A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms

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2011, 1(4): 781-816. doi: 10.3934/naco.2011.1.781

A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms

Demetres D. Kouvatsos ^1, , Jumma S. Alanazi ^1, and Kevin Smith ^1,

1.	Networks and Performance Engineering Research Group, Informatics Research Institute, University of Bradford, Bradford, BD7 1DP, United Kingdom, United Kingdom, United Kingdom

Received June 2011 Revised September 2011 Published November 2011

Abstract
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A generic maximum entropy (ME) product-form approximation is proposed for arbitrary single class open first-come-first-served (FCFS) queueing network models with blocking (QNMs-B), subject to bursty GE-type interarrival and service times and the mixed blocking mechanisms (BMs) of Blocking-After-Service (BAS), Blocking-Before-Service (BBS) and Repetitive-Service (RS) Blocking with Random (RS-RD) and Fixed (RS-FD) destinations. A new GE-type analytic framework is devised, based on the ME analysis of a virtual multiple class GE/GE/1/N+U queueing system with finite capacity, $N (N>1)$ augmented by $U (U\geq1)$ auxiliary-waiting lines, to determine the first two moments of BAS- and BBS-dependent effective service times towards a node-by-node decomposition of the entire network. In this context, a unified ME algorithm is devised for the approximate analysis of arbitrary open FCFS QNMs-B with a mixture of the BMs of BAS, BBS, RS-RD and RS-FD. Typical numerical tests are carried out to assess the credibility of the unified ME algorithm against discrete event simulation and also establish GE-type experimental performance bounds. A critique on the feasibility of ME formalism for QNMs-B and suggested extensions are included.

Keywords: Blocking with Random (RS-RD) and Fixed (RS-FD) destinations, first-come-first-served (FCFS), maximum entropy (ME) principle, generalized exponential (GE) distribution., Blocking-After-Service (BAS), Queueing network models with blocking (QNMs-B), Repetitive-Service (RS), Blocking-Before-Service (BBS).

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Demetres D. Kouvatsos, Jumma S. Alanazi, Kevin Smith. A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 781-816. doi: 10.3934/naco.2011.1.781

References:

