Transient Solution of a Two-Heterogeneous Servers’ Queuing System with Retention of Reneging Customers

  • Rakesh KumarEmail author
  • Sapana Sharma


In this paper, the transient solution of a Markovian queuing system with two heterogeneous servers and retention of reneging customers is obtained. The explicit transient probabilities of system size are obtained using probability generating function technique. The corresponding steady-state probabilities are also derived. Further, the time-dependent mean and variance are obtained. Finally, a numerical example is provided to study the behavior of the system. The numerical results show that the heterogeneous system performs better than its homogeneous counterpart.


Retention of reneging customers Heterogeneous servers Probability generating function Transient solution Steady-state probabilities 

Mathematics Subject Classification




The authors would like to thank the anonymous referees for their constructive comments which helped to bring this paper in the current form. One of the authors (Dr. Rakesh Kumar) would like to thank the UGC, New Delhi, India, for financial support given to him for this research work under the Major Research Project vide Letter No. F.-43-434/2014(SR).


  1. 1.
    Ammar, S.I.: Transient analysis of a two heterogeneous servers queue with impatient behavior. J. Egypt. Math. Soc. 22, 90–95 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ancker Jr., C.J., Gafarian, A.V.: Some queueing problems with balking and reneging I. Oper. Res. 11, 88–100 (1963)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ancker Jr., C.J., Gafarian, A.V.: Some queueing problems with balking and reneging II. Oper. Res. 11, 928–937 (1963)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dharmaraja, S.: Transient solution of a two-processor heterogeneous system. Math. Comput. Model. 32, 1117–1123 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dharmaraja, S., Kumar, R.: Transient solution of a Markovian queuing model with heterogeneous servers and catastrophes. Opsearch 52(4), 810–826 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    El-Paoumy, M.S.: On Poisson bulk arrival queue: \(M^X/M/2/N\) queue balking, reneging and heterogeneous servers. Appl. Math. Sci. 2(24), 1169–1175 (2008)MathSciNetzbMATHGoogle Scholar
  7. 7.
    El-Paoumy, M.S., Nabwey, H.A.: The Poissonian queue with balking function, reneging and two heterogeneous servers. Int. J. Basic Appl. Sci. 11(6), 149–152 (2011)Google Scholar
  8. 8.
    El-Sherbiny, A.A.: The truncated heterogeneous two-server queue: M/M/2/N with reneging and general balk function. Int. J. Math. Arch. 3, 2745–2754 (2012)Google Scholar
  9. 9.
    Haight, F.A.: Queuing with balking I. Biometrika 44, 360–369 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Haight, F.A.: Queueing with reneging. Metrika 2, 186–197 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kasim, E., Gupur, G.: Asymptotic behavior of the time-dependent solution of the M/G/1 queueing model with second optional service. Bull. Malays. Math. Sci. Soc. 39(1), 29–64 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kumar, R., Sharma, S.K.: An M/M/1/N queuing system with retention of reneged customers. Pak. J. Stat. Oper. Res. 8, 859–866 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kumar, R., Sharma, S.K.: An M/M/1/N queuing model with retention of reneged customers and balking. Am. J. Oper. Res. 2, 1–5 (2012)Google Scholar
  14. 14.
    Kumar, R., Sharma, S.K.: An Markovian multi-server queuing model with retention of reneged customers and balking. Int. J. Oper. Res. 20, 427–438 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kumar, B.K., Madheswari, S.P.: An M/M/2 queueing system with heterogeneous servers and multiple vacations. Math. Comput. Model. 41, 1415–1429 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Krishanamoorthy, B.: On Poisson queues with heterogeneous servers. Oper. Res. 11, 321–330 (1963)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Madheswari, S.P., Suganthi, P., Josephine, S.A.: Retrial queueing system with retention of reneging customers. Int. J. Pure Appl. Math. 106, 11–20 (2016)Google Scholar
  18. 18.
    Morse, P.M.: Queues, Inventories and Maintenance. Willey, New York (1958)CrossRefGoogle Scholar
  19. 19.
    Saaty, T.L.: Elements of Queuing Theory with Applications. McGraw Hill, New York (1961)zbMATHGoogle Scholar
  20. 20.
    Sharma, O.P., Dass, J.: Initial busy period analysis for a multichannel Markovian queue. Optimization 20, 317–323 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sharma, O.P., Dass, J.: Limited space double channel Markovian queue with heterogeneous servers. Trabajos De Investigacion Dperativa 5, 73–78 (1990)CrossRefzbMATHGoogle Scholar
  22. 22.
    Singh, V.P.: Two-server Markovian queues with balking: heterogeneous vs. homogeneous servers. Oper. Res. 18, 145–159 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yue, D., Yue, W., Yu, J., Tian, R.: A Heterogeneous two-server queueing system with balking and server breakdowns. In: The Eighth International Symposium on Operations Research and Its Applications (ISORA’09) Zhangjiajie, China, 20–22 September 2009, pp. 230–244 (2009)Google Scholar
  24. 24.
    Wang, Z., Lei, X.: Study on customer retention under dynamic markets. In: Second International Conference on Networks Security, Wireless Communications and Trusted Computing (NSWCTC), 24–25 April 2010, Wuhan, Hubei, pp. 514–517 (2010)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsShri Mata Vaishno Devi UniversityKatraIndia

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