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Performance Analysis of Priority Queueing Systems in Discrete Time

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Network Performance Engineering

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 5233))

Abstract

The integration of different types of traffic in packet-based networks spawns the need for traffic differentiation. In this tutorial paper, we present some analytical techniques to tackle discrete-time queueing systems with priority scheduling. We investigate both preemptive (resume and repeat) and non-preemptive priority scheduling disciplines. Two classes of traffic are considered, high-priority and low-priority traffic, which both generate variable-length packets. A probability generating functions approach leads to performance measures such as moments of system contents and packet delays of both classes.

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References

  1. Miller, R.: Priority queues. Annals of Mathematical Statistics 31, 86–103 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kleinrock, L.: Queueing systems. Computer applications, vol. II. John Wiley & Sons, New York (1976)

    MATH  Google Scholar 

  3. Takagi, H.: Queueing analysis: a foundation of performance evaluation, vacation and priority systems, part 1, vol. 1. North-Holland, Amsterdam (1991)

    MATH  Google Scholar 

  4. Khamisy, A., Sidi, M.: Discrete-time priority queues with two-state Markov Modulated arrivals. Stochastic Models 8(2), 337–357 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Takine, T., Sengupta, B., Hasegawa, T.: An analysis of a discrete-time queue for broadband ISDN with priorities among traffic classes. IEEE Transactions on Communications 42(2-4), 1837–1845 (1994)

    Article  Google Scholar 

  6. Laevens, K., Bruneel, H.: Discrete-time multiserver queues with priorities. Performance Evaluation 33(4), 249–275 (1998)

    Article  Google Scholar 

  7. Choi, B., Choi, D., Lee, Y., Sung, D.: Priority queueing system with fixed-length packet-train arrivals. IEE Proceedings-Communications 145(5), 331–336 (1998)

    Article  Google Scholar 

  8. Walraevens, J., Steyaert, B., Bruneel, H.: Performance analysis of a single-server ATM queue with a priority scheduling. Computers & Operations Research 30(12), 1807–1829 (2003)

    Article  MATH  Google Scholar 

  9. Mehmet Ali, M., Song, X.: A performance analysis of a discrete-time priority queueing system with correlated arrivals. Performance Evaluation 57(3), 307–339 (2004)

    Article  Google Scholar 

  10. Van Velthoven, J., Van Houdt, B., Blondia, C.: The impact of buffer finiteness on the loss rate in a priority queueing system. In: Horváth, A., Telek, M. (eds.) EPEW 2006. LNCS, vol. 4054, pp. 211–225. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  11. Kamoun, F.: Performance analysis of a discrete-time queuing system with a correlated train arrival process. Performance Evaluation 63(4-5), 315–340 (2006)

    Article  Google Scholar 

  12. Walraevens, J., Wittevrongel, S., Bruneel, H.: A discrete-time priority queue with train arrivals. Stochastic Models 23(3), 489–512 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Demoor, T., Walraevens, J., Fiems, D., Bruneel, H.: Mixed finite-/infinite-capacity priority queue with interclass correlation. In: Al-Begain, K., Heindl, A., Telek, M. (eds.) ASMTA 2008. LNCS, vol. 5055, pp. 61–74. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. Walraevens, J., Fiems, D., Bruneel, H.: Time-dependent performance analysis of a discrete-time priority queue. Performance Evaluation 65(9), 641–652 (2008)

    Article  Google Scholar 

  15. Walraevens, J., Wittevrongel, S., Bruneel, H.: Performance analysis of a priority queue with session-based arrivals and its application to E-commerce web servers. International Journal On Advances in Internet Technology 2(1), 46–57 (2009)

    Google Scholar 

  16. Walraevens, J., Fiems, D., Wittevrongel, S., Bruneel, H.: Calculation of output characteristics of a priority queue through a busy period analysis. European Journal of Operational Research 198(3), 891–898 (2009)

    Article  MATH  Google Scholar 

  17. Stanford, D.: Interdeparture-time distributions in the non-preemptive priority ΣMi/Gi/1 queue. Performance Evaluation 12(1), 43–60 (1991)

    Article  MathSciNet  Google Scholar 

  18. Sugahara, A., Takine, T., Takahashi, Y., Hasegawa, T.: Analysis of a nonpreemptive priority queue with SPP arrivals of high class. Performance Evaluation 21(3), 215–238 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Abate, J., Whitt, W.: Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Systems 25(1-4), 173–233 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Takine, T.: The nonpreemptive priority MAP/G/1 queue. Operations Research 47(6), 917–927 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Isotupa, K., Stanford, D.: An infinite-phase quasi-birth-and-death model for the non-preemptive priority M/PH/1 queue. Stochastic Models 18(3), 387–424 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Drekic, S., Stafford, J.: Symbolic computation of moments in priority queues. INFORMS Journal on Computing 14(3), 261–277 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bouallouche-Medjkoune, L., Aissani, D.: Quantitative estimates in an M 2/G 2/1 priority queue with non-preemptive priority: the method of strong stability. Stochastic Models 24, 626–646 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Iftikhar, M., Singh, T., Landfeldt, B., Caglar, M.: Multiclass G/M/1 queuing system with self-similar input and non-preemptive priority. Computer Communications 31, 1012–1027 (2008)