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[2]	J. S. Alanazi and D. D. Kouvatsos, On the experimentation with the unified ME algorithm for arbitrary open QNMs-B,, Technical Report TR7-NetPEn-April 11, (2011). Google Scholar
[3]	J. S. Alanazi and D. D. Kouvatsos, A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms,, in ''Proc. of the IEEE/IPSJ Workshop WS-8: Future Internet Engineering of the SAINT 2011 International Symposium on Applications and the Internet, (2011), 292. doi: 10.1109/SAINT.2011.91. Google Scholar
[4]	T. Altiok and H. G. Perros, Approximate analysis of arbitrary configurations of queueing networks with blocking,, Ann. Oper. Res., 9 (1987), 481. doi: 10.1007/BF02054751. Google Scholar
[5]	S. A. Assi, D. D. Kouvatsos, I. M. Mkwawa and K. Smith, A unified ME decomposition algorithm of open queueing network modes with blocking,, in ''Tech. Proc. of HET-NETs 08 International Working Conference on Performance Modelling and Evaluation of Heterogeneous Networks, (2008). Google Scholar
[6]	S. Balsamo, V. D. Nitto Persone and R. Onvural, "Analysis of Queueing Networks with Blocking,", Kluwer Academic publishers, (2001). Google Scholar
[7]	S. Balsamo, Queueing networks with blocking: snalysis, solution algorithms and properties,, in ''Network Performance Engineering, 5233 (2011), 233. doi: 10.1007/978-3-642-02742-0. Google Scholar
[8]	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, 22 (1975), 248. doi: 10.1145/321879.321887. Google Scholar
[9]	V. E. Benes, "Mathematical Theory of Connecting Networks and Telephone Traffic,", Academic Press, (1965). Google Scholar
[10]	J. Beran, "Statistics for Long-Memory Processes,", Chapman and Hall, (1994). Google Scholar
[11]	R. M. Bryant, A. E. Krzesinski, M. S. Lakshmi and K. M. Chandy, The MVA priority approximation,, T.O.C.S., 2 (1984), 335. Google Scholar
[12]	C. G. Chakrabarti and D. E. Kajal, Boltzmann-Gibbs entropy: axiomatic characterisation and application,, Internat. J. Math. Math. Sci., 23 (2000), 243. doi: 10.1155/S0161171200000375. Google Scholar
[13]	K. M. Chandy, U. Herzog and L. Woo, Approximate analysis of general queuing networks,, IBM J. of Res. Dev., 19 (1975), 43. doi: 10.1147/rd.191.0043. Google Scholar
[14]	Y. L. Chen and C. Chen, Performance analysis of non-preemptive HOL GE/G/1 queue with two priority classes of SIP-T signaling system in carrier grade VoIP network,, J. Chin. Inst. Eng., 33 (2010), 191. doi: 10.1080/02533839.2010.9671610. Google Scholar
[15]	P. J. Courtois, U. Herzog and L. Woo, "Decomposability: Queueing and Computer System Applications,", Academic Press, (1977). Google Scholar
[16]	M. A El-Affendi and D. D Kouvatsos, A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium,, Acta info., 19 (1983), 339. Google Scholar
[17]	A. Ferdinand, A statistical mechanical approach to systems analysis,, IBM J. Res. Dev., 14 (1970), 539. doi: 10.1147/rd.145.0539. Google Scholar
[18]	C. H. Foh, B. Meini, B. Wydrowski and M. Zuerman, Modelling and performance evaluation of GPRS,, in ''Proc. Of IEEE VTC, (2001), 2108. Google Scholar
[19]	E. Gelenbe and G. Pujolle, The behaviour of a single queue in a general queueing network,, Acta info., 7 (1974), 123. Google Scholar
[20]	E. Gelenbe and I. Mitrani, "Analysis and Synthesis of Computer Systems,", Academic Press, (1980). Google Scholar
[21]	J. H. Havrda and F. Charvat, Quantification methods of classificatory processes: concept of structural entropy,, Kybernatica, 3 (1967), 30. Google Scholar
[22]	H. E. Hurst, Long-term storage capacity of reservoirs,, Transactions of the American Society of Civil Engineers, 116 (1951), 770. Google Scholar
