Abstract
In this paper we study a single-server queue where the inter-arrival times and the service times depend on a common discrete time Markov chain. This model generalizes the well-known MAP/G/1 queue by allowing dependencies between inter-arrival and service times. The waiting time process is directly analyzed by solving Lindley's equation by transform methods. The Laplace–Stieltjes transforms (LST) of the steady-state waiting time and queue length distribution are both derived, and used to obtain recursive equations for the calculation of the moments. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times.
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References
E. Arjas, On the use of a fundamental identity in the theory of semi-Markov queues, Adv. in Appl. Probab. 4 (1972) 271–284.
S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993) 365–372.
U.N. Bhat, Queueing systems with first-order dependence, Opsearch 6 (1969) 1–24.
S.C. Borst, O.J. Boxma and M.B. Combé, Collection of customers: A correlated M/G/1 queue, Performance Evaluation Rev. 20 (1992) 47–59.
S.C. Borst, O.J. Boxma and M.B. Combé, An M/G/1 queue with customer collection, Comm. Statist. Stochastic Models 9 (1993) 341–371.
O.J. Boxma and M.B. Combé, The correlated M/G/1 queue, AEÑ 47 (Special Issue on Teletraffic Theory and Engineering in Memory of F. Pollaczek) (1993) 330–335.
O.J. Boxma and D. Perry, A queueing model with dependence between service and interarrival times, European J. Oper. Res. 128 (2001) 611–624.
I. Cidon, R. Guerin, A. Khamisy and M. Sidi, On queues with inter-arrival times proportional to service times, Technical Report EE PUB. No. 811, Technion (1991).
I. Cidon, R. Guerin, A. Khamisy and M. Sidi, Analysis of a correlated queue in communication systems, Technical Report EE PUB. No. 812, Technion (1991).
J.W. Cohen, On periodic Pollaczek waiting time processes, in: Athens Conf. on Applied Probability and Time Series Analysis, Vol. I, 1995, Lecture Notes in Statistics, Vol. 114 (Springer, New York, 1996) pp. 361–378.
M.B. Combé and O.J. Boxma, BMAP modelling of a correlated queue, in: Network Performance Modeling and Simulation, eds. J. Walrand, K. Bagchi and G.W. Zobrist (1998) pp. 177–196.
B. Conolly, The waiting time for a certain correlated queue, Oper. Res. 15 (1968) 1006–1015.
B. Conolly and Q.H. Choo, The waiting process for a generalized correlated queue with exponential demand and service, SIAM J. Appl. Math. 37 (1979) 263–275.
B. Conolly and N. Hadidi, A correlated queue, Appl. Probab. 6 (1969) 122–136.
R.B. Cooper, Introduction to Queueing Theory (North-Holland, New York, 1981).
J.H.A. De Smit, The queue GI/M/s with customers of different types or the queue GI/Hm/s, Adv. in Appl. Probab. 15 (1983) 392–419.
J.H.A. De Smit, The single server semi-Markov queue, Stochastic Process. Appl. 22 (1986) 37–50.
K.W. Fendick, V.R. Saksena and W. Whitt, Dependence in packet queues, IEEE Trans. Commun. 37 (1989) 1173–1183.
N. Hadidi, Queues with partial correlation, SIAM J. Appl. Math. 40 (1981) 467–475.
N. Hadidi, Further results on queues with partial correlation, Oper. Res. 33 (1985) 203–209.
H. Heffes, A class of data traffic processes – Covariance function characterization and related queueing results, Bell Systems Tech. J. 59 (1980) 897–929.
H. Heffes and D.M. Lucantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Selected Areas Commun. 4 (1986) 856–868.
D.P Heyman and M.J. Sobel, Stochastic Models in Operations Research, Vol. I (McGraw-Hill, New York, 1982).
L. Kleinrock, Queueing Systems, Vol. I: Theory (Wiley-Interscience, New York, 1975).
A.J. Lemoine, Waiting time and workload in queues with periodic Poisson input, J. Appl. Probab. 26 (1989) 390–397.
R.M. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Cambridge Philos. Soc. 58 (1962) 497–520.
R.M. Loynes, Stationary waiting time distributions for single-server queues, Ann. Math. Statist. 33 (1962) 1323–1339.
D.M. Lucantoni, New results on the single-server queue with a batch Markovian arrival process, Stochastic Models 7 (1964) 1–46.
D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class of nonrenewal arrival processes, Adv. in Appl. Probab. 22 (1990) 676–705.
M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Allyn and Bacon, Boston, 1964).
C.R. Mitchell, A.S. Paulson and C.A. Beswick, The effect of correlated exponential service times on single server tandem queues, Naval Res. Logistics Quart. 24 (1977) 95–112.
T. Rolski, Approximation of periodic queues, Adv. in Appl. Probab. 19 (1987) 691–707.
T. Rolski, Relationships between characteristics in periodic Poisson queues, Queueing Systems 4 (1989) 17–26.
B. Sengupta, The semi-Markovian queue: Theory and applications, Stochastic Models 6 (1990) 383–413.
K. Sriram and W. Whitt, Characterizing superposition arrival processes in packet multiplexers for voice and data, IEEE J. Selected Areas Commun. 4 (1986) 833–846.
S.R. Smits, M. Wagner and A.G. de Kok, Determination of the order-up-to policies in the stochastic economic lot scheduling problem, Internat. J. Production Economics (2003) to appear.
M. Wagner and S.R. Smits, A local search algorithm for the optimization of the stochastic economic lot scheduling problem, Internat. J. Production Economics (2003) to appear.
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An erratum to this article is available at http://dx.doi.org/10.1007/s11134-006-9860-1.
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Adan, I., Kulkarni, V. Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times. Queueing Syst 45, 113–134 (2003). https://doi.org/10.1023/A:1026093622185
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DOI: https://doi.org/10.1023/A:1026093622185