Skip to main content
SpringerLink
Log in

Queueing models for the analysis of communication systems

  • Invited Paper
  • Published:
TOP Aims and scope Submit manuscript

Abstract

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Altman E, Avrachenkov K, Ayesta U (2006) A survey on discriminatory processor sharing. Queueing Syst 53:53–63

    Article  Google Scholar 

  • Arlitt M, Williamson C (1997) Internet web servers: workload characterization and performance implications. IEEE/ACM Trans Netw 5:631–645

    Article  Google Scholar 

  • Asrin MM, Kamoun F (1998) A transient discrete-time queueing analysis of the ATM multiplexer. Perform Eval 32:153–183

    Article  Google Scholar 

  • Avi-Itzhak B, Naor P (1963) Some queuing problems with the service station subject to breakdown. Oper Res 11:303–319

    Article  Google Scholar 

  • Blondia C, Casals O (1992) Statistical multiplexing of VBR sources: a matrix-analytic approach. Perform Eval 16:5–20

    Article  Google Scholar 

  • Bonomi F, Montagna S, Paglino R (1993) A further look at statistical multiplexing in ATM networks. Comput Netw ISDN Syst 26:119–138

    Article  Google Scholar 

  • Bruneel H, Queueing behavior of statistical multiplexers with correlated inputs, IEEE Transactions on Communications, vol. COM-36 (1988), pp. 1339–1341.

  • Bruneel H (1993) Packet delay and queue length for statistical multiplexers with low-speed access lines. Comput Netw ISDN Syst 25:1267–1277

    Article  Google Scholar 

  • Bruneel H, Kim BG (1993) Discrete-time models for communication systems including ATM. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Bruneel H (1994) Calculation of message delays and message waiting times in switching elements with slow access lines. IEEE Trans Commun 42:255–259

    Article  Google Scholar 

  • Chang J, Harn Y (1992) A discrete-time priority queue with two-class customers and bulk services. Queueing Syst 10:185–212

    Article  Google Scholar 

  • Choi BD, Choi DI, Lee Y, Sung DK (1998) Priority queueing system with fixed-length packet-train arrivals. IEE Proc Commun 145:331–336

    Article  Google Scholar 

  • Choi BD, Kim B, Hwang GU, Kim JK (2004) The analysis of a multiserver queue fed by a discrete autoregressive process of order 1. Oper Res Lett 32:85–93

    Article  Google Scholar 

  • De Turck K, Fiems D, Wittevrongel S, Bruneel H (2011) A Taylor series expansions approach to queues with train arrivals. In: Proceedings of VALUETOOLS, Cachan, May 2011

  • De Vuyst S, Wittevrongel S, Bruneel H (2001) Statistical multiplexing of correlated variable-length packet trains: an analytic performance study. J Oper Res Soc 52:318–327

    Article  Google Scholar 

  • De Vuyst S, Wittevrongel S, Bruneel H (2002) Mean value and tail distribution of the message delay in statistical multiplexers with correlated train arrivals. Perform Eval 48:103–129

    Article  Google Scholar 

  • De Vuyst S, Wittevrongel S, Fiems D, Bruneel H (2008) Controlling the delay trade-off between packet flows using multiple reserved places. Perform Eval 65:484–511

    Article  Google Scholar 

  • Demoor T, Walraevens J, Fiems D, De Vuyst S, Bruneel H (2010) Influence of real-time queue capacity of real-time queue capacity on system contents in Diffserv’s expedited forwarding per-hop behavior. J Ind Manag Optim 6:587–602

    Article  Google Scholar 

  • Demoor T, Fiems D, Walraevens J, Bruneel H (2011) Partially shared buffers with full or mixed priority. J Ind Manag Optim 7:735–751

    Article  Google Scholar 

  • Elsayed K, Perros H (2000) The superposition of discrete-time Markov renewal processes with an application to statistical multiplexing of bursty traffic sources. Appl Math Comput 115:43–62

    Article  Google Scholar 

  • Feyaerts B, De Vuyst S, Bruneel H, Wittevrongel S (2012) Analysis of discrete-time buffers with heterogeneous session-based arrivals and general session lengths. Comput Oper Res 39:2905–2914

    Article  Google Scholar 

  • Fiems D, Bruneel H (2002) A note on the discretization of Little’s result. Oper Res Lett 30:17–18

    Article  Google Scholar 

  • Fiems D, Steyaert B, Bruneel H (2004) Discrete-time queues with generally distributed service times and renewal-type server interruptions. Perform Eval 55:277–298

    Article  Google Scholar 

  • Fiems D, De Turck K (2012) The mean queue content of discrete-time queues with zero-regenerative arrivals. Oper Res Lett 40:235–238

    Article  Google Scholar 

  • Flajolet P, Sedgewick R (2008) Analytic combinatorics. Cambridge University Press, Cambridge

  • Gaver Jr DP (1962) A waiting line with interrupted service, including priorities. J R Stat Soc B24:73–90

  • Herrmann C (2001) The complete analysis of the discrete time finite DBMAP/G/1/N queue. Perform Eval 43:95–121

    Article  Google Scholar 

  • Hirano M, Watanabe N (1990) Traffic characteristics and a congestion control scheme for an ATM network. Int J Digit Analog Commun Syst 3:211–217

    Article  Google Scholar 

  • Hoflack L, De Vuyst S, Wittevrongel S, Bruneel H (2008) Analytic traffic model of web server. Electron Lett 44:61–63

    Article  Google Scholar 

  • Hoflack L, De Vuyst S, Wittevrongel S, Bruneel H (2010) Discrete-time buffer systems with session-based arrival streams. Perform Eval 67:432–450

    Article  Google Scholar 

  • Hunter JJ (1983) Mathematical techniques of applied probability, vol 2. In: Discrete time models: techniques and applications. Academic Press, New York

  • Hwang GU, Choi BD, Kim JK (2002) The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1. J Appl Probab 39:619–629

    Article  Google Scholar 

  • Hwang GU, Sohraby K (2003) On the exact analysis of a discrete-time queueing system with autoregressive inputs. Queueing Syst 43:29–41

    Article  Google Scholar 

  • Inai H, Yamakita J (1998) A two-layer queueing model to predict performance of packet transfer in broadband networks. Ann Oper Res 79:349–371

    Article  Google Scholar 

  • Kamoun F (2006) Performance analysis of a discrete-time queuing system with a correlated train arrival process. Perform Eval 63:315–340

    Article  Google Scholar 

  • Kamoun F (2006) The discrete-time queue with autoregressive inputs revisited. Queueing Syst 54:185–192

    Article  Google Scholar 

  • Kamoun F (2009) Performance evaluation of a queuing system with correlated packet-trains and server interruption. Telecommun Syst 41:267–277

    Article  Google Scholar 

  • Khamisy A, Sidi M (1992) Discrete-time priority queues with two-state Markov modulated arrivals. Stoch Models 8:337–357

    Google Scholar 

  • Kim B, Sohraby K (2006) Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals. Adv Appl Probab 38:1116–1131

    Article  Google Scholar 

  • Kim B, Chang Y, Kim YC, Choi BD (2007) A queueing system with discrete autoregressive arrivals. Perform Eval 64:148–161

    Article  Google Scholar 

  • Kim J, Kim B, Sohraby K (2008) Mean queue size in a queue with discrete autoregressive arrivals of order \(p\). Ann Oper Res 162:69–83

    Article  Google Scholar 

  • Kim B, Kim J (2010) Queue size distribution in a discrete-time D-BMAP/G/1 retrial queue. Comput Oper Res 37:1220–1227

    Article  Google Scholar 

  • Laevens K, Bruneel H (1998) Discrete-time multiserver queues with priorities. Perform Eval 33:249–275

    Article  Google Scholar 

  • Maertens T, Walraevens J, Bruneel H (2006) On priority queues with priority jumps. Perform Eval 63:1235–1252

    Article  Google Scholar 

  • Massoulié L, Roberts JW (2000) Bandwidth sharing and admission control for elastic traffic. Telecommun Syst 15:185–201

    Article  Google Scholar 

  • Massoulié L, Roberts JW (1999) Bandwidth sharing: objectives and algorithms. In: Proceedings of the IEEE INFOCOM ’99, New York, March 1999, pp 1395–1403

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

    Article  Google Scholar 

  • Miao DWC, Chen H (2011) On the variances of system size and soujourn time in a discrete-time DAR(1)/D/1 queue. Probab Eng Inform Sci 25:519–535

    Article  Google Scholar 

  • Miao DWC, Lee HC (2013) Second-order performance analysis of discrete-time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter. Appl Stoch Models Bus Ind 29:45–60

    Article  Google Scholar 

  • Mitrani I (1987) Modelling of Computer and Communication Systems. Cambridge University Press, Cambridge

    Google Scholar 

  • Morozov E, Fiems D, Bruneel H (2011) Stability analysis of multiserver discrete-time queueing systems with renewal type server interruptions. Perform Eval 68:1261–1275

    Article  Google Scholar 

  • Nabe M, Murata M, Miyahara H (1998) Analysis and modeling of world wide web traffic for capacity dimensioning of internet access lines. Perform Eval 34:249–271

    Article  Google Scholar 

  • Nain P (1983) Queueing systems with service interruptions: an approximate model. Perform Eval 3:123–129

    Article  Google Scholar 

  • Pieloor J, Lewis DJH (1996) Variance of a discrete-time \(G/D/1\) queue fed by two-state on-off sources. Electron Lett 32:19–20

    Article  Google Scholar 

  • Roberts JW (2004) A survey on statistical bandwidth sharing. Comput Netw 45:319–332

    Article  Google Scholar 

  • Sidi M, Segall A (1983) Structured priority queueing systems with applications to packet-radio networks. Perform Eval 3:265–275

    Article  Google Scholar 

  • Sohraby K (1993) On the theory of general ON-OFF sources with applications in high-speed networks. In: Proceedings of the IEEE INFOCOM ’93, San Francisco, March/April 1993, pp 401–410

