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Asymptotic Expansions for Large Closed Queueing Networks with Multiple Job Classes
April 1992 (vol. 41 no. 4)
pp. 480-488

A closed BCMP queuing network consisting of R job classes (chains), K+1 single-server, fixed-rate nodes, and M/sub j/ class j jobs (j=1, 2, . . ., R) is considered. Asymptotic expansions are constructed for the partition function under assumptions (1) K<<1, (2) M/sub j/<<1 for each j, and (3) K/M/sub j/=O(1). Analytic expressions for performance measures such as the mean queue length are also given. The approach employs the ray method and the method of matched asymptotic expansions. Numerical comparisons illustrate the accuracy of the approximations.

[1] J. R. Jackson, "Jobshop-like queueing systems,"Management. Sci., vol. 10, pp. 131-142, 1963.
[2] W. J. Gordon and G. F. Newell, "Closed queueing systems with exponential servers,"Oper. Res., vol. 15, pp. 254-265, 1967.
[3] F. Baskett, K. M. Chandy, R. R. Muntz, and F. G. Palacios, "Open, closed, and mixed networks of queues with different classes of customers,"J. ACM, vol. 22, no. 2, pp. 248-260, 1975.
[4] J. P. Buzen, "Queueing network models of multiprogramming," Ph.D. dissertation, Div. Eng. Appl. Phys., Harvard Univ., Cambridge, MA., 1971.
[5] J. P. Buzen, "Computational algorithms for closed queueing networks with exponential servers,"Commun. ACM, vol. 16, no. 9, Sept. 1973.
[6] A. E. Conway and N. D. Georganas, "RECAL, A new efficient algorithm for the exact analysis of multiple-chain closed queueing networks,"J. ACM, vol. 33, pp. 768-791, 1986.
[7] J. McKenna, "Extensions and applications of RECAL in the solution of closed product-form queueing networks,"Commun. in Statist.: Stochastic Models, vol. 4, pp. 235-276, 1988.
[8] J. McKenna and D. Mitra, "Integral representations and asymptotic expansions for closed Markovian queueing networks: Normal usage,"Bell Syst. Tech. J., vol. 61, pp. 661-683, 1982.
[9] J. McKenna and D. Mitra, "Asymptotic expansions and integral representations of moments of queue lengths in closed Markovian networks,"J. ACM, vol. 31, pp. 346-360, 1984.
[10] D. Mitra and J. McKenna, "Asymptotic expansions for closed Markovian networks with state dependent rates,"J. ACM, vol. 33, pp. 568-592, 1986.
[11] C. Knessl and C. Tier, "Asymptotic expansions for large closed queueing networks,"J. ACM, vol. 37, pp. 144-174, 1990.
[12] J. B. Keller, "Rays, waves and asymptotics,"Bull. Amer. Math. Soc., vol. 44, pp. 727-750, 1978.
[13] R. Courant and D. Hilbert,Methods of Mathematical Physics, Vol. 2. New York: Interscience, 1962.
[14] R. W. Hamming,Numerical Methods for Scientists and Engineers, 2nd Ed. New York: McGraw-Hill, 1973.
[15] E. De Souza e Silva and S. S. Lavenberg, "Calculating joint queue length distribution in product-form queueing networks,"J. ACM, vol. 36, pp. 194-207, 1989.
[16] S.S. Lavenberg,Computer Performance Modeling Handbook, Academic Press, New York, 1983.
[17] W. Whitt, "Open and closed models for networks of queues,"AT&T Bell Labs. Tech. J., vol. 63, pp. 1911-1979, 1984.
[18] M. Reiser and H. Kobayashi, "Queueing networks with multiple closed chains: Theory and computational algorithms,"IBM J. Res. Develop., vol. 19, pp. 283-294, 1975.
[19] J. Zahorjan, K. C. Sevcik, D. L. Eager, and B. Geller, "Balanced job bound analysis of queueing networks,"Commun. ACM, vol. 25, pp. 134-141, 1982.
[20] J. D. Mei and C. Tier, "Asymptotic analysis of multiple class queueing networks,"Appl. Math. Lett., vol. 4, pp. 35-38, 1991.

Index Terms:
large closed queueing networks; multiple job classes; BCMP queuing network; queueing theory.
C. Knessl, C. Tier, "Asymptotic Expansions for Large Closed Queueing Networks with Multiple Job Classes," IEEE Transactions on Computers, vol. 41, no. 4, pp. 480-488, April 1992, doi:10.1109/12.135560
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