×

zbMATH — the first resource for mathematics

A tandem queue with coupled processors: Computational issues. (English) Zbl 1094.60065
The paper considers a two-stage queue, where jobs arrive at the first station according to a Poisson process. After receiving service at this station, they move to the second station, and upon completion of service at the second station they leave the system. The amount of work that a job requires at each of the station is an exponentially distributed random variable, and the total service capacity of the two stations together is constant. When both stations are nonempty, a given proportion of the capacity is allocated to station 1, and the remaining proportion is allocated to station 2. If one of the stations is empty, however, the total service capacity of the stations is allocated to the nonempty station. The paper investigates the two-dimensional Markov process representing the numbers of jobs at the two stations. It is known that the problem of finding the bivariate generating function of the stationary distribution can be reduced to a Riemann-Hilbert boundary value problem. In general, obtaining performance measures from the formal solution of a Riemann-Hilbert boundary value problem is not straightforward. The paper discusses the computational issues that arise when obtaining performance measures.

MSC:
60K25 Queueing theory (aspects of probability theory)
30E25 Boundary value problems in the complex plane
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Andradottir, H. Ayhan and D. Down, Server assignment policies for maximizing the steady-state throughput of finite state queueing systems, Management Science 47 (2001) 1421–1439. · Zbl 1232.90135 · doi:10.1287/mnsc.47.10.1421.10262
[2] J.P.C. Blanc, The relaxation time of two queueing systems in series, Stochastic Models 1 (1985) 1–16. · Zbl 0554.60091 · doi:10.1080/15326348508807001
[3] J.P.C. Blanc, Asymptotic analysis of a queueing system with a two-dimensional state space, Journal of Applied Probability 21 (1984) 870–886. · Zbl 0554.60086 · doi:10.2307/3213703
[4] O.J. Boxma and W.P. Groenendijk, Two queues with alternating service and switching times, in: Queueing Theory and its Applications, Liber Amicorum for J.W. Cohen, eds. O.J. Boxma and R. Syski, pp. 261–282, 1988.
[5] J.W. Cohen, The Single Server Queue. North-Holland, Amsterdam, 1982. · Zbl 0481.60003
[6] J.W. Cohen and O.J. Boxma, Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam, 1983. · Zbl 0515.60092
[7] G. Fayolle and R. Iasnogorodski, Two coupled processors: The reduction to a Riemann-Hilbert problem, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 47 (1979) 325–351. · Zbl 0395.68032 · doi:10.1007/BF00535168
[8] W. Feng, M. Kowada and K. Adachi, A two-queue model with Bernouilli service schedule and switching times, Queueing Systems 30 (1998) 405–434. · Zbl 0919.90065 · doi:10.1023/A:1019185509235
[9] D. Gaier, Konstruktive Methoden der Konformen Abbildung. Springer-Verlag, Berlin, 1964.
[10] S.J. de Klein, Fredholm Integral Equations in Queueing Analysis. Ph.D. thesis, Rijksuniversiteit Utrecht, 1988.
[11] N. Mikou, A two-node Jackson’s network subject to breakdowns, Commun. Statist.-Stochastic Models 4 (1988) 523–552. · Zbl 0695.60094 · doi:10.1080/15326348808807093
[12] N. Mikou, O. Idrissi-Kacimi and S. Saadi, Two processes interacting only during breakdown: The case where the load is not lost, Queueing Systems 19 (1995) 301–317. · Zbl 0843.68011 · doi:10.1007/BF01150415
[13] N.I. Muskhelishvili, Singular Integral Equations. Dover Publications, New York, 1992. · Zbl 0108.29203
[14] H. Nauta, Ergodicity Conditions for a Class of Two-dimensional Queueing Problems. Ph.D. thesis, Rijksuniversiteit Utrecht, 1988.
[15] G. Pólya and G. Szegö, Problems and Theorems in Analysis. Volume I. Springer-Verlag, New York, 1972.
[16] J. Resing and L. Örmeci, A tandem queueing model with coupled processors, Operations Research Letters 31 (2003) 383–389. · Zbl 1033.90017 · doi:10.1016/S0167-6377(03)00046-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.