[23]	E. T. Jaynes, Information theory and statistical mechanics,, Phys. Rev., 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar
[24]	E. T. Jaynes, Information theory and statistical mechanics II,, Phys. Rev., 108 (1957), 171. doi: 10.1103/PhysRev.108.171. Google Scholar
[25]	R. Johnson, Properties of cross-entropy minimization,, IEEE Trans. Info. Theory, 27 (1981), 472. doi: 10.1109/TIT.1981.1056373. Google Scholar
[26]	J. N. Kapur, "Maximum-entropy Models in Science and Engineering,", John Wiley, (1989). Google Scholar
[27]	J. N. Kapur and H. K. Kesavan, "Entropy Optimization Principles with Applications,", Academic Press, (1992). Google Scholar
[28]	F. P. Kelly, "Reversibility and Stochastic Networks,", Wiley, (1979). Google Scholar
[29]	D. D. Kouvatsos, Maximum entropy methods for general queueing networks,, in ''Modelling Techniques and Tools for Performance Analysis, (1985), 589. Google Scholar
[30]	D. D. Kouvatsos, Maximum entropy and the G/G/1/N queue,, Acta info., 23 (1986), 545. Google Scholar
[31]	D. D. Kouvatsos, A universal maximum entropy algorithm for the analysis of general closed networks,, in ''Computer Networks and Performance Evaluation, (1986), 113. Google Scholar
[32]	D. D. Kouvatsos, A maximum entropy analysis of the G/G/1 queue at equilibrium,, J. Opl. Res. Soc., 39 (1988), 183. Google Scholar
[33]	D. D. Kouvatsos and N. P. Xenios, MEM for arbitrary queueing networks with multiple general servers and repetitive service blocking,, Performance Evaluation, 10 (1989), 169. doi: 10.1016/0166-5316(89)90009-6. Google Scholar
[34]	D. D. Kouvatsos, P. H. Georgatsos and N. Tabet-Aouel, A universal maximum entropy algorithm for general multiple class open networks with mixed service disciplines,, in ''Modelling Techniques and Tools for Computer Performance Evaluation, (1989), 397. doi: 10.1007/978-1-4613-0533-0_26. Google Scholar
[35]	D. D. Kouvatsos and N. M. Tabet-Aouel, A maximum entropy priority approximation for a stable G/G/1 queue,, Acta info., 27 (1989), 247. Google Scholar
[36]	D. D. Kouvatsos and N. M. Tabet-Aouel, Product-form approximations for an extended class of general closed queueing networks,, in ''Performance '90', (1990), 301. Google Scholar
[37]	D. D. Kouvatsos and S. G. Denazis, Entropy maximised queueing networks with blocking and multiple job classes,, Performance Evaluation, 17 (1993), 189. doi: 10.1016/0166-5316(93)90041-R. Google Scholar
[38]	D. D. Kouvatsos, Entropy maximisation and queueing network models,, Annals of Oper. Res., 48 (1994), 63. doi: 10.1007/BF02023095. Google Scholar
[39]	D. D. Kouvatsos and I. U. Awan, MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes,, Annals of Oper. Res., 79 (1998), 231. doi: 10.1023/A:1018922705462. Google Scholar
[40]	D. D. Kouvatsos and I. U. Awan, Entropy maximisation and open queueing networks with priorities and blocking,, Performance Evaluation, 51 (2003), 191. doi: 10.1016/S0166-5316(02)00092-5. Google Scholar
[41]	D. D. Kouvatsos, I. U. Awan, R. J. Fretwell and R. Dimakopoulos, G. A cost-effective approximation for SRD traffic in arbitrary multi-buffered networks,, Computer Networks, 34 (2000), 97. doi: 10.1016/S1389-1286(00)00099-2. Google Scholar
[42]	D. D. Kouvatsos, Y. Li and W. Xi, Performance modelling and analysis of a 4G handoff priority scheme for cellular networks,, in ''Performance Modelling and Analysis of Heterogeneous Networks, (2009), 215. Google Scholar
