Abstract
We consider a multistage single server queueing model with one infinite queue and two finite buffers. The primary customer on arrival joins an infinite queue. The head of the queue finding an idle server, enter into Stage I of service. A customer who found fit, pass on to Stage II with probability p, \(0\le p\le 1\) and those who found unfit pass on to Buffer I with complimentary probability \(1-p\). The customer from Buffer I after an exponential duration of time move to Buffer II. A customer in Buffer II who seems unfit again move to Buffer I. Buffer I and Buffer II individually and collectively should not exceed capacity M. On every service completion epoch, the head of Buffer II is selected for service. We also assume customer reneges from both infinite queue and from Buffer II within an exponential duration of time intervals. Customers arrive according to a Markovian arrival process (MAP). The service time follows phase type distribution. Stability condition of the system is established. Steady-state system size distribution is obtained. Some performance measures of the system are evaluated.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Baumann, H., Sandmann, W.: Multi-server tandem queue with Markovian arrival process, phase-type service times, and finite buffers. Eur. J. Oper. Res. 256(1), 187–195 (2017)
Bright, L., Taylor, P.G.: Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Commun. Statist.-Stochastic Models 11(3), 497–525 (1995)
Dudin, A.N., Jacob, V., Krishnamoorthy, A.: A multi-server queueing system with service interruption, partial protection and repetition of service. Ann. Oper. Res. 233(1), 101–121 (2005)
Drekic, S., Stanford, D.A., Woolford, D.G., McAlister, M.D.V.C.: A model for deceased-donor transplant queue waiting times. Queueing Syst. 79(1), 87–115 (2015)
Green, L.: Queueing analysis in healthcare. In: Hall, R.W. (ed.) Patient Flow Reducing Delay in Healthcare Delivery. ISOR, vol. 91, pp. 281–307. Springer, Boston (2006). https://doi.org/10.1007/978-0-387-33636-7_10
Kharoufeh, J.P.: Level-dependent quasi-birth-and-death processes. Wiley Encycl. Oper. Res. Manag. Sci. (2011). https://doi.org/10.1002/9780470400531.eorms0460
Klimenok, V., Dudin, A.: Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Syst. 54(4), 245–259 (2006)
Klimenok, V., Dudin, A., Vishnevsky, V.: On the stationary distribution of tandem queue consisting of a finite number of stations. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2012. CCIS, vol. 291, pp. 383–392. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31217-5_40
Klimenok, V., Dudin, A., Vishnevsky, V.: Tandem queueing system with correlated input and cross-traffic. In: Kwiecień, A., Gaj, P., Stera, P. (eds.) CN 2013. CCIS, vol. 370, pp. 416–425. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38865-1_42
Krishnamoorthy, A., Deepak, T.G., Joshua, V.C.: Queues with postponed work. Top 12, 375–398 (2004)
Krishnamoorthy, A., Joshua, V.C.: Excursion between classical and retrial queue. Inf. Syst. Technol. 315–323 (2002)
Latouche, G., Ramaswami, V.: Introduction to Matrix analytic Methods in Stochastic Modelling. ASA/SIAM Series on Statistics and Applied Probability (1999)
Neuts, M.F.: Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin, Department of Mathematics, University of Louvain, pp. 173–206 (1975)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach. The Johns Hopkins University Press, Baltimore and London (1981)
Ramaswami, V., Taylor, P.G.: Some properties of the rate perators in level dependent quasi-birth-and-death processes with countable number of phases. Commun. Statist. Stochast. Models 12(1), 143–164 (1996)
Stanford, D.A., Lee, J.M., Chandok, N., McAlister, V.: A queuing model to address waiting time inconsistency in solid-organ transplantation. Oper. Res. Health Care 3(1), 40–45 (2014)
Zenios, A.S.: Modeling the transplant waiting list: a queueing model with reneging. Queueing Syst. 31(3–4), 239–251 (1999)
Acknowledgement
The work of the first author is supported by Maulana Azad National Fellowship \([F1-17.1/2015-16/MANF-2015-17-KER-65493]\) of University Grants Commission, India.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Babu, D., Joshua, V.C., Krishnamoorthy, A. (2019). A Multistage Queueing Model with Priority for Customers Become Fit. In: Vishnevskiy, V., Samouylov, K., Kozyrev, D. (eds) Distributed Computer and Communication Networks. DCCN 2019. Lecture Notes in Computer Science(), vol 11965. Springer, Cham. https://doi.org/10.1007/978-3-030-36614-8_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-36614-8_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36613-1
Online ISBN: 978-3-030-36614-8
eBook Packages: Computer ScienceComputer Science (R0)