    Article  Google Scholar 

  25. Al-Begain, K., Dudin, A., Kazimirsky, A., Yerima, S.: Investigation of the M 2/G 2/1/ ∞ ,N queue with restricted admission of priority customers and its application to HSDPA mobile systems. Computer Networks 53, 1186–1201 (2009)

    Article  MATH  Google Scholar 

  26. Chen, Y., Chen, C.: Performance analysis of non-preemptive GE/G/1 priority queueing of LER system with bulk arrivals. Computers and Electrical Engineering 35, 764–789 (2009)

    Article  MATH  Google Scholar 

  27. Rubin, I., Tsai, Z.: Message delay analysis of multiclass priority TDMA, FDMA, and discrete-time queueing systems. IEEE Transactions on Information Theory 35(3), 637–647 (1989)

    Article  MathSciNet  Google Scholar 

  28. Hashida, O., Takahashi, Y.: A discrete-time priority queue with switched batch Bernoulli process inputs and constant service time. In: Proceedings of ITC 13, Copenhagen, pp. 521–526 (1991)

    Google Scholar 

  29. Takine, T., Matsumoto, Y., Suda, T., Hasegawa, T.: Mean waiting times in nonpreemptive priority queues with Markovian arrival and i.i.d. service processes. Performance Evaluation 20, 131–149 (1994)

    Article  Google Scholar 

  30. Takine, T.: A nonpreemptive priority MAP/G/1 queue with two classes of customers. Journal of Operations Research Society of Japan 39(2), 266–290 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Walraevens, J., Steyaert, B., Bruneel, H.: Performance analysis of the system contents in a discrete-time non-preemptive priority queue with general service times. Belgian Journal of Operations Research, Statistics and Computer Science (JORBEL) 40(1-2), 91–103 (2000)

    MathSciNet  MATH  Google Scholar 

  32. Walraevens, J., Steyaert, B., Bruneel, H.: Delay characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline. Performance Evaluation 50(1), 53–75 (2002)

    Article  Google Scholar 

  33. Walraevens, J., Steyaert, B., Moeneclaey, M., Bruneel, H.: Delay analysis of a HOL priority queue. Telecommunication Systems 30(1-3), 81–98 (2005)

    Article  Google Scholar 

  34. Maertens, T., Walraevens, J., Bruneel, H.: Priority queueing systems: from probability generating functions to tail probabilities. Queueing Systems 55(1), 27–39 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Demoor, T., Walraevens, J., Fiems, D., De Vuyst, S., Bruneel, H.: Analysis of a non-preemptive priority queue with finite high-priority capacity and general service times. In: Proceedings of the 4th International Conference on Queueing Theory and Applications (QTNA 2009), Singapore, ID12 (2009)

    Google Scholar 

  36. Miller, D.: Computation of steady-state probabilities for M/M/1 priority queues. Operations Research 29(5), 945–958 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sandhu, D., Posner, M.: A priority M/G/1 queue with application to voice/data communication. European Journal of Operational Research 40(1), 99–108 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Takine, T., Hasegawa, T.: The workload in the MAP/G/1 queue with state-dependent services: its application to a queue with preemptive resume priority. Communications in Statistics - Stochastic Models 10(1), 183–204 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Takahashi, Y., Miyazawa, M.: Relationship between queue-length and waiting time distributions in a priority queue with batch arrivals. Journal of the Operations Research Society of Japan 37(1), 48–63 (1994)

    MathSciNet  MATH  Google Scholar 

  40. Boxma, O., Cohen, J., Deng, Q.: Heavy-traffic analysis of the M/G/1 queue with priority classes. In: Proceedings of ITC 16, Edinburgh, pp. 1157–1167 (1999)

    Google Scholar 

  41. Sharma, V., Virtamo, J.: A finite buffer queue with priorities. Performance Evaluation 47(1), 1–22 (2002)

    Article  MATH  Google Scholar 

  42. Takada, H., Miyazawa, M.: A Markov Modulated fluid queue with batch arrivals and preemptions. Stochastic Models 18(4), 529–652 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  43. Liu, Y., Gong, W.: On fluid queueing systems with strict priority. IEEE Transactions on Automatic Control 48(12), 2079–2088 (2003)