[43]	D. D. Kouvatsos and S. A. Assi, Generalised entropy maximisation and queues with bursty and/or heavy tails,, in ''Network Performance Engineering, 5233 (2011), 357. doi: 10.1007/978-3-642-02742-0. Google Scholar
[44]	D. D. Kouvatsos and S. A. Assi, On the analysis of queues with heavy tails: a non-extensive maximum entropy formalism and a generalisation of the Zipf-mandelbrot distribution,, in ''Special IFIP LNCS issue in Honour of Guenter Haring, (2011). Google Scholar
[45]	R. A. Marie, An approximate analytical method for general queueing networks,, IEEE Trans. Software Eng., 5 (1979), 530. doi: 10.1109/TSE.1979.234214. Google Scholar
[46]	B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman, (1982). Google Scholar
[47]	R. O. Onvural and I. F. Akyildiz, "Queueing Networks with Finite Capacity,", Elsevier Science publishers, (1993). Google Scholar
[48]	R. O. Onvural, Survey of closed queueing networks with blocking,, ACM Comput. Surv., 22 (1990), 83. doi: 10.1145/78919.78920. Google Scholar
[49]	H. G. Perros and T. Altiok, "Queueing Networks with Blocking,", Elsevier Science publishers, (1989). Google Scholar
[50]	H. G. Perros, Approximate algorithms for open queueing networks with blocking,, in ''Stochastic Analysis of Computer and Communication Systems, (1990), 451. Google Scholar
[51]	H. G. Perros, "Queueing Networks with Blocking,", Oxford University Press, (1994). Google Scholar
[52]	E. Pinsky and Y. Yemini, A statistical mechanics of some interconnection networks,, in ''Performance '48', (1984), 147. Google Scholar
[53]	E. Pinsky and Y. Yemini, The canonical approximation in performance analysis,, in ''Computer Networking and Performance Evaluation, (1986), 125. Google Scholar
[54]	M. Reiser and H. Kobayashi, Accuracy of the diffusion approximation for some queuing systems,, IBM J. Res. Dev., 18 (1974), 110. doi: 10.1147/rd.182.0110. Google Scholar
[55]	K. C. Sevcik, A. I. Levy, S. Tripathi and J. L. Zahorjan, Improving approximations of aggregated queuing network subsystems,, in ''Computer Performance, (1977), 1. Google Scholar
[56]	C. E. Shannon and W. Weaver, A mathematical theory of communication,, Bell Syst. Tech. J., 27 (1948), 379. Google Scholar
[57]	J. Shore and R. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,, IEEE Trans. Info. Theory, 26 (1980), 26. doi: 10.1109/TIT.1980.1056144. Google Scholar
[58]	J. E. Shore, Information theoretic approximations for M/G/1 and G/G/1 queuing systems,, Acta info., 17 (1982), 43. Google Scholar
[59]	K. Smith and D. D. Kouvatsos, "Entropy Maximisation and QNM with Blocking after Service,", Research report RS-08-01, (2001), 08. Google Scholar
[60]	K. Smith and D.D. Kouvatsos, Entropy maximisation and QNM with blocking before service,, in ''Proc. of the 2nd Annual Postgraduate Symposium on Convergence of Telecommunications, (2001), 78. Google Scholar
[61]	Y. Takahashi and H. Miyahara, An approximation method for open restricted queueing networks,, Oper. Res., 28 (1980), 594. doi: 10.1287/opre.28.3.594. Google Scholar
[62]	C. Tsallis, Possible generalisation of boltzmann-gibbs statistics,, Journal of Statistical Physics, 52 (1988), 479. doi: 10.1007/BF01016429. Google Scholar
[63]	M. Tribus, "Rational Descriptions, Decisions and Designs,", Pergamon, (1969). Google Scholar
[64]	B. R. Walstra, "Iterative Analysis of Networks of Queues,", Ph.D thesis, (1984). Google Scholar
[65]	D. Yao and J. A Buzacott, Modelling a class of state dependent routing in flexible manufacturing systems,, Annals of Oper. Res., 3 (1985), 153. doi: 10.1007/BF02024744. Google Scholar
[66]	, "Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions,", Notebook, (2007). Google Scholar

show all references

References:

[1]	I. F Akyildiz and C. C Huang, Exact analysis of multi-job class networks of queues with blocking-after-sevice,, in ''Proc. of the 2nd Inter. WS on Queueing Networks with Finite Capacity, (1992), 258. Google Scholar
[2]	J. S. Alanazi and D. D. Kouvatsos, On the experimentation with the unified ME algorithm for arbitrary open QNMs-B,, Technical Report TR7-NetPEn-April 11, (2011). Google Scholar
[3]	J. S. Alanazi and D. D. Kouvatsos, A unified ME algorithm for arbitrary open QNMs with mixed blocking mechanisms,, in ''Proc. of the IEEE/IPSJ Workshop WS-8: Future Internet Engineering of the SAINT 2011 International Symposium on Applications and the Internet, (2011), 292. doi: 10.1109/SAINT.2011.91. Google Scholar
[4]	T. Altiok and H. G. Perros, Approximate analysis of arbitrary configurations of queueing networks with blocking,, Ann. Oper. Res., 9 (1987), 481. doi: 10.1007/BF02054751. Google Scholar
[5]	S. A. Assi, D. D. Kouvatsos, I. M. Mkwawa and K. Smith, A unified ME decomposition algorithm of open queueing network modes with blocking,, in ''Tech. Proc. of HET-NETs 08 International Working Conference on Performance Modelling and Evaluation of Heterogeneous Networks, (2008). Google Scholar
[6]	S. Balsamo, V. D. Nitto Persone and R. Onvural, "Analysis of Queueing Networks with Blocking,", Kluwer Academic publishers, (2001). Google Scholar
[7]	S. Balsamo, Queueing networks with blocking: snalysis, solution algorithms and properties,, in ''Network Performance Engineering, 5233 (2011), 233. doi: 10.1007/978-3-642-02742-0. Google Scholar
[8]	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, 22 (1975), 248. doi: 10.1145/321879.321887. Google Scholar
[9]	V. E. Benes, "Mathematical Theory of Connecting Networks and Telephone Traffic,", Academic Press, (1965). Google Scholar
[10]	J. Beran, "Statistics for Long-Memory Processes,", Chapman and Hall, (1994). Google Scholar
[11]	R. M. Bryant, A. E. Krzesinski, M. S. Lakshmi and K. M. Chandy, The MVA priority approximation,, T.O.C.S., 2 (1984), 335. Google Scholar
[12]	C. G. Chakrabarti and D. E. Kajal, Boltzmann-Gibbs entropy: axiomatic characterisation and application,, Internat. J. Math. Math. Sci., 23 (2000), 243. doi: 10.1155/S0161171200000375. Google Scholar
[13]	K. M. Chandy, U. Herzog and L. Woo, Approximate analysis of general queuing networks,, IBM J. of Res. Dev., 19 (1975), 43. doi: 10.1147/rd.191.0043. Google Scholar
[14]	Y. L. Chen and C. Chen, Performance analysis of non-preemptive HOL GE/G/1 queue with two priority classes of SIP-T signaling system in carrier grade VoIP network,, J. Chin. Inst. Eng., 33 (2010), 191. doi: 10.1080/02533839.2010.9671610. Google Scholar
[15]	P. J. Courtois, U. Herzog and L. Woo, "Decomposability: Queueing and Computer System Applications,", Academic Press, (1977). Google Scholar
[16]	M. A El-Affendi and D. D Kouvatsos, A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium,, Acta info., 19 (1983), 339. Google Scholar
[17]	A. Ferdinand, A statistical mechanical approach to systems analysis,, IBM J. Res. Dev., 14 (1970), 539. doi: 10.1147/rd.145.0539. Google Scholar
[18]	C. H. Foh, B. Meini, B. Wydrowski and M. Zuerman, Modelling and performance evaluation of GPRS,, in ''Proc. Of IEEE VTC, (2001), 2108. Google Scholar
[19]	E. Gelenbe and G. Pujolle, The behaviour of a single queue in a general queueing network,, Acta info., 7 (1974), 123. Google Scholar
[20]	E. Gelenbe and I. Mitrani, "Analysis and Synthesis of Computer Systems,", Academic Press, (1980). Google Scholar
[21]	J. H. Havrda and F. Charvat, Quantification methods of classificatory processes: concept of structural entropy,, Kybernatica, 3 (1967), 30. Google Scholar
