Wide sense regenerative processes with applications to multi-channel queues and networks. (English) Zbl 0799.60089
The general theory of regenerative processes is used to describe functioning of general multiserver queueing systems with FIFO discipline and networks of such queues. This allows to obtain some rate conservation laws for separate nodes which generalize such classical results of queueing theory as Pollaczek-Khinchin formula and Little’s formula.
Reviewer: G.Falin (Moskva)
MSC:
| 60K15 | Markov renewal processes, semi-Markov processes |
| 60K25 | Queueing theory (aspects of probability theory) |
| 60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |
| 60K99 | Special processes |
Keywords:
regenerative processes; multiserver queueing systems; Pollaczek-Khinchin formula; Little’s formulaReferences:
| [1] | Asmussen, S.:Applied Probability and Queues, Wiley, Chichester, New York, Brisbane, Toronto, Singapore, 1987. · Zbl 0624.60098 |
| [2] | Borovkov, A. A.: Some limit theorems in the theory of mass service, II,Theory Probab. Appl. 10 (1965), 375-400. · doi:10.1137/1110046 |
| [3] | Borovkov, A. A.:Asymptotic Methods in Queueing Theory, Wiley, Chichester, New York, Brisbane, Toronto, Singapore, 1984. · Zbl 0544.60085 |
| [4] | Borovkov, A. A.: Limit theorems for queueing networks,Theory Probab. Appl. 31 (1986), 474-490. |
| [5] | Feller, W.:An Introduction to Probability Theory and its Applications, Vol. 2, Wiley, New York, 1971. · Zbl 0219.60003 |
| [6] | Foss, S. G.: The ergodicity of queueing networks,Siberian Math. J. 32 (1991), 184-203 (in Russian). · Zbl 0758.60098 |
| [7] | Glynn, P. W. and Whitt, W.: Limit theorems for cumulative processes,Stochastic Proc. Appl. (1993) (to appear). · Zbl 0779.60021 |
| [8] | Foss, S. G. and Kalashnikov, V. V.: Regeneration and renovation in queues,Queueing Systems 8 (1991), 211-224. · Zbl 0728.60090 · doi:10.1007/BF02412251 |
| [9] | Iglehart, D. L. and Whitt, W: Multiple channel queues in heavy traffic, I,Adv. Appl. Probab. 2 (1970), 150-170. · Zbl 0218.60098 · doi:10.2307/3518347 |
| [10] | Iglehart, D. L. and Shedler, G. S.:Regenerative Simulation of Response Times in Networks of Queues, Springer-Verlag, Berlin, Heidelberg, 1980. · Zbl 0424.90016 |
| [11] | Kalahne, U.: Existence, uniqueness and some invariance properties of stationary distributions for general single server queues,Math. Operationsforsch. Statist. 7 (1976), 557-575. · Zbl 0353.60093 |
| [12] | Kalashnikov, V. V. and Rachev, S.:Mathematical Methods for Construction of Queueing Models, Wadsworth and Brooks/Cole, 1990. · Zbl 0709.60096 |
| [13] | Kaspi, H. and Mandelbaum, A.: Regenerative closed queueing networks,Stochastics and Stochastics Rep. 39 (1992), 239-258. · Zbl 0767.60094 |
| [14] | Miyazawa, M.: A formal approach to queueing processes in the steady state and their applications,J. Appl. Probab. 16 (1979), 332-346. · Zbl 0403.60089 · doi:10.2307/3212901 |
| [15] | Morozov, E. V.: Some results for continuous-time processes in queue GI/GI/I with losses, I,Izv. Akad. Nauk Byelorus. 2 (1983), 51-55 (in Russian). · Zbl 0524.60089 |
| [16] | Morozov, E. V.: Queueing system GI/GI/m with losses and non-identical channels,Eng. Cybern. No. 2 (1985), 84-90. |