    Article  MathSciNet  Google Scholar 

  44. Jin, X., Min, G.: Performance analysis of priority scheduling mechanisms under heterogeneous network traffic. Journal of Computer and System Sciences 73, 1207–1220 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tarabia, A.: Two-class priority queueing system with restricted number of priority customers. AEÜ-International Journal of Electronics and Communications 61(8), 534–539 (2007)

    Article  Google Scholar 

  46. Tzenova, E., Adan, I., Kulkarni, V.: Output analysis of multiclass fluid models with static priorities. Performance Evaluation 65(1), 71–81 (2008)

    Article  Google Scholar 

  47. Horvath, A., Horvath, G., Telek, M.: A traffic based decomposition of two-class queueing networks with priority service. Computer Networks 53, 1235–1248 (2009)

    Article  MATH  Google Scholar 

  48. Lee, Y.: Discrete-time Geo x/G/1 queue with preemptive resume priority. Mathematical and Computer Modelling 34(3-4), 243–250 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  49. Walraevens, J., Steyaert, B., Bruneel, H.: Performance analysis of a GI-Geo-1 buffer with a preemptive resume priority scheduling discipline. European Journal of Operational Research 157(1), 130–151 (2004)

    Article  MATH  Google Scholar 

  50. Walraevens, J., Steyaert, B., Bruneel, H.: A packet switch with a priority scheduling discipline: Performance analysis. Telecommunication Systems 28(1), 53–77 (2005)

    Article  Google Scholar 

  51. Van Houdt, B., Blondia, C.: Analyzing priority queues with 3 classes using tree-like processes. Queueing Systems 54 (2), 99–109 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  52. Ndreca, S., Scoppola, B.: Discrete-time GI/Geom/1 queueing system with priority. European Journal of Operational Research 189, 1403–1408 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  53. Walraevens, J., Steyaert, B., Bruneel, H.: Analysis of a discrete-time preemptive resume priority buffer. European Journal of Operational Research 186(1), 182–201 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  54. Sumita, U., Sheng, O.: Analysis of query processing in distributed database systems with fully replicated files: a hierarchical approach. Performance Evaluation 8(3), 223–238 (1988)

    Article  MATH  Google Scholar 

  55. Yoon, C., Un, C.: Unslotted 1- and p i -persistent CSMA-CD protocols for fiber optic bus networks. IEEE Transactions on Communications 42(2-4), 158–465 (1994)

    Google Scholar 

  56. Mukherjee, S., Saha, D., Tripathi, S.: A preemptive protocol for voice-data integration in ring-based LAN: performance analysis and comparison. Performance Evaluation 11(3), 339–354 (1995)

    MATH  Google Scholar 

  57. Walraevens, J., Steyaert, B., Bruneel, H.: A preemptive repeat priority queue with resampling: performance analysis. Annals of Operations Research 146(1), 189–202 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  58. Walraevens, J., Fiems, D., Bruneel, H.: The discrete-time preemptive repeat identical queue. Queueing Systems 53(4), 231–243 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  59. Hong, S., Takagi, H.: Analysis of transmission delay for a structured-priority packet-switching system. Computer Networks and ISDN Systems 29(6), 701–715 (1997)

    Article  Google Scholar 

  60. Kim, K., Chae, K.: Discrete-time queues with discretionary priorities. European Journal of Operational Research 200(2), 473–485 (2010)

    Article  MATH  Google Scholar 

  61. Fidler, M., Persaud, R.: M/G/1 priority scheduling with discrete pre-emption points: on the impacts of fragmentation on IP QoS. Computer Communications 27(12), 1183–1196 (2004)

    Article  Google Scholar 

  62. Fiems, D., Maertens, T., Bruneel, H.: Queueing systems with different types of server interruptions. European Journal of Operational Research 188(3), 838–845 (2008)

    Article  MATH  Google Scholar 

  63. Hsu, J.: Buffer behavior with Poisson arrival and geometric output processes. IEEE Transactions on Communications 22, 1940–1941 (1974)

    Article  Google Scholar 

  64. Heines, T.: Buffer behavior in computer communication systems. IEEE Transactions on Communications 28, 573–576 (1979)

    MATH  Google Scholar 

  65. Bruneel, H.: A general treatment of discrete-time buffers with one randomly interrupted output line. European Journal of Operational Research 27(1), 67–81 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  66. Woodside, C., Ho, E.: Engineering calculation of overflow probabilities in buffers with Markov-interrupted service. IEEE Transactions on Communications 35(12), 1272–1277 (1987)