  • Steyaert B, Bruneel H (1995) On the performance of multiplexers with three-state bursty sources: analytical results. In: IEEE transactions on communications, vol COM-43, pp 1299–1303

  • Takagi H (1993) Queueing analysis—a foundation of performance evaluation, vol 3. In: Discrete-time systems. North-Holland, Amsterdam

  • Thiruvengadam K (1963) Queuing with breakdowns. Oper Res 11:62–71

    Article  Google Scholar 

  • Vinck B, Bruneel H (1995) Relationship between delay and buffer contents in ATM queues. Electron Lett 31:952–954

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Walraevens J, Steyaert B, Bruneel H (2008) Analysis of a discrete-time preemptive resume priority buffer. Eur J Oper Res 186:182–201

    Article  Google Scholar 

  • Walraevens J, Wittevrongel S, Bruneel H (2009) Performance analysis of a priority queue with session-based arrivals and its application to E-commerce web servers. Int J Adv Internet Technol 2:46–57

    Google Scholar 

  • Walraevens J, van Leeuwaarden JSH, Boxma OJ (2010) Power series approximations for generalized processor sharing systems. Queueing Syst 66:107–130

    Article  Google Scholar 

  • Wang J, Huang Y, Dai Z (2011) A discrete-time on-off source queueing system with negative customers. Computers & Industrial Engineering 61:1226–1232

    Article  Google Scholar 

  • White H, Christie L (1958) Queuing with preemptive priorities or with breakdown. Oper Res 6:79–95

    Article  Google Scholar 

  • Wittevrongel S, Bruneel H (1996) Exact calculation of buffer-contents variance and delay jitter in a discrete-time queue with correlated input traffic. Electronics Letters 32:1258–1259

    Article  Google Scholar 

  • Wittevrongel S, Bruneel H (1996) Variance of buffer contents and delay in ATM queues. Electron Lett 32:1974–1975

    Article  Google Scholar 

  • Wittevrongel S (1998) Discrete-time buffers with variable-length train arrivals. Electron Lett 34:1719–1721

    Article  Google Scholar 

  • Wittevrongel S, Bruneel H (1998) Correlation effects in ATM queues due to data format conversions. Perform Eval 32:35–56

    Article  Google Scholar 

  • Wittevrongel S, Bruneel H (1995) Effect of the on-period distribution on the performance of an ATM multiplexer fed by on/off sources: an analytical study. In: Proceedings of PCN ’95, Istanbul, October 1995, pp 33–47

  • Wittevrongel S, Bruneel H (1997) Deriving the tail distribution of the buffer contents in a statistical multiplexer with general heterogeneous on/off sources. In: Proceedings of the PMCCN ’97, Tsukuba, November 1997, pp 37–56

  • Xiong Y, Bruneel H (1993) Buffer contents and delay for statistical multiplexers with fixed-length packet-train arrivals. Perform Eval 17:31–42

    Article  Google Scholar 

  • Xiong Y, Bruneel H (1994) An analytic approach to obtain tail distributions of buffer contents and delay in a discrete-time single-server queue with bursty arrivals. JORBEL 34:3–13

    Google Scholar 

  • Xiong Y, Bruneel H (1997) Buffer behavior of statistical multiplexers with correlated train arrivals. Int J Electron Commun (AEÜ) 51:178–186

    Google Scholar 

  • Yashkov SF, Yashkova AS (2007) Processor sharing: a survey of the mathematical theory. Autom Remote Control 68:1662–1731

    Article  Google Scholar 

  • Zhao JA, Li B, Cao XR, Ahmad I (2006) A matrix-analytic solution for the DBMAP/PH/1 priority queue. Queueing Syst 53:127–145

    Article  Google Scholar 

  • Zhou WH, Wang AH (2008) Discrete-time queue with Bernoulli bursty source arrival and generally distributed service times. Appl Math Model 32:2233–2240

    Article  Google Scholar 

Download references

Acknowledgments

This research has been funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

Author information

Authors and Affiliations

  1. Stochastic Modeling and Analysis of Communication Systems (SMACS) Research Group, Department of Telecommunications and Information Processing (TELIN), Ghent University, Sint-Pietersnieuwstraat 41, 9000 , Gent, Belgium

    Herwig Bruneel, Dieter Fiems, Joris Walraevens & Sabine Wittevrongel

Authors
  1. Herwig Bruneel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dieter Fiems
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joris Walraevens
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Sabine Wittevrongel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sabine Wittevrongel.

Additional information

This invited paper is discussed in the comments available at: doi:10.1007/s11750-014-0327-y; doi:10.1007/s11750-014-0328-x; doi:10.1007/s11750-014-0329-9.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bruneel, H., Fiems, D., Walraevens, J. et al. Queueing models for the analysis of communication systems. TOP 22, 421–448 (2014). https://doi.org/10.1007/s11750-014-0330-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11750-014-0330-3

Keywords

Mathematics Subject Classification

Search

Navigation