Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Queueing Systems
Published in: Queueing Systems 3-4/2017

15-02-2017

Fitting correlated arrival and service times and related queueing performance

Authors: Peter Buchholz, Jan Kriege

Published in: Queueing Systems | Issue 3-4/2017

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we consider a queue where the inter-arrival times are correlated and, additionally, service times are also correlated with inter-arrival times. We show that the resulting model can be interpreted as an MMAP[K]/PH[K]/1 queue for which matrix geometric solution algorithms are available. The major result of this paper is the presentation of approaches to fit the parameters of the model, namely the MMAP, the PH distribution and the parameters introducing correlation between inter-arrival and service times, according to some trace of inter-arrival and corresponding service times. Two different algorithms are presented. The first algorithm is based on available methods to compute a MAP from the inter-arrival times and a PH distribution from the service times. Afterward, the correlation between inter-arrival and service times is integrated by solving a quadratic programming problem over some joint moments. The second algorithm is of the expectation maximization type and computes all parameters of the MAP and the PH distribution in an iterative way. It is shown that both algorithms yield sufficiently accurate results with an acceptable effort.
previous article The analysis of batch sojourn-times in polling systems
next article A general workload conservation law with applications to queueing systems

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
2.
3.
4.
5.
6.
Adan, I.J.B.F., Kulkarni, V.G.: Single-server queue with Markov-dependent inter-arrival and service times. Queueing Syst. 45(2), 113–134 (2003)CrossRef
7.
Alfa, A.S., Neuts, M.F.: Modelling vehicular traffic using the discrete time Markovian arrival process. Transp. Sci. 29(2), 109–117 (1995)CrossRef
8.
Asmussen, S., Nerman, O., Olsson, M.: Fitting phase type distributions via the EM algorithm. Scand. J. Stat. 23, 419–441 (1996)
9.
Bhat, U.N.: Sixty years of queueing theory. Manag. Sci. 15(6), B280–B294 (1969)CrossRef
10.
Biller, B., Nelson, B.L.: Modeling and generating multivariate time-series input processes using a vector autoregressive technique. ACM Trans. Model. Comput. Simul. 13(3), 211–237 (2003)CrossRef
11.
Bini, D., Meini, B., Steffé, S., Pérez, J.F., Houdt, B.V.: Smcsolver and Q-MAM: tools for matrix-analytic methods. SIGMETRICS Perform. Eval. Rev. 39(4), 46 (2012)CrossRef
12.
Bini, D.A., Latouche, G., Meini, B.: Solving nonlinear matrix equations arising in tree-like stochastic processes. Linear Algebra Appl. 366, 39–64 (2003)CrossRef
13.
Bobbio, A., Horváth, A., Scarpa, M., Telek, M.: Acyclic discrete phase type distributions: properties and a parameter estimation algorithm. Perform. Eval. 54(1), 1–32 (2003)CrossRef
14.
Bobbio, A., Horváth, A., Telek, M.: The scale factor: a new degree of freedom in phase-type approximation. Perform. Eval. 56(1–4), 121–144 (2004)CrossRef
15.
Boxma, O.J., Perry, D.: A queueing model with dependence between service and interarrival time. Eur. J. Oper. Res. 128(13), 611–624 (2001)CrossRef
16.
Breuer, L.: An EM algorithm for batch Markovian arrival processes and its comparison to a simpler estimation procedure. Ann. OR 112(1–4), 123–138 (2002)CrossRef
17.
Buchholz, P., Kemper, P., Kriege, J.: Multi-class Markovian arrival processes and their parameter fitting. Perform. Eval. 67(11), 1092–1106 (2010)CrossRef
18.
19.
Buchholz, P., Kriege, J., Felko, I.: Input Modeling with Phase-Type Distributions and Markov Models—Theory and Applications. Briefs in Mathematics. Springer, Berlin (2014)CrossRef
20.
Buchholz, P., Telek, M.: On minimal representations of rational arrival processes. Ann. OR 202(1), 35–58 (2013)CrossRef
21.
Casale, G., Zhang, E.Z., Smirni, E.: Trace data characterization and fitting for Markov modeling. Perform. Eval. 67(2), 61–79 (2010)CrossRef
22.
Combé, M.B., Boxma, O.J.: BMAP modeling of a correlated queue. In: Walrand, J., Bagchi, K., Zobrist, G.W. (eds.) Network Performance Modeling and Simulation, pp. 177–196. Gordon and Breach Science Publishers, Philadelphia (1999)
23.
Conolly, B.W.: The waiting time process for a certain correlated queue. Oper. Res. 16(5), 1006–1015 (1968)CrossRef
24.
Conolly, B.W., Haidi, N.: A correlated queue. J. Appl. Probab. 6(1), 122–136 (1969)CrossRef
25.
Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum-likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. 39(1), 1–38 (1977)
26.
Fendick, K.W., Saksena, V.R., Whitt, W.: Dependence in packet queues. IEEE Trans. Commun. 37(11), 1173–1183 (1989)CrossRef
