Analysing Impatience in Multiserver Markovian Queues

Document Type: Research Paper


Department of Statistics, Gauhati University, Guwahati, Assam, India.


Reneging and Balking are practical aspects of customer impatience. In traditional queuing literature, customer impatience was rarely considered. However, for the last few decades, queuing theorists have been trying to integrate aspect of balking and reneging into modeling of queues. This paper is an extension of the work in the same direction. We consider a multi-server Markovian queuing model under the assumption that customers are state aware and the reneging rates increase as the state of the system goes up. We also assume that the customer may balk with balking probabilities proportional to the state of the system. Under this setup performance measures are derived and presented.


Agarwal, S.K. and Singh B.K., (2018). Influence of reneging and jockeying on various queuing characteristics of tri-cum biserial based queue model. International Journal of Mechanical Engineering and Technology, Vol. 9(10), pp. 1062–1073.

Allen, A., (2005). Probability, Statistics and Queuing Theory with Computer Science Application, (2nd ed). San Diego, California: Academic Press.

Ancker, C.J., Jr. and Gafarian, A.V., (1963). Queuing problems with balking and reneging I*. Operations Research, Vol. 11(1), pp. 88–100.

Choudhury, A. and Medhi, P., (2011(A)). A simple analysis of customer impatience in multi-server queues. International Journal of Applied Management Science, Vol. 3(3), pp. 294-315.

Choudhury, A. and Medhi, P., (2011(B)).Some aspects of balking and reneging in finite buffer queues.  RAIRO-Operations Research, Vol. 45(3), pp. 223-240.

Choudhury, A. and Medhi, P., (2012). Aspects of impatience in a finite buffer queue. RAIRO Operation Research, Vol. 46(3), pp. 189-209.

Haight, F.A., (1957). Queuing with balking.  Biometrika, Vol. 44(3-4), pp. 360–369.

Barrer, D.Y., (1957). Queuing with impatient customers and ordered service. Operational Research, Vol. 5(5),pp.650–656.

Haight, F.A., (1959). Queuing with reneging.  Metrika, Vol. 2(1), 186–197.

Haghighi, A.M., Medhi, J. and Mohanty, S.G., 1986. On a multi-server Markovian queuing system with balking and reneging. Computer and Operational Research, Vol. 13(4), pp. 421–425.

Jain, N.K., Kumer, R., and Som, B.K., (2014). An M/M/1/N queuing system with reverse balking. American Journal of Operational Research, Vol. 2(2), pp. 17-20.

Ke J.C. and Wang K.H., (1999). Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns.  Journal of the Operational Research Society, Vol. 50, pp. 275-282

Kuila, M., (2013). Balking and reneging in queuing system.  IOSR Journal of Mathematics, Vol. 1(1), pp. 35-37.

Kumer, R., Som, B.K. and Jain, S., (2015). An M/M/1/N feedback queuing system with reverse balking. Journal of Reliability and Statistical Studies, Vol. 8(1), pp. 31-38.

Rao, S.S., (1965). Queuing Models with Balking, Reneging, and Interruptions. Operations Research, Vol. 13(4), pp. 596-608.

Rao, S.S., (1969). Balking and reneging in M/G/1 systems with post-ponable interruptions. Metrika, Vol. 14, 101–116.

Som, B.K. and Kumer, R., (2017).  A heterogeneous queuing system with reverse balking and reneging. Journal of Industrial and Production Engineering, Vol. 2(2), pp. 1-5.

Sharma, J.K., (2013). Operation research theory and applications. (5th ed). Chennai, India: Macmillan publishers India Ltd.

Swarup, K., Gupta, P.K., and Mohan, M., (2014). An introduction to management science operation research. (16th ed) New Delhi , India: Sultan Chand & Sons educational publishers.

Wang K.H. and Chang Y.C., (2002). Cost analysis of a finite M/M/R queueing system with balking, reneging, and server breakdowns. Mathematical methods in operations research, Vol. 56, pp. 169-180.

Wang Q. and Zhang B., (2018). Analysis of a busy period queuing system with balking, reneging and motivating.  Applied Mathematical Modelling, Vol. 58, pp. 480-488.