×

Analysis and optimization of systems with heterogeneous servers and jump priorities. (English. Russian original) Zbl 1440.93159

J. Comput. Syst. Sci. Int. 58, No. 5, 718-735 (2019); translation from Izv. Ross. Akad. Nauk, Teor. Sist. Upr. 2019, No. 5, 56-74 (2019).
Summary: Markov models of systems with heterogeneous servers, various types of requests, and jump priorities are proposed. It is assumed that there are high and low priority requests; the high priority requests are assigned to the server with the high service rate, and the low priority requests are assigned to the server with the low service rate. Models of two types are studied: with separate queues and the shared queue for requests of different types. Jump priorities determine the rules according to which the type of low priority requests changes depending on the state of the queue. Methods for calculating the distribution of the probability of the system states are developed, formulas for the calculation of the characteristics of the system are derived, and the problem of the optimization of these characteristics is solved. The results of numerical experiments are presented.

MSC:

93C83 Control/observation systems involving computers (process control, etc.)
93E03 Stochastic systems in control theory (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Gumbel, “Waiting lines with heterogeneous servers,” Oper. Res. 8, 504-511 (1960). · Zbl 0097.13101 · doi:10.1287/opre.8.4.504
[2] V. S. Singh, “Two-server markovian queues with balking: heterogeneous vs homogeneous servers,” Oper. Res. 18, 145-159 (1970). · Zbl 0186.24703 · doi:10.1287/opre.18.1.145
[3] V. S. Singh, “Markovian queues with three servers,” IIE Trans. 3, 45-48 (1971).
[4] D. Fakinos, “The M/G/K blocking system with heterogeneous servers,” J. Oper. Res. Soc. 31, 919-927 (1980). · Zbl 0439.60095 · doi:10.1057/jors.1980.167
[5] D. Fakinos, “The generalized M/G/K blocking system with heterogeneous servers,” J. Oper. Res. Soc. 33, 801-809 (1982). · Zbl 0492.90033 · doi:10.1057/jors.1982.175
[6] G. Nath and E. Enns, “Optimal service rates in the multiserver loss system with heterogeneous servers,” J. Appl. Probab. 18, 776-781 (1981). · doi:10.2307/3213336
[7] F. Alpaslan and A. Shahbazov, “An analysis and optimization of stochastic service with heterogeneous channels and Poisson arrivals,” Pure Appl. Math. Sci. 43, 15-20 (1996). · Zbl 0881.60084
[8] B. W. Lin and E. A. Elsayed, “A general solution for multichannel queuing systems with ordered entry,” Comput. Oper. Res. 5, 219-225 (1978). · doi:10.1016/0305-0548(78)90031-X
[9] E. A. Elsayed, “Multichannel queuing systems with ordered entry and finite source,” Comput. Oper. Res. 10, 213-222 (1983). · doi:10.1016/0305-0548(83)90014-X
[10] D. D. Yao, “The arrangement of servers in an ordered entry system,” Oper. Res. 35, 759-763 (1987). · Zbl 0635.90036 · doi:10.1287/opre.35.5.759
[11] B. Pourbabai and D. Sonderman, “Server utilization factors in queuing loss systems with ordered entry and heterogeneous servers,” J. Appl. Prob. 23, 236-242 (1986). · Zbl 0589.60078 · doi:10.2307/3214135
[12] B. Pourbabai, “Markovian queuing systems with retrials and heterogeneous servers,” Comput. Math. Appl. 13, 917-923 (1987). · Zbl 0642.60086 · doi:10.1016/0898-1221(87)90064-2
[13] W. M. Nawijn, “On a two-server finite queuing system with ordered entry and deterministic arrivals,” Eur. J. Oper. Res. 18, 388-395 (1984). · Zbl 0552.60091 · doi:10.1016/0377-2217(84)90161-9
[14] W. M. Nawijn, “A note on many-server queuing systems with ordered entry with an application to conveyor theory,” J. Appl. Prob. 20, 144-152 (1983). · Zbl 0528.60095 · doi:10.2307/3213728
[15] D. D. Yao, “Convexity properties of the overflow in an ordered entry system with heterogeneous servers,” Oper. Res. Lett. 5, 145-147 (1986). · Zbl 0614.90038 · doi:10.1016/0167-6377(86)90087-8
[16] H. O. Isguder and U. U. Kocer, “Analysis of GI/M/N/N queuing system with ordered entry and no waiting line,” Appl. Math. Model. 38, 1024-1032 (2014). · Zbl 1427.90089 · doi:10.1016/j.apm.2013.07.029
[17] B. K. Kumar, S. P. Madheswari, and K. S. Venkatakrishnan, “Transient solution of an M/M/2 queue with heterogeneous servers subject to catastrophes,” Int. J. Inform. Manage. Sci. 18, 63-80 (2007). · Zbl 1148.60073
[18] S. Dharmaraja and R. Kumar, “Transient solution of a markovian queuing models with heterogeneous servers and catastrophes,” OPSEARCH 52, 810-8217 (2015). · Zbl 1365.90099 · doi:10.1007/s12597-015-0209-6
[19] S. I. Ammar, “Transient behavior of a two-processor heterogeneous systems with catastrophes, server failures and repairs,” Appl. Math. Model. 38, 2224-2234 (2014). · Zbl 1427.60185 · doi:10.1016/j.apm.2013.10.033
[20] B. K. Kumar and D. Arivudainambi, “Transient solution of an M/M/C queue with heterogeneous servers and balking,” Inform. Manage. Sci. 12, 15-27 (2001). · Zbl 1009.90024
[21] D. Selvamuthu, “Transient solution of a two-processor heterogeneous systems,” Math. Comput. Model. 32, 1117-1123 (2000). · Zbl 0970.60101 · doi:10.1016/S0895-7177(00)00194-1
[22] A. Krishnamoorthy and C. Sreenivasan, “An M/M/2 queuing systems with heterogeneous servers including one with working vacation,” Int. J. Stoch. Anal. 2012, 145867 (2012). · Zbl 1252.60091
[23] A. Sridhar and R. A. Pitchai, “Analysis of a markovian queue with two heterogeneous servers and working vacation,” Int. J. Appl. Oper. Res. 5 (4), 1-15 (2015).
[24] J. Xu, L. Liu, and T. Zhu, “Transient analysis of two-heterogeneous server queue with impatient behavior and multiple vacations,” J. Syst. Sci. Inform. 6, 69-84 (2018).
[25] D. Yue, J. Yu, and W. Yue, “A markovian queue with two-heterogeneous servers and multiple vacations,” J. Ind. Manage. Optim. 5, 453-465 (2009). · Zbl 1187.60079 · doi:10.3934/jimo.2009.5.453
[26] B. Bakmaz, Z. Bojkovic, and M. Bakmaz, “Queuing loss models with more alternative heterogeneous groups,” Int. J. Commun. Syst. 31, 1724-1735 (2018). · doi:10.1002/dac.3522
[27] Y. C. Chow and W. H. Kohler, “Models for dynamic load balancing in a heterogeneous multiple processor system,” IEEE Trans. Comput. 28, 354-361 (1979). · Zbl 0407.68018 · doi:10.1109/TC.1979.1675365
[28] M. Armony, “Dynamic routing in large-scale service systems with heterogeneous servers,” Queuing Syst. 51, 287-329 (2005). · Zbl 1094.60058 · doi:10.1007/s11134-005-3760-7
[29] M. Armony and A. R. Ward, “Fair dynamic routing in large-scale heterogeneous servers systems,” Oper. Res. 58, 624-637 (2010). · Zbl 1231.90133 · doi:10.1287/opre.1090.0777
[30] M. F. Neuts and Y. Takahashi, “Asymptotic behavior of the stationary distributions in the GI/PH/C queues with heterogeneous servers,” Appl. Math. Inst. Tech. Rep. No. 57B (Univ. of Delaware, Neware, 1980). · Zbl 0451.60085
[31] B. Legros and O. Jouini, “Routing in a queuing system with two heterogeneous servers in speed and in quality of resolution,” Stoch. Models 33, 392-410 (2017). · Zbl 1380.90092 · doi:10.1080/15326349.2017.1303615
[32] H. O. Isguder and U. U. Kocer, “Analysis of K-capacity queuing system with two heterogeneous server,” Lect. Notes Comput. Sci. 10684, 23-30 (2017). · Zbl 1383.60082 · doi:10.1007/978-3-319-71504-9_3
[33] V. Saglam and A. Shahbazov, “Minimizing of loss probabilty in queuing systems with heterogeneous servers,” Iran. J. Sci. Technol. Trans., Ser. A 31, 199-206 (2007). · Zbl 1244.90069
[34] A. A. Bouchentouf and A. Messabihi, “A heterogeneous two-server queuing system with reneging and no waiting line,” ProbStat Forum 11, 67-76 (2018). · Zbl 1395.60106
[35] D. Efrosinin, Controlled Queuing Systems with Heterogeneous Servers (VDM Verlag, Saarbrucken, 2008).
[36] Yu. E. Malashenko and I. A. Nazarova, “Normative dynamic analysis of extreme operational modes of a heterogeneous computing system,” J. Comput. Syst. Sci. Int. 54, 738 (2015). · Zbl 1384.68013 · doi:10.1134/S1064230715050081
[37] Yu. E. Malashenko and I. A. Nazarova, “Control model of the phased upgrade of a heterogeneous computing system,” J. Comput. Syst. Sci. Int. 55, 924 (2016). · Zbl 1384.93081 · doi:10.1134/S1064230716050117
[38] T. Maertens, J. Walraevens, and H. Bruneel, “On priority queues with priority jumps,” Perform. Eval. 63, 1235-1252 (2006). · Zbl 1152.90392 · doi:10.1016/j.peva.2005.12.003
[39] T. Maertens, J. Walraevens, and H. Bruneel, “A modified HOL priority scheduling discipline: performance analysis,” Eur. J. Oper. Res. 180, 1168-1185 (2007). · Zbl 1121.90037 · doi:10.1016/j.ejor.2006.05.004
[40] T. Maertens, J. Walraevens, and H. Bruneel, “Performance comparison of several priority schemes with priority jumps,” Ann. Oper. Res. 162, 109-125 (2008). · Zbl 1152.90392 · doi:10.1007/s10479-008-0314-5
[41] A. Z. Melikov, L. A. Ponomarenko, and Ch. S. Kim, “Approximate method for analysis of queuing models with jump priorities,” Autom. Remote Control 74, 62 (2013). · Zbl 1276.90020 · doi:10.1134/S0005117913010062
[42] A. Z. Melikov, A. M. Rustamov, J. Sztrik, and T. I. Jafarzade, “Methods to analysis of queueing models with state-dependent jump priorities,” Ann. Math. Inform. 46, 143-163 (2016). · Zbl 1374.60184
[43] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (John Hopkins Univ. Press, Baltimore, 1981). · Zbl 0469.60002
[44] I. Mitrani and R. Chakka, “Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method,” Perform. Eval. 23, 241-260 (1995). · Zbl 0875.68103 · doi:10.1016/0166-5316(94)00025-F
[45] R. Chakka, “Spectral expansion solution for some finite capacity queues,” Ann. Oper. Res. 79, 27-44 (1998). · Zbl 0896.90094 · doi:10.1023/A:1018974722301
[46] A. Z. Melikov and M. O. Shakhmalyev, “Markov models of inventory management systems with a positive service time,” J. Comput. Syst. Sci. Int. 57, 766 (2018). · Zbl 1411.90034 · doi:10.1134/S106423071805009X
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.