Skip to main content
SpringerLink
Log in

Applications in Queueing Theory

  • Chapter
  • First Online:
Fundamentals of Matrix-Analytic Methods

  • 3005 Accesses

Abstract

This chapter focuses on the applications of the matrix-analytic methods developed in Chaps. 1, 2, and 3 in queueing theory. The emphasis is on both the introduction of analytically and numerically tractable stochastic models and the analysis of such models. The first part of this chapter deals with a number of simple and classical queueing models. The second part analyzes a few queueing models with multiple types of customers. Algorithms for computing performance measures are developed.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Ahn S, Ramaswami V (2003) Fluid flow models and queues – a connection by stochastic coupling. Stoch Models 19:325–348

    Article  MathSciNet  MATH  Google Scholar 

  • Alfa AS (1998) Matrix geometric solution of discrete time MAP/PH/1 priority queue. Nav Res Logist 45:23–50

    Article  MathSciNet  MATH  Google Scholar 

  • Alfa AS (2010) Queueing theory for telecommunications: discrete time modelling of a single node system. Springer, New York

    Book  Google Scholar 

  • Alfa AS, Liu B, He QM (2003) Discrete-time analysis of MAP/PH/1 multiclass general preemptive priority queue. Nav Res Logist 50:662–682

    Article  MathSciNet  MATH  Google Scholar 

  • Artalejo JR (1999) Accessible bibliography on retrial queues. Math Comput Model 30:1–6

    Article  Google Scholar 

  • Artalejo JR, Gomez-Corral A (2008) Retrial queueing systems: a computational approach. Springer, Berlin

    Book  MATH  Google Scholar 

  • Asmussen S (1992) Phase-type representation in random walk and queueing problems. Ann Prob 20:772–789

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S (1995a) Stationary distributions for fluid flow models with or without Brownian noise. Stoch Models 11:21–49

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S (1995b) Stationary distributions via first passage times. In: Dshalalow J (ed) Advances in queueing: models, methods and problems. CRC Press, Boca Raton, pp 79–102

    Google Scholar 

  • Asmussen S (2003) Applied probability and queues, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Asmussen S, Møller JR (2001) Calculation of the steady state waiting time distribution in GI/PH/c and MAP/PH/c queues. Queueing Syst 37:9–29

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S, O’Cinneide C (1998) Representation for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues. Stoch Models 14:369–387

    Article  MathSciNet  MATH  Google Scholar 

  • Bean NG, O’Reilly MM, Taylor PG (2005a) Algorithms for return probabilities for stochastic fluid flows. Stoch Models 21:149–184

    Article  MathSciNet  MATH  Google Scholar 

  • Bean NG, O’Reilly MM, Taylor PG (2005b) Hitting probabilities and hitting times for stochastic fluid flows. Stoch Processes Appl 115:1530–1556

    Article  MathSciNet  MATH  Google Scholar 

  • Breuer L, Baum D (2005) An introduction to queueing theory and matrix-analytic methods. Springer, Dordrecht

    MATH  Google Scholar 

  • Brill PH (2008) Level crossing methods in stochastic models. Springer, New York

    Book  MATH  Google Scholar 

  • Buzacott JA, Shanthikumar JG (1993) Stochastic models of manufacturing systems. Prentice-Hall, New York

    MATH  Google Scholar 

  • C’inlar E (1969) Markov renewal theory. Adv Appl Prob 1:123–187

    Article  MathSciNet  Google Scholar 

  • Chakravarthy S (2009) A disaster queue with Markovian arrivals and impatient customers. Appl Math Comput 214:48–59

    Article  MathSciNet  MATH  Google Scholar 

  • Chaudhry ML, Templeton JGC (1983) A first course in bulk queues. Wiley, New York

    MATH  Google Scholar 

  • Chen H, Yao DD (2001) Fundamentals of queueing networks: performance, asymptotics and optimization, vol 46, Applications of mathematics. Springer, New York

