Analysis Of A Tandem Queue With Bernoulli Routing And Load-Dependent Service Mechanism

Dr. Chakrala. Sreelatha; Bammidi. Anil Kumar

doi:10.29284/h94gs403

Authors

Dr. Chakrala. Sreelatha Assistant Professor, Statistics Department, Rajendra University, Balangir, Odisha, India, (ORCID No- 0000-0002-7512-5516)
Bammidi. Anil Kumar Research Scholar Statistics Department, Rajendra University, Balangir, Odisha, India (ORCID No- 0009-0009-7093-614X)

DOI:

https://doi.org/10.29284/h94gs403

Keywords:

Tandem queue; Bernoulli routing; Load-dependent service; Poisson process; Steady-state analysis; Performance measures.

Abstract

Queueing theory provides an essential framework for analyzing congestion and resource utilization in multi-stage service systems. This study examines a tandem queueing model incorporating Bernoulli routing and load-dependent service mechanisms, which better represent practical scenarios where customer flow is probabilistic and service rates vary with system load.

Customers arrive according to a stochastic process, and after service at the first node, they proceed to the next node with probability Ɵ or leave the system with probability 1−Ɵ. The service rates at each node depend on the number of customers present.

Key performance measures, including expected queue length, waiting time, and system throughput, are derived. A sensitivity analysis is also conducted to evaluate the impact of routing probabilities and load-dependent service parameters on system performance.

The results demonstrate that probabilistic routing and state-dependent service significantly influence system behaviour, making the proposed model more suitable for analyzing real-world multi-stage service systems.

References

1. Sreelatha , D. C. ., & Kumar , B. A. . (2025). A STUDY ON THREE-STAGE TANDEM QUEUEING SYSTEMS WITH POISSON INPUT AND LOAD-DEPENDENT SERVICE MECHANISMS. International Journal of Basic and Applied Sciences, 14(2), 260-280. https://doi.org/10.14419/zcgkw986

2. Aparajitha, J. D., Sreelatha, C., & Srinivasa Rao, K. (2025). A hybrid parallel-sequential ‎service model for tandem communication networks with load-dependent and time-variant ‎behaviour. Journal of Information Systems Engineering and Management, 10(46s)‎. https://doi.org/10.52783/jisem.v10i46s.8788

3. Rao, K. S., & Aparajitha, J. D. (2019). On two node tandem queueing model with time dependent service rates. International Journal of System Assurance Engineering and Management, 10(1), 19–34. https://doi.org/10.1007/s13198-018-0731-z

4. Abry P, Baraniuk R, Flandrin P, Riedi R, Veitch D (2002) Multiscale nature of network ‎traffic. IEEE Signal Process Mag 19(3):28–46 ‎ https://doi.org/10.1109/79.998080.

5. Aparajitha, J. D., & Srinivasa Rao, K. (2023). Parallel and series queueing model with state ‎and time dependent service. OPSEARCH, 60(4), 1626–1658. ‎ https://doi.org/10.1007/s12597-023-00667-8.

6. Badrinath S, Balakrishnan H (2017) Control of a non-stationary tandem queue model of the ‎airport surface. In: American control conference. IEEE, pp 655–661 ‎ https://doi.org/10.23919/ACC.2017.7963027.

7. Cappe O, Moulines E, Pesquet JC, Petropulu A, Yang X (2002) Long range dependence and ‎heavy-trail modeling for tele traffic data. IEEE Signal Process Mag 19:14–27 ‎ https://doi.org/10.1109/79.998079

8. Chan WKV, Schruben LW (2008) Mathematical programming models of closed tandem ‎queueing networks. J ACM Trans Model Comput Simul 19(1) ‎ https://doi.org/10.1145/1456645.1456648

9. Che Soong K, Klimennok V, Taramin O (2010) A tandem retrial queueing system with two ‎Markovian flows and reservation of channels. Comput Oper Res 37(7):1238–1246 ‎ https://doi.org/10.1016/j.cor.2009.03.030

10. Che Soong K, Klimenok VI, Dudin AN (2016) Priority tandem queueing system with retrials ‎and reservation of channels as a model of call center. Comput Ind Eng 96:61–71 ‎ https://doi.org/10.1016/j.cie.2016.03.012

11. Crovella ME, Bestarros A (1997) Self similarly in world wide traffic evidence and possible ‎causes. IEEE/ACM Trans Netw 5(6):835–846 ‎ https://doi.org/10.1109/90.650143.

12. D’Auria B, Kanta S, D’Auria B, Kanta S (2015) Pure threshold strategies for a two-node ‎tandem network under partial infor mation. Oper Res Lett 43(5):467–470 ‎ https://doi.org/10.1016/j.orl.2015.06.014.

13. Dinda PA (2006) Design implementation and performance of an extensible toolkit in resource ‎prediction in distribution system. IEEE Trans Parallel Distrib Syst 17(2):160–173 ‎ https://doi.org/10.1109/TPDS.2006.24.

14. Hayashida Y (1993) Throughput analysis of tandem type ggo-back NARQ scheme for ‎satellite communications. IEEE Trans Commun 41(10):1517–1524 https://doi.org/10.1109/26.237886.‎

15. Jackson RRP (1954) Queueing systems with phase type service. Oper Res Q 5(4):109–120 ‎ https://doi.org/10.1057/jors.1954.23.

16. Jackson RRP (1956) Random queueing process with phase type service. J R Stat Soc Ser B ‎‎18(1):129–132 Leland et al (1994) Onthe self-similar nature of Ethernet traffic (extended ‎version). IEEE/ACM Trans Netw 2(1):1–15 ‎ https://doi.org/10.1109/90.282603.

17. Li Y, Cai X, Tu F, Xiuli S (2004) Optimization of Tandem queue systems with finite buffers. ‎Comput Oper Res 31:963–984 ‎ https://doi.org/10.1016/S0305-0548(03)00046-7.

18. Lieshout P, Mandje M (2006) Tandem Brownian queues. J Probab Netw Algorithms. Report ‎PNA-RO604 4:1-20 Mandelbaum A, Massey WA (1995) Strong approximations for time ‎dependent queues. MOR 20(1):33–64‎ https://doi.org/10.1287/moor.20.1.33.

19. Murali Krishna P, Gadre VM, Desai UB (2003) Multi fractal based network traffic modelling. ‎Kluwer Academic Publisher, Dor drecht. ISBN 1-4020-7566-9 ‎ https://doi.org/10.1007/978-1-4615-0499-3.

20. Newell GF (1968) Queues with time-dependent arrival rates (parts I III). J Appl Probab ‎‎5:436–451(I), 579–590 (II), 591–606 (III) O’Brien GG (1954) The solution of some queueing ‎problems. J Soc Ind Appl Math 2:132–142 ‎ https://doi.org/10.1017/S0021900200114433.

21. Ramasundri MV, Srinivasa Rao K, Srinivasa Rao P, Suresh Varma PS (2011) Three node ‎communication network model with dynamic bandwidth allocation and non-homogeneous ‎Poisson arrivals. Int J Comput Appl 31(1):19–27 ‎ https://doi.org/10.5120/3789-5217.

22. Sitaratnam G, Srinivas Rao K, Srinivas Rao P (2015) Two node communication network ‎model with weibull inter arrival times and dynamic band width allocation. Int J Comput Appl ‎‎128(5):19–29‎ https://doi.org/10.5120/ijca2015906542.