b).Queueing Systems with Varying Modes of Operation and its Optimization. Minsk-Tomsk.Google Scholar
  56. Dudin A. (1997). Optimal Control for an MX/G/1 Queue with two Operation Modes.Probability in the Engineering and Informational Sciences 11, 255–265.CrossRefGoogle Scholar
  57. Evans R.V. (1971). Programming Problems and Changes in the Stable Behavior of a Class of Markov Chains.Journal of Applied Probability 8, 543–550.CrossRefGoogle Scholar
  58. Falin G.I. (1990). A Survey on Retrial Queues.Queueing Systems 7, 127–168.CrossRefGoogle Scholar
  59. Falin G.I. and Templeton J.G.C. (1997).Retrial Queues. Chapman and Hall.Google Scholar
  60. Farahmad K. (1990). Single Line Queue with Repeated Demands.QUESTA 7, 223–228.Google Scholar
  61. Federgruen A. and So K.C. (1990). Optimal Maintenance Policies for Single-Server Queueing Systems Subject to Breakdowns.Operations Research 38, 330–343.Google Scholar
  62. Federgruen A. and So K.C. (1991). Optimality of Threshold Policies in Single-Server Queueing Systems with Server Vacations.Advances in Applied Probability 23, 388–405.CrossRefGoogle Scholar
  63. Feinberg E.A. and Kella O. (2002). Optimality of D-Policies for an M/G/1 Queue with a Removable Server.Queueing Systems 42, 355–376.CrossRefGoogle Scholar
  64. Feinberg E.A. and Kim D.J. (1994). Optimal Switching Policies for M/G/1 Queues with Two Performance Criteria, In: Degris U., Bachem A. and Drexi A. (eds.),Operations Research Proceedings. Springer Verlag, 227–232.Google Scholar
  65. Feinberg E.A. and Kim D.J. (1996). Bicriterion Optimization of an M/G/1 Queue with a Removable Server.Probability in the Engineering and Informational Sciences 10, 57–73.Google Scholar
  66. Gebhard R.F. (1967). A Queueing Process with Bilevel Hysteretic Service-Rate Control.Naval Research Logistics Quarterly 14, 55–68.Google Scholar
  67. Grassman W.K., Chen X. and Kashyap B.R.K. (2001). Optimal Service Rates for the State-Dependent M/G/1 Queues in Steady State.Operations Research Letters 29, 57–63.CrossRefGoogle Scholar
  68. Gross D. and Harris C.M. (1998).Fundamentals of Queueing Theory (3rd edition). Wiley.Google Scholar
  69. Gupta S.M. (1995).N-policy Queueing System with Finite Population.Transactions on Operational Research 7, 45–62.Google Scholar
  70. Hersh M. and Brosh I. (1980). The Optimal Strategy Structure of an Intermittently Operated Service Channel.European Journal of the Operational Research Society 5, 133–141.CrossRefGoogle Scholar
  71. Heyman D.P. (1968). Optimal Operating Policies for M/G/1 Queueing Systems.Operations Research 16, 362–382.Google Scholar
  72. Heyman D.P. (1977). The T-Policy for the M/G/1 Queue.Management Science 23, 775–778.Google Scholar
  73. Heyman D.P. and Marshall K.T. (1968). Bounds on the Optimal Operating Policy for a Class of Single Server Queues.Operations Research 16, 1138–1146.Google Scholar
  74. Hillier F.S. (1963). Economic Models for Industrial Waiting Line Problems.Management Science 10, 119–130.Google Scholar
  75. Hur S. and Paik S.-J. (1999). The Effect of Different Arrival Rates on the N-Policy of M/G/1 with Server Setup.Applied Mathematical Modelling 23, 289–299.CrossRefGoogle Scholar
  76. Hur S., Kim J. and Kang C. (2003). An Analysis of the M/G/1 System with N and T Policy.Applied Mathematical Modelling 27, 665–675.CrossRefGoogle Scholar
  77. Igaki N. (1992). Exponential two Server Queue with N-policy and General Vacations.Queueing Systems 10, 279–294.CrossRefGoogle Scholar
  78. Ignall E. and Kolesar P. (1972). Operating Characteristic of a Simple Shuttle under Local Dispatching Rules.Operations Research 20, 1077–1088.Google Scholar
  79. Jaiswal N.K. and Sinha P.S. (1972). Optimal Operating Policies for the Finite-Source Queueing Process.Operations Research 20, 698–707.Google Scholar
  80. Karaesman F. and Gupta S.M. (1997). Duality Relations for Queues with Arrival and Service Control.Computers and Operations Research 24, 529–538.CrossRefGoogle Scholar
  81. Kasahara S., Takine T., Takahashi Y. and Hasegawa T. (1996). MAP/G/1 Queues under N-policy with and without Vacations.Journal of the Operations Research Society of Japan 39, 188–212.Google Scholar
  82. Ke J.-C. (2001). The Control Policy of an M[x]/G/1 Queueing System with Server Startup and Two Vacation Types.Mathematical Methods of Operations Research 54, 471–490.CrossRefGoogle Scholar
  83. Ke J.-C. (2003a). The Optimal Control of an M/G/1 Queueing System with Server Startup and Two Vacation Types.Applied Mathematical Modelling 27, 437–450.CrossRefGoogle Scholar
  84. Ke J.-C. (2003b). The Control Policy in Batch Arrival Queue with Server Vacations, Startups and Breakdowns.Computers and Industrial Engineering 44, 567–579.CrossRefGoogle Scholar
  85. Ke J.-C. and Pearn W.L. (2004). Optimal Management Policy for Heterogeneous Arrival Queueing Systems with Server Breakdowns and Vacations.Quality Technology and Quantitative Management 1, 149–162.Google Scholar
  86. Ke J.-C. and Wang K.-H. (1999). Cost Analysis of the M/M/R Machine Repair Problem with Balking, Reneging, and Server Breakdown.Journal of the Operational Research Society 50, 275–282.CrossRefGoogle Scholar
  87. Kella O. (1989). The Threshold Policies in the M/G/1 Queue with Server Vacations.Naval Research Logistics 36, 111–123.Google Scholar
  88. Kella O. (1990). Optimal Control of the Vacation Scheme in an M/G/1 Queue.Operations Research 38, 724–728.Google Scholar
  89. Kim S.-K. and Dshalalow J.H. (2002). Stochastic Disaster Recovery Systems with External Resources.Mathematical and Computer Modelling 36, 1235–1257.CrossRefGoogle Scholar
  90. Kimura T. (1981). Optimal Control of an M/G/1 Queueing System with Removable Server via Diffusion Approximation.European Journal of Operational Research 8, 390–398.CrossRefGoogle Scholar
  91. Köchel P. (2004). Finite Queueing System—structural Investigations and Optimal Design.International Journal of Production Economics 88, 157–171.CrossRefGoogle Scholar
  92. Kosten L. (1967). The Custodian Problem. In: Cruon O.R. (ed.), Queueing Theory, Recent Developments and Applications. English University Press, 65–70.Google Scholar
  93. Krishnamoorthy A. and Deepak T.G. (2002). Modified N-Policy for M/G/1 Queues.Computers and Operations Research 29, 1611–1620.CrossRefGoogle Scholar
  94. Krishnamoorthy A. and Ushakumari P.V. (2000). A Queueing System with Single Arrival Bulk Service and Single Departure.Mathematical and Computer Modelling 31, 99–108.CrossRefGoogle Scholar
  95. Krishna Reddy G.V., Nadarajan R. and Arumuganathan R. (1998). Analysis of a Bulk Queue with N-Policy Multiple Vacations and Setup Timed.Computers and Operations Research 25, 957–967.CrossRefGoogle Scholar
  96. Kulkarni V.G. and Liang H.M. (1997). Retrial Queues Revisited. In: J.H. Dshalalow J.H. (ed.),Frontiers in Queueing. CRC Press, 19–34.Google Scholar
  97. Lee H.S. (1995). Optimal Control of the MX/G/1/K Queue with Multiple Server Vacations.Computers and Operations Research 22, 543–552.CrossRefGoogle Scholar
  98. Lee H.S. and Srinivasan M.M. (1989). Control Policies for the Mx/G/1 Queueing System.Management Science 35, 708–721.Google Scholar
  99. Lee H.W. and Ahn B.Y. (2002). Operational Behavior of the MAP/G/1 Queue under N-Policy with Single Vacation and Set-Up.Journal of Applied Mathematics and Stochastic Analysis 15, 167–196.CrossRefGoogle Scholar
  100. Lee H.W. and Baek J.W. (2005). BMAP/G/1 Queue under D-Policy: Queue Length Analysis.Stochastic Models 21, 485–505.CrossRefGoogle Scholar
  101. Lee H.W. and Song K.S. (2004). Quene Length Analysis of MAP/G/1 Queue under D-Policy.Stochastic Models 20, 363–380.CrossRefGoogle Scholar
  102. Lee H.W. and Park J.O. (1997). Optimal Strategy in N-Policy Production System with Early Set-Up.Journal of the Operational Research Society 48, 306–313.CrossRefGoogle Scholar
  103. Lee H.W. and Park N.I. (2004). Using Factorization for Waiting Times in BMAP/G/1 Queues with N-Policy and Vacations.Stochastic Analysis and Applications 22, 755–773.CrossRefGoogle Scholar
  104. Lee H.W., Ahn B.Y. and Park N.I. (2001). Decomposition of the Queue length Distributions in the MAP/G/1 Quene under Multiple and Single Vacations with N-policy.Stochastic Models 17, 157–190.CrossRefGoogle Scholar
  105. Lee H.W., Baek J.W. and Jeon J. (2005). Analysis of the MX/G/1 Queue under D-Policy.Stochastic Analysis and Applications (to appear).Google Scholar
  106. Lee H.W., Lee S.S. and Chae K.C. (1994a). Operating Characteristics of MX/G/1 Queue with N-Policy.Queueing Systems 15, 387–399.CrossRefGoogle Scholar
  107. Lee H.W., Yoon S.H. and Seo W.J. (1999). Start-Up Class Models in Multiple-Class Queues with N-Policy.Queueing Systems 31, 101–124.CrossRefGoogle Scholar
  108. Lee H.W., Cheon S.H., Lee E.Y. and Chae K.C. (2004). Workload and Waiting Time Analysis of MAP/G/1 Queue under D-Policy.Queueing Systems 48, 421–443.CrossRefGoogle Scholar
  109. Lee H.W., Lee S.S., Park J.O. and Chae K.C. (1994b). Analysis of the MX/G/1 Queue with N-policy and Multiple Vacations.Journal of Applied Probability 31, 467–496.CrossRefGoogle Scholar
  110. Lee S.S., Lee H.W. and Chae K.C. (1995). On a Batch Arrival Queue with N-Policy and Single Vacation.Computers and Operations Research 22, 173–189.CrossRefGoogle Scholar
  111. Levy Y. and Yechiali U. (1975). Utilization of Idle Time in an M/G/1 Queueing System.Management Science 22, 202–211.Google Scholar
  112. Li W. and Alfa A.S. (2000). Optimal Policies for M/M/m Queue with Two Different Kinds of (N,T) Policies.Naval Research Logistics Quarterly 47, 240–258.CrossRefGoogle Scholar
  113. Liao C.J. (1992). Optimal Release Time on a Stochastic Single Machine.International Journal of Systems Science 23, 1693–1701.Google Scholar
  114. Lillo R.E. and Martin M. (2000). On Optimal Exhaustive Policies for the M/G/1 Queue.Operations Research Letters 27, 39–46.CrossRefGoogle Scholar
  115. Martin M. and Artalejo J.R. (1995). Analysis of an M/G/1 Queue with Two Types of Impatient Units.Advances in Applied Probability 27, 840–861.CrossRefGoogle Scholar
  116. Medhi J. (1984). Bulk Service Queueing Models and Associated Control Problems in Statistics: Applications and New Directions, 369–377, Indian Statistical Institute, Calcutta.Google Scholar
  117. Medhi J. (1997). Single Server Queueing System with Poisson Input: A Review of Some Recent Developments. In: Balakrishnan N. (ed.),Advances in Combinatorial Methods and Applications to Probability and Statistics. Birkhäuser, 317–338.Google Scholar
  118. Mitra D. and Mitrani I. (1991).Special Issue of Queueing Systems 9 (numbers 1 and 2). Springer Verlag.Google Scholar
  119. Moder J.J. and Phillips Jr. C.R. (1962). Queueing with Fixed and Variable Channels.Operations Research 10, 218–231.Google Scholar
  120. Morse P.M. (1958).Queues, Inventories and Maintenance. Wiley.Google Scholar
  121. Nobel R.D. and Tijms H.C. (1999). Optimal Control for an MX/G/1 Queue with Two Service Modes.European Journal of Operational Research 113, 610–619.CrossRefGoogle Scholar
  122. Nobel R.D. and Tijms H.C. (2000). Optimal Control of a Queueing System with Heterogeneous Servers and Setup Costs.IEEE Transaction on Automatic Control 40, 780–784.CrossRefGoogle Scholar
  123. Okamura H., Dohi T. and Osaki S. (1996). Optimal Timing Strategies for Controlled M/G/1 Queueing System via Diffusion Approximation. Proceedings of the 28th ISCIE International Symposium on Stochastic Systems Theory and Its Applications, 143–148.Google Scholar
  124. Okamura H., Dohi T. and Osaki S. (2000). Optimal Policies for a Controlled Queueing System with Removable Server under a Random Vacation Circumstance.Computers and Mathematics with Applications 39, 215–227.CrossRefGoogle Scholar
  125. Pearn W.L. and Chang Y.C. (2004). Optimal Management of the N-Policy M/Ek/1 Queueing System with a Removable Service Station: A Sensitivity Investigation.Computers and Operations Research 31, 1001–1015.CrossRefGoogle Scholar
  126. Pearn W.L., Ke J.-C. and Chang Y.C. (2004). Sensitivity Analysis of the Optimal Management Policy for a Queueing System with a Removable Service Station:Computers and Industrial Engineering 46, 87–99.CrossRefGoogle Scholar
  127. Pegden C.D. and Rosenshine M. (1990). Scheduling Arrivals to Queues.Computers and Operations Research 17, 343–348.CrossRefGoogle Scholar
  128. Piunovskiy A.B. (2004). Bicriteria Optimization of a Queue with a Controlled Input Stream.Queueing Systems 48, 159–184.CrossRefGoogle Scholar
  129. Powell W.B. and Humblet P. (1986). The Bulk Service Queue with a General Control Strategy: Theoretical Analysis and a New Computational Procedure.Operations Research 34, 267–275.Google Scholar
  130. Ragatz G.L. and Mabert V.A. (1988). An Evaluation of Order Release Mechanisms in a Job-Shop Environment.Decision Science 19, 167–189.Google Scholar
  131. Romani J. (1957). Un Modelo de la Teoría de Colas con Número Variable de Canales.Trabajos de Estadística 8, 175–189.Google Scholar
  132. Rykov V. and Efrosinin D. (2004). Optimal Control of Queueing Systems with Heterogeneous Servers.Queueing Systems 46, 389–438.CrossRefGoogle Scholar
  133. Selim S.Z. (1997). Time-dependent Solution and Optimal Control of a Bulk Service Queue.Journal of Applied Probability 34, 258–266.CrossRefGoogle Scholar
  134. Shanthikumar J. (1980). Analysis on the Control of Queues with Shortest Processing Time Service Discipline.Journal of the Operations Research Society of Japan 23, 341–352.Google Scholar
  135. Shanthikumar J. (1981a). M/G/1 Queues with Scheduling within Generations and Removable Server.Operations research 29, 1010–1018.Google Scholar
  136. Shanthikumar J. (1981b). Optimal Control of an M/G/1 Priority Queue via N-control.American Journal of Mathematical and Management Science 1, 192–212.Google Scholar
  137. Skinner C.E. (1967). A Priority Queueing System with Server-Walking Time.Operations Research 15, 278–285.Google Scholar
  138. Sivazlian B.D. (1979). Approximate Optimal Solution for a D-Policy in an M/G/1 Queueing System.AIIE Transactions 11, 341–343.Google Scholar
  139. Sobel M.J. (1969). Optimal Average-Cost Policy for a Queue with Start-up and Shut-Down Costs.Operations Research 17, 145–162.Google Scholar
  140. Sobel M.J. (1974). Optimal operation of Queues. In: Clarke A.B. (ed.),Mathematical Methods in Queueing Theory, Lecture Notes in Economics and Mathematical Systems, 231–261.Google Scholar
  141. Srinivasan M.M. and Lee H.-S. (1970). Random Review Production/Inventory Systems with Compound Poisson Demands and Arbitrary Processing Times.Management Science 37, 813–833.Google Scholar
  142. Stidham Jr. S. (1970). On the Optimality of Single-Server Queueing Systems.Operations Research 18, 708–732.Google Scholar
  143. Stidham Jr. S. (1977). Optimal Control of Stochastic Service Systems.Advances in Applied Probability 10, 277–278.CrossRefGoogle Scholar
  144. Stidham Jr. S. and Prabhu N.U. (1974). Optimal Control of Queueing Systems. In: Clarke A.B. (ed.),Mathematical Methods in Queueing Theory, Lecture Notes in Economics and Mathematical Systems, 263–294.Google Scholar
  145. Tadj L. (2003a). A Quorum Queueing System under D-Policy.Applied Mathematics and Computation 144, 325–336.CrossRefGoogle Scholar
  146. Tadj L. (2003b). A quorum Queueing System under T-Policy.Journal of the Operational Research Society 54, 466–471.CrossRefGoogle Scholar
  147. Tadj L., Choudhury G. and Tadj C. (2005a). A Quorum Queueing System with a Random Setup Time Under N-Policy and with Bernoulli Vacation Schedule.Quality Technology and Quantitative Management (to appear).Google Scholar
  148. Tadj L., Choudhury G. and Tadj C. (2005b). A Bulk Quorum Queueing System with a Random Setup Time under N-Policy and with Bernoulli Vacation Schedule.Stochastics (to appear).Google Scholar
  149. Tadj L. and Ke J.-C. (2003). Control Policy of a Hysteretic Queueing System.Mathematical Methods of Operations Research 57, 367–376.Google Scholar
  150. Tadj L. and Ke J.-C. (2005). Control Policy of a Hysteretic Bulk Queueing System.Mathematical and Computer Modelling 41, 571–579.CrossRefGoogle Scholar
  151. Tadj L. and Sarhan A. (2003). Effect of the Server Capacity Distribution on the Optimal Control of a Bulk Service Queueing System.Chaos, Fractals, and Solitons 18, 1101–1110.CrossRefGoogle Scholar
  152. Tadj L. and Tadj C. (2003). On an M/Dr/1 Queueing System.Journal of Statistical Theory and Applications 2, 17–32.Google Scholar
  153. Takagi H. (1991).Queueing Analysis: A Foundation of Performance Evaluation (volumen 1). North-Holland.Google Scholar
  154. Talman A.J.J. (1979). A Simple Proof of the Optimality of the Best N-Policy in the M/G/1 Queueing Control Problem with Removable Server.Statistica Neerlandica 33, 143–150.Google Scholar
  155. Teghem Jr. J. (1986). Control of the Service Process in a Queueing System.European Journal of Operational Research 23, 141–158.CrossRefGoogle Scholar
  156. Teghem Jr. J. (1987). Optimal Control of a Removable Server in an M/G/1 Queue with Finite Capacity.European Journal of Operational Research 31, 358–367.CrossRefGoogle Scholar
  157. Tijms H. (1974). A Control Policy for a Priority Queue with Removable Server.Operation Research 22, 833–837.Google Scholar
  158. Wang K.-H. (1995). Optimal Operation of a Markovian Queueing System with a Removable and Non-Reliable Server.Microelectronics and Reliability 35, 1131–1136.CrossRefGoogle Scholar
  159. Wang K.-H. (1997). Optimal Control of an M/Ek/1 Queueing System with Removable Service Station Subject to Breakdowns.Journal of the Operations Research Society 48, 936–942.CrossRefGoogle Scholar
  160. Wang K.-H., Chang K.-W. and Sivazlian B.D. (1999). Optimal Control of a Removable and Non-Reliable Server in an Infinite and a Finite M/H2/1 Queueing System.Applied Mathematical Modelling 23, 651–666.CrossRefGoogle Scholar
  161. Wang K.-H. and Hsieh W.-F. (1995). Optimal Control of a Removable and Non-Reliable Server in a Markovian Queueing System with Finite Capacity.Microelectronics and Reliability 35, 189–196.CrossRefGoogle Scholar
  162. Wang K.-H. and Huang H.-M. (1995a). Optimal Control of an M/Ek/1 Queueing System with a Removable Service Station.Journal of the Operational Research Society 46, 1014–1022.CrossRefGoogle Scholar
  163. Wang K.-H. and Huang H.-M. (1995b). Optimal Control of an M/Ek/1 Queueing System with Finite Capacity.Microelectronics and Reliability 35, 1023–1030.CrossRefGoogle Scholar
  164. Wang K.-H., Kao H.-T. and Chen G. (2004). Optimal Management of a Removable and Non-Reliable Server in an Infinite and a Finite M/Hk/1 Queueing System.Quality Technology and Quantitative Management 1, 325–339.Google Scholar
  165. Wang K.-H. and Wang Y.-L. (2002). Optimal Control of an M/M/2 Queueing System with Finite Capacity Operating under the Triadic (0,Q, N, M) Policy.Mathematical Methods of Operations Research 55, 447–460.CrossRefGoogle Scholar
  166. Wang P.P. (1996). Markovian Queueing Models with Periodic Review.Computers and Operations Research 23, 741–754.CrossRefGoogle Scholar
  167. Weiss J. (1979a). The Computation of Optimal Control Limits for a Queue with Batch Services.Management Science 25, 320–328.Google Scholar
  168. Weiss J. (1979b). Optimal Control of Batch Service Queues with Nonlinear Waiting costs.Modeling and Simulation 10, 305–309.Google Scholar
  169. Weiss J. (1981). Further Results on an Infinite Capacity Shuttle with Control at a Single Terminal.Operations Research 29, 1212–1217.Google Scholar
  170. Winston W. (1978). Optimality of Monotonic Policies for Multiple Server Exponential Queueing Systems with State-Dependent Arrival Rates.Operations Research 26, 1089–1094.Google Scholar
  171. Yadin M. and Naor P. (1963). Queueing Systems with a Removable Service Station.Operational Research Quarterly 14, 393–405.CrossRefGoogle Scholar
  172. Yadin M. and Naor P. (1967). On Queueing Systems with Variable Service Capacities.Naval Research Logistics Quarterly 14, 43–54.Google Scholar
  173. Yang T. and Templeton J.G.C. (1987). A Survey on Retrial Queues.Queueing Systems 2, 201–233.CrossRefGoogle Scholar
  174. Zhang Z.G. and Love C.E. (1998). The Threshold Policy in an M/G/1 Queue with an Exceptional First Vacation.INFOR 36, 193–204.Google Scholar
  175. Zhang Z.G., Vickson R.G. and Love C.E. (2001). The Optimal Service Policies in an M/G/1 Queueing System with Multiple Vacation Types.INFOR 39, 357–366.Google Scholar
  176. Zhang Z.G., Vickson R.G. and van Eengie M.J.A. (1997). Optimal Two Threshold Policies in an M/G/1 Queue with two Vacation Types.Performance Evaluation 29, 63–80.CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2005

Authors and Affiliations

  • Lotfi Tadj
    • 1
  • Gautam Choudhury
    • 2
  1. 1.Department of Statistics and Operations ResearchCollege of Science King Saud UniversityRiyadhSaudi Arabia
  2. 2.Mathematical Sciences DivisionInstitute of Advanced Study in Science and TechnologyGuwahatiIndia

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