    Book  Google Scholar 

  • Cohen JW (1982) The single server queues. North-Holland, Amsterdam

    Google Scholar 

  • da Silva Soares A, Latouche G (2002) Further results on the similarity between fluid queues and QBDs. In: Latouche G, Taylor P (eds) Matrix-analytic methods theory and applications – proceedings of the 4th international conference on matrix-analytic methods. World Scientific, New Jersey, pp 89–106

    Google Scholar 

  • da Silva Soares A, Latouche G (2006) Matrix-analytic methods for fluid queues with finite buffers. Perform Eval 63:295–314

    Article  Google Scholar 

  • da Silva Soares A, Latouche G (2009) Fluid queues with level dependent evolution. Eur J Oper Res 196:1041–1048

    Article  MATH  Google Scholar 

  • Diamond JE, Alfa AS (1999) Matrix analytical methods for a multi-server retrial queue with buffer. TOP 7:249–266

    Article  MathSciNet  MATH  Google Scholar 

  • Erlang AK (1917) Solution of some problems in the theory of probabilities of significance in automatic telephone exchange. Post Office Elec Eng J 10:189–197

    Google Scholar 

  • Falin GI, Templeton JGC (1997) Retrial queues. Chapman & Hall, New York

    MATH  Google Scholar 

  • Foley R, McDonald D (2001) Join the shortest queue: stability and exact asymptotics. Ann Appl Prob 11:569–607

    MathSciNet  MATH  Google Scholar 

  • Gross D, Harris CM (1985) Fundamentals of queueing theory, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Hassin R, Haviv M (1999) To queue or not to queue: equilibrium behavior in queueing systems. Kluwer Academic, Boston

    Google Scholar 

  • He QM (1996) Queues with marked customers. Adv Appl Prob 28:567–587

    Article  MATH  Google Scholar 

  • He QM (2000) Quasi-birth-and-death Markov processes with a tree structure and the MMAP[K]/PH[K]/1 queue. Eur J Oper Res 120:641–656

    Article  MATH  Google Scholar 

  • He QM (2001) The versatility of MMAP[K] and the MMAP[K]/G[K]/1 queue. Queueing Syst 38:397–418

    Article  MathSciNet  MATH  Google Scholar 

  • He QM (2003) Age process, total workload, sojourn times, and waiting times in a discrete time SM[K]/PH[K]/1/FCFS queue. Queueing Syst 49:363–403

    Article  Google Scholar 

  • He QM (2004) Workload process, waiting times, and sojourn times in a discrete time MMAP[K]/SM[K]/1/FCFS queue. Stoch Models 20:415–437

    Article  MathSciNet  MATH  Google Scholar 

  • He QM, Alfa AS (1998) The MMAP[K]/PH[K]/1 queue with a last-come-first-served preemptive service discipline. Queueing Syst 28:269–291

    Article  MathSciNet  Google Scholar 

  • He QM, Jewkes EM (1996) A level crossing analysis of the MAP/G/1 queue. In: Chakravarthy SR, Alfa AS (eds) Matrix-analytic methods in stochastic models. Marcel Dekker, New York, pp 107–116

    Google Scholar 

  • He QM, Neuts MF (2002) Two M/M/1 queues with transfers of customers. Queueing Syst 42:377–400

    Article  MathSciNet  MATH  Google Scholar 

  • He QM, Li H, Zhao Q (2000) Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times. Queueing Syst 35:323–347

    Article  MathSciNet  MATH  Google Scholar 

  • Hsu GH (1988) Stochastic service systems. Academic, Beijing

    Google Scholar 

  • Hsu GH, He QM (1994) The matched queueing system GIâ—¦PH/PH/1. Acta Math Appl Sin 17:34–47

    Google Scholar 

  • Jaiswal NK (1968) Priority queues. Academic, New York

    MATH  Google Scholar 

  • Kendall DG (1953) Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann Math Stat 24:338–354

    Article  MathSciNet  MATH  Google Scholar 

  • Kleinrock L (1975) Queueing systems, volume I: theory. Wiley Interscience, New York