| [17] | Morozov, E. V.: Renovation of multi-channel queues,Dokl. Akad. Nauk BSSR 2 (1987), 120-123 (in Russian). · Zbl 0627.60093 |
| [18] | Morozov, E. V.: Conservation of regenerative input in the acyclic queueing network, inStability Problems for Stochastic Models, Proceedings, VNIISI, Moscow, 1988, pp. 115-119 (in Russian). |
| [19] | Morozov, E. V.: Harris regeneration in queueing networks, inAbstracts of Communications, Fifth Internat. Vilnius Conf. on Probab. Theory and Math. Statist., 1989, pp. 65-66 (in Russian). |
| [20] | Morozov, E. V.: Regeneration of closed queueing network,Stability Problems for Stochastic Models, Proceedings, VNIISI, Moscow, 1990, pp. 61-69 (in Russian). |
| [21] | Morozov, E. V.: A comparison theorem for queueing system with non-identical channels, inStability Problems for Stochastic Models, Springer-Verlag, New York, 1993, pp. 130-133. · Zbl 0788.60114 |
| [22] | Nummelin, E.: A conservation property for general GI|G|1 queues with application to tandem queues,Adv. Appl. Probab. 11 (1979), 660-672. · Zbl 0411.60093 · doi:10.2307/1426960 |
| [23] | Nummelin, E.: Regeneration in tandem queues,Adv. Appl. Probab. 13 (1981), 221-230. · Zbl 0452.60094 · doi:10.2307/1426476 |
| [24] | Rootzen, H.: Maxima and exceedances of stationary Markov chains,Adv. Appl. Probab. 20 (1988), 371-390. · Zbl 0654.60023 · doi:10.2307/1427395 |
| [25] | Shanbhag, D. N.: Some extentions of Takacs’s limit theorems,J. Appl. Probab. 11 (1974), 752-761. · Zbl 0315.60045 · doi:10.2307/3212558 |
| [26] | Shedler, G. S.:Regeneration and Networks of Queues, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1987. · Zbl 0607.60083 |
| [27] | Sigman, K.: Queues as Harris recurrent Markov chains,Queueing Systems 3 (1988), 179-198. · Zbl 0652.60096 · doi:10.1007/BF01189048 |
| [28] | Sigman, K.: Regeneration in tandem queues with multiserver stations,J. Appl. Probab. 25 (1988), 391-403. · Zbl 0662.60101 · doi:10.2307/3214447 |
| [29] | Sigman, K.: Notes on the stability of closed queueing networks,J. Appl. Probab. 26 (1989), 678-682. · Zbl 0714.60081 · doi:10.2307/3214428 |
| [30] | Sigman, K.: The stability of open queueing networks,Stoch. Proc. Appl. 35 (1990), 11-25. · Zbl 0697.60087 · doi:10.1016/0304-4149(90)90119-D |
| [31] | Sigman, K.: One-dependent regenerative processes and queues in continious time,Math. Oper. Res. 15 (1991), 175-189. · Zbl 0713.60104 · doi:10.1287/moor.15.1.175 |
| [32] | Smith, W. L.: Regenerative stochastic processes,Proc. Roy. Soc., Ser. A 232 (1955), 6-31. · Zbl 0067.36301 · doi:10.1098/rspa.1955.0198 |
| [33] | Smith, W. L.: Renewal theory and its ramifications,J. Roy. Stat. Soc., Ser. B 20 (1958), 243-302. · Zbl 0091.30101 |
| [34] | Stidham, S.: Regenerative processes in the theory of queues, with applications to the alternating-priority queue,Adv. Appl. Probab. 4 (1972), 542-577. · Zbl 0251.60059 · doi:10.2307/1425993 |
| [35] | Wolff, R. L.: An upper bound for multi-channel queues,J. Appl. Probab. 14 (1977), 884-888. · Zbl 0378.60097 · doi:10.2307/3213363 |
| [36] | Wolff, R. L.: Upper bounds on work in system for multi-channel queue,J. Appl. Probab. 24 (1987), 547-551. · Zbl 0625.60112 · doi:10.2307/3214279 |
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