    Article  Google Scholar 

  67. Yang, O., Mark, J.: Performance analysis of integrated services on a single server system. Performance Evaluation 11, 79–92 (1990)

    Article  Google Scholar 

  68. Lee, D.: Analysis of a single server queue with semi-Markovian service interruption. Queueing Systems 27(1–2), 153–178 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  69. Bruneel, H.: Buffers with stochastic output interruptions. Electronics Letters 19(18), 735–737 (1983)

    Article  Google Scholar 

  70. Georganas, N.: Buffer behavior with Poisson arrivals and bulk geometric output processes. IEEE Transactions on Communications 24(8), 938–940 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  71. Bruneel, H.: A general model for the behaviour of infinite buffers with periodic service opportunities. European Journal of Operational Research 16, 98–106 (1984)

    Article  MATH  Google Scholar 

  72. Laevens, K., Bruneel, H.: Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers. European Journal of Operational Research 85, 161–177 (1995)

    Article  MATH  Google Scholar 

  73. Bruneel, H.: A discrete-time queueing system with a stochastic number of servers subjected to random interruptions. Opsearch 22(4), 215–231 (1985)

    MathSciNet  MATH  Google Scholar 

  74. Bruneel, H.: On buffers with stochastic input and output interruptions. International Journal of Electronics and Communications (AEU) 38(4), 265–271 (1984)

    MATH  Google Scholar 

  75. Ali, M., Zhang, X., Hayes, J.: A discrete-time queueing analysis of the wireless ATM multiplexing system. In: Lorenz, P. (ed.) ICN 2001. LNCS, vol. 2093, pp. 429–438. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  76. Kamoun, F.: Performance evaluation of a queuing system with correlated packet-trains and server interruption. Telecommunication Systems 41(4), 267–277 (2009)

    Article  Google Scholar 

  77. Inghelbrecht, V., Laevens, K., Bruneel, H., Steyaert, B.: Queueing of fixed-length messages in the presence of server interruptions. In: Proceedings Symposium on Performance Evaluation of Computer and Telecommunication Systems, SPECTS 2k, Vancouver, Canada (July 2000)

    Google Scholar 

  78. Fiems, D., Steyaert, B., Bruneel, H.: Performance evaluation of CAI and RAI transmission modes in a GI-G-1 queue. Computers and Operations Research 28(13), 1299–1313 (2001)

    Article  MATH  Google Scholar 

  79. Fiems, D., Steyaert, B., Bruneel, H.: Randomly interrupted GI-G-1 queues, service strategies and stability issues. Annals of Operations Research 112, 171–183 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  80. Fiems, D., Steyaert, B., Bruneel, H.: Analysis of a discrete-time GI-G-1 queueing model subjected to bursty interruptions. Computers and Operations Research 30(1), 139–153 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  81. Fiems, D., Steyaert, B., Bruneel, H.: Discrete-time queues with generally distributed service times and renewal-type server interruptions. Performance Evaluation 55(3-4), 277–298 (2004)

    Article  Google Scholar 

  82. Adan, I., Van Leeuwaarden, J., Winands, E.: On the application of Rouché’s theorem in queueing theory. Operations Research Letters 34(3), 355–360 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  83. Bruneel, H., Kim, B.: Discrete-time models for communication systems including ATM. Kluwer Academic Publisher, Boston (1993)

    Book  Google Scholar 

  84. Fiems, D., Bruneel, H.: A note on the discretization of Little’s result. Operations Research Letters 30(1), 17–18 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  85. Drmota, M.: Systems of functional equations. Random Structures & Algorithms 10(1-2), 103–124 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  86. Flajolet, P., Odlyzko, A.: Singularity analysis of generating functions. SIAM Journal on discrete mathematics 3(2), 216–240 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  87. Takagi, H.: Queueing Analysis; A foundation of performance evaluation. Discrete-time systems, vol. 3. Elsevier Science Publishers, Amsterdam (1993)

    Google Scholar 

  88. Hunter, J.J.: Mathematical Techniques of Applied Probability. Operations Research and Industrial Engineering, vol. 2. Academic Press, New York (1983)

    MATH  Google Scholar 

  89. Bruneel, H.: Performance of discrete-time queuing systems. Computers and Operations Research 20, 303–320 (1993)

    Article  MATH  Google Scholar 

  90. Kleinrock, L.: Queueing systems. Theory, vol. I. John Wiley & Sons, New York (1975)

    MATH  Google Scholar 

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Walraevens, J., Fiems, D., Bruneel, H. (2011). Performance Analysis of Priority Queueing Systems in Discrete Time. In: Kouvatsos, D.D. (eds) Network Performance Engineering. Lecture Notes in Computer Science, vol 5233. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02742-0_10

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