    Google Scholar 

  • Kroese DP, Scheinhardt WRW, Taylor PG (2004) Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Ann Appl Prob 14:2057–2089

    Article  MathSciNet  MATH  Google Scholar 

  • Li H, Zhao YQ (2009) Exact tail asymptotics in a priority queue – characterization of the preemptive model. Queueing Syst 63:355–381

    Article  MathSciNet  MATH  Google Scholar 

  • Liu B, Wang X, Zhao YQ (2011) Tail asymptotics for M/M/c retrial queues with non-persistent customers. Oper Res Int J 12:173–188

    Article  Google Scholar 

  • Lucantoni DM (1991) New results on the single server queue with a batch Markovian arrival process. Stoch Models 7:1–46

    Article  MathSciNet  MATH  Google Scholar 

  • Lucantoni DM (1998) Further transient analysis of the BMAP/G/1 queue. Stoch Models 14:461–478

    Article  MathSciNet  MATH  Google Scholar 

  • Lucantoni DM, Neuts MF (1994) Some steady-state distributions for the MAP/SM/1 queue. Stoch Models 10:575–598

    Article  MathSciNet  MATH  Google Scholar 

  • Lucantoni DM, Meier-Hellstern KS, Neuts MF (1990) A single server queue with server vacations and a class of non-renewal arrival processes. Adv Appl Prob 22:676–705

    Article  MathSciNet  MATH  Google Scholar 

  • Lucantoni DM, Choudhury GL, Whitt W (1994) The transient BMAP/G/1 queue. Stoch Models 10:145–182

    Article  MathSciNet  MATH  Google Scholar 

  • Miller DG (1981) Computation of steady-state probabilities for M/M/1 priority queues. Oper Res 29:945–958

    Article  MathSciNet  MATH  Google Scholar 

  • Miyazawa M (2009) Tail decay rates in double QBD processes and related reflected random walks. Math Oper Res 34:547–575

    Article  MathSciNet  MATH  Google Scholar 

  • Miyazawa M, Zhao YQ (2004) The stationary tail asymptotics in the GI/G/1 type queue with countably many background states. Adv Appl Prob 36:1231–1251

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts MF (1978) Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector. Adv Appl Prob 10:185–212

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts MF (1981) Matrix-geometric solutions in stochastic models – an algorithmic approach. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Neuts MF (1986a) The caudal characteristic curve of queues. Adv Appl Prob 18:221–254

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts MF (1986b) Generalizations of the Pollaczek-Khinchin integral method in the theory of queues. Adv Appl Prob 18:952–990

    Article  MathSciNet  MATH  Google Scholar 

  • Neuts MF (1989a) The fundamental period of the queue with Markov-modulated arrivals. In: Probability, statistics and mathematics: papers in honor of Samuel Karlin. Academic, Boston, pp 187–200

    Google Scholar 

  • Neuts MF (1989b) Structured stochastic matrices of M/G/1 type and their applications. Marcel Dekker, New York

    Google Scholar 

  • Ozawa T (2006) Sojourn time distributions in the queue defined by a general QBD process. Queueing Syst 53:203–211

    Article  MathSciNet  MATH  Google Scholar 

  • Prabhu NU (1998) Stochastic storage processes: queues, insurance risk, dams, and data communication, 2nd edn. Springer, New York

    Book  Google Scholar 

  • Ramaswami V (1980) The N/G/1 queue and its detailed analysis. Adv Appl Prob 12:222–261

    Article  MathSciNet  MATH  Google Scholar 

  • Ramaswami V (1982) The busy period of queues which have a matrix-geometric steady state probability vector. Opsearch 19:265–281

    MathSciNet  Google Scholar 

  • Ramaswami V, Taylor PG (1996) Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch Models 12:143–164

    Article  MathSciNet  MATH  Google Scholar 

  • Ross SM (2010) Introduction to probability models, 10th edn. Academic, Amsterdam/Boston

    MATH  Google Scholar 

  • Sengupta B (1989) Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue. Adv Appl Prob 21:159–180