[22]	H. E. Hurst, Long-term storage capacity of reservoirs,, Transactions of the American Society of Civil Engineers, 116 (1951), 770. Google Scholar
[23]	E. T. Jaynes, Information theory and statistical mechanics,, Phys. Rev., 106 (1957), 620. doi: 10.1103/PhysRev.106.620. Google Scholar
[24]	E. T. Jaynes, Information theory and statistical mechanics II,, Phys. Rev., 108 (1957), 171. doi: 10.1103/PhysRev.108.171. Google Scholar
[25]	R. Johnson, Properties of cross-entropy minimization,, IEEE Trans. Info. Theory, 27 (1981), 472. doi: 10.1109/TIT.1981.1056373. Google Scholar
[26]	J. N. Kapur, "Maximum-entropy Models in Science and Engineering,", John Wiley, (1989). Google Scholar
[27]	J. N. Kapur and H. K. Kesavan, "Entropy Optimization Principles with Applications,", Academic Press, (1992). Google Scholar
[28]	F. P. Kelly, "Reversibility and Stochastic Networks,", Wiley, (1979). Google Scholar
[29]	D. D. Kouvatsos, Maximum entropy methods for general queueing networks,, in ''Modelling Techniques and Tools for Performance Analysis, (1985), 589. Google Scholar
[30]	D. D. Kouvatsos, Maximum entropy and the G/G/1/N queue,, Acta info., 23 (1986), 545. Google Scholar
[31]	D. D. Kouvatsos, A universal maximum entropy algorithm for the analysis of general closed networks,, in ''Computer Networks and Performance Evaluation, (1986), 113. Google Scholar
[32]	D. D. Kouvatsos, A maximum entropy analysis of the G/G/1 queue at equilibrium,, J. Opl. Res. Soc., 39 (1988), 183. Google Scholar
[33]	D. D. Kouvatsos and N. P. Xenios, MEM for arbitrary queueing networks with multiple general servers and repetitive service blocking,, Performance Evaluation, 10 (1989), 169. doi: 10.1016/0166-5316(89)90009-6. Google Scholar
[34]	D. D. Kouvatsos, P. H. Georgatsos and N. Tabet-Aouel, A universal maximum entropy algorithm for general multiple class open networks with mixed service disciplines,, in ''Modelling Techniques and Tools for Computer Performance Evaluation, (1989), 397. doi: 10.1007/978-1-4613-0533-0_26. Google Scholar
[35]	D. D. Kouvatsos and N. M. Tabet-Aouel, A maximum entropy priority approximation for a stable G/G/1 queue,, Acta info., 27 (1989), 247. Google Scholar
[36]	D. D. Kouvatsos and N. M. Tabet-Aouel, Product-form approximations for an extended class of general closed queueing networks,, in ''Performance '90', (1990), 301. Google Scholar
[37]	D. D. Kouvatsos and S. G. Denazis, Entropy maximised queueing networks with blocking and multiple job classes,, Performance Evaluation, 17 (1993), 189. doi: 10.1016/0166-5316(93)90041-R. Google Scholar
[38]	D. D. Kouvatsos, Entropy maximisation and queueing network models,, Annals of Oper. Res., 48 (1994), 63. doi: 10.1007/BF02023095. Google Scholar
[39]	D. D. Kouvatsos and I. U. Awan, MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes,, Annals of Oper. Res., 79 (1998), 231. doi: 10.1023/A:1018922705462. Google Scholar
[40]	D. D. Kouvatsos and I. U. Awan, Entropy maximisation and open queueing networks with priorities and blocking,, Performance Evaluation, 51 (2003), 191. doi: 10.1016/S0166-5316(02)00092-5. Google Scholar
[41]	D. D. Kouvatsos, I. U. Awan, R. J. Fretwell and R. Dimakopoulos, G. A cost-effective approximation for SRD traffic in arbitrary multi-buffered networks,, Computer Networks, 34 (2000), 97. doi: 10.1016/S1389-1286(00)00099-2. Google Scholar
[42]	D. D. Kouvatsos, Y. Li and W. Xi, Performance modelling and analysis of a 4G handoff priority scheme for cellular networks,, in ''Performance Modelling and Analysis of Heterogeneous Networks, (2009), 215. Google Scholar