27.
He, Q.M.: Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: age process, sojourn times, and queue lengths. J. Syst. Sci. Complex. 25(1), 133–155 (2012)CrossRef
28.
He, Q.M., Alfa, A.S.: The discrete time MMAP[K]/PH[K]/1/LCFS-GPR queue and its variants. In: Proceedings of the 3rd International Conference on Matrix Analytic Methods, pp. 167–190 (2000)
29.
He, Q.M., Neuts, M.: Markov arrival processes with marked transitions. Stoch. Process. Appl. 74, 37–52 (1998)CrossRef
30.
Horváth, A., Telek, M.: Markovian modeling of real data traffic: heuristic phase type and MAP fitting of heavy tailed and fractal like samples. In: Calzarossa, M., Tucci, S. (eds.) Performance Evaluation of Complex Systems: Techniques and Tools, Performance 2002, Tutorial Lectures. Lecture Notes in Computer Science, vol. 2459, pp. 405–434. Springer, Berlin (2002)CrossRef
31.
Horváth, G.: Efficient analysis of the queue length moments of the MMAP/MAP/1 preemptive priority queue. Perform. Eval. 69(12), 684–700 (2012)CrossRef
32.
Horváth, G., Buchholz, P., Telek, M.: A MAP fitting approach with independent approximation of the inter-arrival time distribution and the lag-correlation. In: Proceedings of 2nd International Conference on the Quantitative Analysis of Systems. IEEE CS Press, Washington (2005)
33.
Iyer, S.K., Manjunath, D.: Queues with dependency between interarrival and service times using mixtures of bivariates. Stoch. Models 22(1), 3–20 (2006)CrossRef
34.
Klemm, A., Lindemann, C., Lohmann, M.: Modeling IP traffic using the batch Markovian arrival process. Perform. Eval. 54(2), 149–173 (2003)CrossRef
35.
Kriege, J., Buchholz, P.: An empirical comparison of MAP fitting algorithms. In: Measurement, Modelling, and Evaluation of Computing Systems and Dependability and Fault Tolerance, 15th International GI/ITG Conference, MMB&DFT 2010, Essen, Germany, March 15–17, 2010. Proceedings, Lecture Notes in Computer Science, vol. 5987, pp. 259–273. Springer, Berlin (2010)
36.
Lambert, J., van Houdt, B., Blondia, C.: Queue with correlated service and inter-arrival times and its application to optical buffers. Stoch. Models 22(2), 233–251 (2006)CrossRef
37.
Lütkepohl, H.: Introduction to Multiple Time Series Analysis. Springer, Berlin (1993)CrossRef
38.
Machihara, F.: A BMAP/SM/1 queue with service times depending on the arrival process. Queueing Syst. 33(4), 277–291 (1999)CrossRef
39.
Okamura, H., Dohi, T., Trivedi, K.S.: A refined EM algorithm for PH distributions. Perform. Eval. 68(10), 938–954 (2011)CrossRef
40.
Pérez, J.F., van Velthoven, J., van Houdt, B.: Q-MAM: a tool for solving infinite queues using matrix-analytic methods. In: 3rd International ICST Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2008, Athens, Greece, October 20–24, 2008, p. 16 (2008)
41.
Takine, T.: Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions. Queueing Syst. 39(4), 349–375 (2001)CrossRef
42.
Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9–12), 1153–1168 (2007)CrossRef
43.
Thümmler, A., Buchholz, P., Telek, M.: A novel approach for phase-type fitting with the EM algorithm. IEEE Trans. Dependable Secure Comput. 3(3), 245–258 (2006)CrossRef
44.
van Houdt, B.: Analysis of the adaptive MMAP[K]/PH[K]/1 queue: a multi-type queue with adaptive arrivals and general impatience. Eur. J. Oper. Res. 220(3), 695–704 (2012)CrossRef
45.
van Houdt, B.: A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues. Perform. Eval. 69(7–8), 299–314 (2012)CrossRef
46.
van Houdt, B., Blondia, C.: The waiting time distribution of a type k customer in a discrete-time FCFS MMAP[K]/PH[K]/c (c \(=\) 1, 2) queue using QBDs. Stoch. Models 20(1), 55–69 (2004)CrossRef
47.
van Houdt, B., van Leeuwaarden, J.: Triangular M/G/1-type and tree-like quasi-birth-death Markov chains. INFORMS J. Comput. 23(1), 165–171 (2011)CrossRef
Metadata
Title
Fitting correlated arrival and service times and related queueing performance
Authors
Peter Buchholz
Jan Kriege
Publication date
15-02-2017
Publisher
Springer US
Published in
Queueing Systems / Issue 3-4/2017
Print ISSN: 0257-0130
Electronic ISSN: 1572-9443
DOI
https://doi.org/10.1007/s11134-017-9514-5

Other articles of this Issue 3-4/2017

Queueing Systems 3-4/2017 Go to the issue

Letter

Correction to “Queue-proportional rate allocation with per-link information in multihop wireless networks”

OriginalPaper

A general workload conservation law with applications to queueing systems

OriginalPaper

Large deviations for the total queue size in non-Markovian tandem queues

OriginalPaper

Transient error approximation in a Lévy queue

OriginalPaper

An asymptotic approximation for TCP compound

OriginalPaper

The analysis of batch sojourn-times in polling systems

Premium Partner

    Image Credits