    Article  MATH  Google Scholar 

  • Sengupta B (1990a) Phase-type representations for matrix-geometric solutions. Stoch Models 6:163–167

    Article  MATH  Google Scholar 

  • Sengupta B (1990b) The semi-Markovian queue: theory and applications. Stoch Models 6:383–413

    Article  MATH  Google Scholar 

  • Takahashi Y (1981) Asymptotic exponentiality of the tail of the waiting time distribution in a PH/PH/c queue. Adv Appl Prob 13:619–630

    Article  MATH  Google Scholar 

  • Takahashi Y, Fujimoto K, Makimoto N (2001) Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases. Stoch Models 17:1–24

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (1996) A nonpreemptive priority MAP/G/1 queue with two classes of customers. J Oper Res Soc Jpn 39:266–290

    MathSciNet  MATH  Google Scholar 

  • Takine T (1999) The nonpreemptive priority MAP/G/1 queue. Oper Res 47:917–927

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (2001a) A recent progress in algorithmic analysis of FIFO queues with Markovian arrival streams. J Korean Math Soc 38:807–842

    MathSciNet  MATH  Google Scholar 

  • Takine T (2001b) Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams. Stoch Models 17:429–448

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (2001c) Distributional form of Little’s law for FIFO queues with multiple Markovian arrival streams and its application to queues with vacations. Queueing Syst 37:31–63

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (2001d) Queue length distribution in a FIFO single-sever queue with multiple arrival streams having different service time distributions. Queueing Syst 39:349–375

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (2002) Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain. Queueing Syst 42:131–151

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T (2004) Geometric and subexponential asymptotics of Markov chains of M/G/1 type. Math Oper Res 29:624–648

    Article  MathSciNet  MATH  Google Scholar 

  • Takine T, Hasegawa T (1994) The workload in a MAP/G/1 queue with state-dependent services: its applications to a queue with preemptive resume priority. Stoch Models 10:183–204

    Article  MathSciNet  MATH  Google Scholar 

  • Tang J, Zhao YQ (2008) Stationary tail asymptotics of a tandem queue with feedback. Ann Oper Res 160:173–189

    Article  MathSciNet  MATH  Google Scholar 

  • Tian NS, Zhang ZG (2003) Stationary distributions of GI/M/c queue with PH type vacations. Queueing Syst 44:183–202

    Article  MathSciNet  MATH  Google Scholar 

  • Tian NS, Zhang ZG (2006) Vacation queueing models: theory and applications. Springer, New York

    Google Scholar 

  • van Houdt B, Blondia C (2002) The delay distribution of a type k customer in a FCFS MMAP[K]/PH[K]/1 queue. J Appl Prob 39:213–222

    Article  MATH  Google Scholar 

  • van Houdt B, Blondia C (2004) The waiting time distribution of a type k customer in a discrete time MMAP[K]/PH[K]/c (c = 1, 2) queue using QBDs. Stoch Models 20:55–69

    Article  MathSciNet  MATH  Google Scholar 

  • Whitt W (2002) Stochastic-process limits. Springer, New York

    MATH  Google Scholar 

  • Yeung RW, Sengupta B (1994) Matrix product-form solutions for Markov chains with a tree structure. Adv Appl Prob 26:965–987

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang ZG, Tian NS (2001) Discrete time Geo/G/1 queue with multiple adaptive vacations. Queueing Syst 38:419–429

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao YQ, Grassmann WK (1990) The shortest queue model with jockeying. Nav Res Logist 37:773–787

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao YQ, Grassmann WK (1995) Queueing analysis of a jockeying model. Oper Res 43:520–529

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management Sciences, University of Waterloo, Waterloo, ON, Canada

    Qi-Ming He

Authors
  1. Qi-Ming He
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media New York

About this chapter

Cite this chapter

He, QM. (2014). Applications in Queueing Theory. In: Fundamentals of Matrix-Analytic Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7330-5_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4614-7330-5_4

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-7329-9

  • Online ISBN: 978-1-4614-7330-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation