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Optimization via trunk reservation in single resource loss systems under heavy traffic. (English) Zbl 0890.60088

Summary: Trunk reservation is a simple, robust and extremely effective mechanism for controlling loss systems which allows priority to be given to chosen traffic streams. We consider the control of a single resource under a limiting regime in which capacity and arrival rates increase together. We obtain trunk reservation control policies which are asymptotically optimal when calls have differing capacity requirements, holding times, arrival rates and reward rates. The priority levels associated with these trunk reservation policies arise from an attainable bound on the performance of any control policy.

MSC:

60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
90B22 Queues and service in operations research
93E20 Optimal stochastic control
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[1] BEAN, N. G., GIBBENS, R. J. and ZACHARY, S. 1994. Dy namic and equilibrium behaviour of controlled loss networks. Research Report 94-31, Statistical Laboratory, Univ. Cambridge.
[2] BEAN, N. G., GIBBENS, R. J. and ZACHARY, S. 1995. Asy mptotic analysis of single resource loss sy stems in heavy traffic, with applications to integrated networks. Adv. in Appl. Probab. 27 273 292. JSTOR: · Zbl 0823.60081 · doi:10.2307/1428107
[3] BILLINGSLEY, P. 1986. Probability and Measure, 2nd ed. Wiley, New York. · Zbl 0649.60001
[4] COHEN, J. E. and KELLY, F. P. 1990. A paradox of congestion in a queueing network. J. Appl. Probab. 27 730 734. JSTOR: · Zbl 0718.60105 · doi:10.2307/3214558
[5] GIBBENS, R. J. and HUNT, P. J. 1991. Effective bandwidths for the multi-ty pe UAS channel. Queueing Sy stems Theory Appl. 9 17 28. · Zbl 0738.90028
[6] GIBBENS, R. J., HUNT, P. J. and KELLY, F. P. 1990. Bistability in communication networks. Z. In Disorder in physical Sy stems G. R. Grimmett and D. J. A. Welsh, eds. 113 127. Oxford Univ. Press. · Zbl 0721.60103
[7] GIBBENS, R. J. and KELLY, F. P. 1990. Dy namic routing in fully connected networks. IMA J. Math. Control Inform. 7 77 111. · Zbl 0697.94017 · doi:10.1093/imamci/7.1.77
[8] GIBBENS, R. J., KELLY, F. P. and TURNER, S. R. E. 1993. Dy namic routing in multiparented networks. IEEE ACM Transactions on Networking 1 261 270.
[9] GIBBENS, R. J. and REICHL, P. C. 1995. Performance bounds applied to loss networks. In Z. Complex Stochastic Sy stems and Engineering D. M. Titterington, ed. 267 279. Oxford Univ. Press. · Zbl 0833.60090
[10] HUI, J. Y. 1988. Resource allocation for broadband networks. IEEE J. Selected Areas in Communications 6 1598 1608.
[11] HUNT, P. J. 1995. Pathological behaviour in loss networks. J. Appl. Prob. 32 519 533. JSTOR: · Zbl 0823.60093 · doi:10.2307/3215305
[12] HUNT, P. J. and KURTZ, T. G. 1994. Large loss networks. Stochastic Process. Appl. 53 363 378. · Zbl 0810.60087 · doi:10.1016/0304-4149(94)90071-X
[13] HUNT, P. J. and LAWS, C. N. 1993. Asy mptotically optimal loss network control. Math. Oper. Res. 18 880 900. JSTOR: · Zbl 0804.93051 · doi:10.1287/moor.18.4.880
[14] KELLY, F. P. 1986. Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 473 505. JSTOR: · Zbl 0597.60092 · doi:10.2307/1427309
[15] KELLY, F. P. 1991. Effective bandwidths at multi-class queues. Queueing Sy stems Theory Appl. 9 5 16. · Zbl 0737.60084 · doi:10.1007/BF01158789
[16] KELLY, F. P. 1991. Loss networks. Ann. Appl. Prob. 1 319 378. · Zbl 0743.60099 · doi:10.1214/aoap/1177005872
[17] KELLY, F. P. 1994. Bounds on the performance of dy namic routing schemes for highly connected networks. Math. Oper. Res. 19 1 20. JSTOR: · Zbl 0809.90048 · doi:10.1287/moor.19.1.1
[18] KEY, P. B. 1990. Optimal control and trunk reservation in loss networks. Probab. Engrg. Inform. Sci. 4 203 242. · Zbl 1134.90329 · doi:10.1017/S0269964800001558
[19] KEY, P. B. 1994. Some control issues in telecommunications networks. In Probability, Z. Statistics and Optimisation F. P. Kelly, ed. 383 395. Wiley, New York. · Zbl 0869.60084
[20] LAWS, C. N. 1995. On trunk reservation in loss networks. In Stochastic Networks F. P.. Kelly and R. J. Williams, eds. 187 198. Springer, New York. · Zbl 0823.60095
[21] LIPPMAN, S. A. 1975. Applying a new device in the optimization of exponential queueing sy stems. Oper. Res. 23 686 710. JSTOR: · Zbl 0312.60048 · doi:10.1287/opre.23.4.687
[22] MEy N, S. P. and TWEEDIE, R. L. 1993. Stability of Markovian processes III: Foster Ly apunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 518 548. JSTOR: · Zbl 0781.60053 · doi:10.2307/1427522
[23] MILLER, B. L. 1969. A queueing reward sy stem with several customer classes. Management Science 16 234 245. · Zbl 0186.24701 · doi:10.1287/mnsc.16.3.234
[24] MITRA, D. and GIBBENS, R. J. 1992. State-dependent routing on sy mmetric loss networks with trunk reservations, II: asy mptotics, optimal design. Ann. Oper. Res. 35 3 30. · Zbl 0768.90024 · doi:10.1007/BF02023088
[25] MITRA, D., GIBBENS, R. J. and HUANG, B. D. 1993. State-dependent routing on sy mmetric loss networks with trunk reservations, I. IEEE Transactions on Communications 41 400 411. · Zbl 0775.94153 · doi:10.1109/26.216515
[26] NGUy EN, V. 1991. On the optimality of trunk reservation in overflow processes. Probab. Engrg. Inform. Sci. 5 369 390. · Zbl 1134.60404 · doi:10.1017/S0269964800002163
[27] NOTEBAERT, N. 1992. Asy mptotical convergence of a trunk reservation policy. Technical report, Statistical Laboratory, Univ. Cambridge and Dept. de Mathematiques Ap\' ṕliquees, Ecole Poly technique. \'
[28] REIMAN, M. I. 1991. Optimal trunk reservation for a critically loaded link. In Proceedings Z. of the 13th International Teletraffic Congress A. Jensen and V. B. Iversen, eds. 247 252. North-Holland, Amsterdam.
[29] ROGERS, L. C. G. and WILLIAMS, D. 1994. Diffusions, Markov Processes, and Martingales, 2nd ed. Wiley, New York. · Zbl 0826.60002
[30] ROSS, S. M. 1970. Applied Probability Models with Optimization Applications. Holden-Day, San Francisco. · Zbl 0213.19101
[31] ROSS, K. W. and TSANG, D. H. K. 1989. Optimal circuit access policies in an ISDN environment: a Markov decision approach. IEEE Transactions on Communications 37 934 939. · Zbl 0675.90066 · doi:10.1109/26.31166
[32] WHITT, W. 1985. Blocking when service is required from several facilities simultaneously. AT & T Technical Journal 64 1807 1856. · Zbl 0591.90035
[33] WHITTLE, P. 1983. Optimization over Time 2. Wiley, New York. · Zbl 0577.90046
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