[43]	D. D. Kouvatsos and S. A. Assi, Generalised entropy maximisation and queues with bursty and/or heavy tails,, in ''Network Performance Engineering, 5233 (2011), 357. doi: 10.1007/978-3-642-02742-0. Google Scholar
[44]	D. D. Kouvatsos and S. A. Assi, On the analysis of queues with heavy tails: a non-extensive maximum entropy formalism and a generalisation of the Zipf-mandelbrot distribution,, in ''Special IFIP LNCS issue in Honour of Guenter Haring, (2011). Google Scholar
[45]	R. A. Marie, An approximate analytical method for general queueing networks,, IEEE Trans. Software Eng., 5 (1979), 530. doi: 10.1109/TSE.1979.234214. Google Scholar
[46]	B. B. Mandelbrot, "The Fractal Geometry of Nature,", W. H. Freeman, (1982). Google Scholar
[47]	R. O. Onvural and I. F. Akyildiz, "Queueing Networks with Finite Capacity,", Elsevier Science publishers, (1993). Google Scholar
[48]	R. O. Onvural, Survey of closed queueing networks with blocking,, ACM Comput. Surv., 22 (1990), 83. doi: 10.1145/78919.78920. Google Scholar
[49]	H. G. Perros and T. Altiok, "Queueing Networks with Blocking,", Elsevier Science publishers, (1989). Google Scholar
[50]	H. G. Perros, Approximate algorithms for open queueing networks with blocking,, in ''Stochastic Analysis of Computer and Communication Systems, (1990), 451. Google Scholar
[51]	H. G. Perros, "Queueing Networks with Blocking,", Oxford University Press, (1994). Google Scholar
[52]	E. Pinsky and Y. Yemini, A statistical mechanics of some interconnection networks,, in ''Performance '48', (1984), 147. Google Scholar
[53]	E. Pinsky and Y. Yemini, The canonical approximation in performance analysis,, in ''Computer Networking and Performance Evaluation, (1986), 125. Google Scholar
[54]	M. Reiser and H. Kobayashi, Accuracy of the diffusion approximation for some queuing systems,, IBM J. Res. Dev., 18 (1974), 110. doi: 10.1147/rd.182.0110. Google Scholar
[55]	K. C. Sevcik, A. I. Levy, S. Tripathi and J. L. Zahorjan, Improving approximations of aggregated queuing network subsystems,, in ''Computer Performance, (1977), 1. Google Scholar
[56]	C. E. Shannon and W. Weaver, A mathematical theory of communication,, Bell Syst. Tech. J., 27 (1948), 379. Google Scholar
[57]	J. Shore and R. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,, IEEE Trans. Info. Theory, 26 (1980), 26. doi: 10.1109/TIT.1980.1056144. Google Scholar
[58]	J. E. Shore, Information theoretic approximations for M/G/1 and G/G/1 queuing systems,, Acta info., 17 (1982), 43. Google Scholar
[59]	K. Smith and D. D. Kouvatsos, "Entropy Maximisation and QNM with Blocking after Service,", Research report RS-08-01, (2001), 08. Google Scholar
[60]	K. Smith and D.D. Kouvatsos, Entropy maximisation and QNM with blocking before service,, in ''Proc. of the 2nd Annual Postgraduate Symposium on Convergence of Telecommunications, (2001), 78. Google Scholar
[61]	Y. Takahashi and H. Miyahara, An approximation method for open restricted queueing networks,, Oper. Res., 28 (1980), 594. doi: 10.1287/opre.28.3.594. Google Scholar
[62]	C. Tsallis, Possible generalisation of boltzmann-gibbs statistics,, Journal of Statistical Physics, 52 (1988), 479. doi: 10.1007/BF01016429. Google Scholar
[63]	M. Tribus, "Rational Descriptions, Decisions and Designs,", Pergamon, (1969). Google Scholar
[64]	B. R. Walstra, "Iterative Analysis of Networks of Queues,", Ph.D thesis, (1984). Google Scholar
[65]	D. Yao and J. A Buzacott, Modelling a class of state dependent routing in flexible manufacturing systems,, Annals of Oper. Res., 3 (1985), 153. doi: 10.1007/BF02024744. Google Scholar
[66]	, "Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions,", Notebook, (2007). Google Scholar

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