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Bivariate distributions of maximum remaining service times in fork-join infinite-server queues. (English. Russian original) Zbl 1447.60135
Probl. Inf. Transm. 56, No. 1, 73-90 (2020); translation from Probl. Peredachi Inf. 56, No. 1, 80-98 (2020).
Summary: We study the maximum remaining service time in \(M^{(2)} \vert G_2 \vert \infty\) fork-join queueing systems where an incoming task forks on arrival for service into two subtasks, each of them being served in one of two infinite-sever subsystems. The following cases for the arrival rate are considered: (1) time-independent, (2) given by a function of time, (3) given by a stochastic process. As examples of service time distributions, we consider exponential, hyperexponential, Pareto, and uniform distributions. In a number of cases we find copula functions and the Blomqvist coefficient. We prove asymptotic independence of maximum remaining service times under high load conditions.
60K25 Queueing theory (aspects of probability theory)
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
Full Text: DOI
[1] Andrews, GR, Foundations of Multithreaded, Parallel, and Distributed Programming (1999), Reading, MA: Addison-Wesley, Reading, MA
[2] Toporkov, VV, Modeli raspredelennykh vychislenii (2004), Moscow: Fizmatlit, Moscow
[3] Alkasem, A.; Liu, H., A Survey of Fault-Tolerance in Cloud Computing: Concepts and Practice, Res. J. Appl. Sci. Eng. Tech., 11, 12, 1365-1377 (2015)
[4] Kumari, P. and Kaur, P., A Survey of Fault Tolerance in Cloud Computing, J. King Saud Univ., Comp. & Info. Sci. (in press, corrected proof, 2018). Available online at 10.1016/j.jksuci.2018.09.021.
[5] Riordan, J., Telephone Traffic Time Averages, Bell Syst. Tech. J., 30, 4, 1129-1144 (1951)
[6] Afanas’eva, LG; Bulinskaya, EV, Sluchainye protsessy v teorii massovogo obsluzhivaniya i upravleniya zapasami (1980), Moscow: Moscow State Univ., Moscow
[7] Bocharov, PP; Pechinkin, AV, Teoriya massovogo obsluzhivaniya (1995), Moscow: Ross. Univ. Druzhby Narodov, Moscow
[8] Lebedev, AV, Asymptotics of Maxima in an Infinite Server Queue with Bounded Batch Sizes, Fundam. Prikl. Mat., 2, 4, 1107-1115 (1996) · Zbl 0902.60076
[9] Lebedev, AV, Extrema of Some Queueing Processes, Cand. Sci. (Math.) Dissertation (1997), Moscow: Moscow State Univ., Moscow
[10] Lebedev, A.V., Maxima in the M^X∣G∣∞ System with “Heavy Tails” of Group Sizes, Avtomat. i Telemekh., 2000, no. 12, pp. 115-121 [Autom. Remote Control (Engl. Transl.), 2000, vol. 61, no. 12, pp. 2039-2044]. · Zbl 1057.90510
[11] Nelson, R.; Tantawi, AN, Approximate Analysis of Fork/Join Synchronization in Parallel Queues, IEEE Trans. Comput., 37, 6, 739-743 (1988)
[12] Thomasian, A., Analysis of Fork-Join and Related Queueing Systems, ACM Comput. Surv., 2014, vol. 47, no. 2. Article no. 17 (71 pp.).
[13] Gorbunova, AV; Zaryadov, IS; Matyushenko, SI; Samouylov, KE; Shorgin, SYa, The Approximation of Response Time of a Cloud Computing System, Inform. Primen., 9, 3, 32-38 (2015)
[14] Zhidkova, LA; Moiseeva, SP, Investigation of the Parallel Service System with Multiple Claims of the Poisson Process, Vestn. Tomsk. Gos. Univ. Upravlen. Vychisl. Tekh. Inf., 4, 17, 49-54 (2011)
[15] Moiseeva, SP; Zakhorol’naya, IA, Mathematical Model of Parallel Retrial Queueing of Multiple Requests, Avtometriya, 47, 6, 51-58 (2011)
[16] Ivanovskaya, IA; Moiseeva, SP, Studying of Mathematical Model of Mixed Type Claim Parallel Service, Izv. Tomsk. Polytech. Univ., 317, 5, 32-34 (2010)
[17] Sinyakova, IA, Mathematical Methods and Models for Analysis of Parallel Service Systems with Double Requests of Random Flows, Cand. Sci. (Math.) Dissertation (2013), Tomsk: Tomsk State Univ., Tomsk
[18] Gorbunova, AV; Zaryadov, IS; Samouylov, KE; Sopin, ES, Survey on Queuing Systems with Parallel Servingof Customers, Vestn. Ross. Univ. Druzhby Narodov Ser. Mat. Inform. Fiz., 25, 4, 350-362 (2017)
[19] Gorbunova, AV; Zaryadov, IS; Samouylov, KE, A Survey on Queuing Systems with Parallel Serving of Customers. Part II, Vestn. Ross. Univ. Druzhby Narodov Ser. Mat. Inform. Fiz., 26, 1, 13-27 (2018)
[20] Glukhova, E.V. and Orlov, A.B., Mean Busy Period of Infinite Multilinear Queues with a Doubly Stochastic Input Flow, Izv. Vuzov, Ser. Fiz., 2003, no. 3, pp. 62-68 [Russian Phys. J. (Engl. Transl.), 2003, vol. 46, no. 3, pp. 287-295]. · Zbl 1063.60509
[21] Orlov, AB, Probability Density of the Maximum Remaining Service Time on Busy Servers, Vychisl. Tekhnol., 13, SpecialIssue5, 93-98 (2008) · Zbl 1212.60173
[22] Lebedev, AV, Maximum Remaining Service Time in Infinite-Server Queues, Probl. Peredachi Inf., 54, 2, 86-102 (2018)
[23] Nelsen, R., An Introduction to Copulas (2006), New York: Springer, New York · Zbl 1152.62030
[24] Chernavskaya, EA, Limit Theorems for an Infinite-Server Queuing System, Mat. Zametki, 98, 4, 590-605 (2015)
[25] Chernavskaya, EA, Limit Theorems for Infinite-Server Queues with Heavy-Tail Service Time Distributions, Cand. Sci. (Math.) Dissertation (2017), Moscow: Moscow State Univ., Moscow
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