Abstract
In the area of optimal design and control of queues, the N-policy has received great attention. A single server queueing system with system disaster is considered where the server waits till N customers accumulate in the queue and upon the arrival of Nth customer the server begins to serve the customers until the system becomes idle or the occurrence of disaster whichever happens earlier. The system size probabilities in transient state are obtained in closed form using generating functions and steady-state system size probabilities are derived in closed form using generating functions and continued fractions. Further, the mean and variance for the number of customers in the system are derived for both transient and steady states and these results are deduced for the specific models. Time-dependent busy period distribution is also obtained. Numerical illustrations are also shown to visualize the system effect.
Similar content being viewed by others
References
Abate J, Whitt W (1988) Simple spectral representations for the \(M/M/1\) queue. Queueing Syst 3(4):321–346
Al-Hassan QM (2005) On inverses of tridiagonal matrices. J Discrete Math Sci Cryptogr 8(1):49–58
Arumuganathan R, Jeyakumar S (2005) Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl Math Model 29(10):972–986
Baumann H, Sandmann W (2012) Steady state analysis of level dependent quasi-birth-and-death processes with catastrophes. Comput Oper Res 39(2):413–423
Bhagat A, Jain M (2013) Unreliable \(M^X/G/1\) retrial queue with multi-optional services and impatient customers. Int J Oper Res 17(2):248–273
Boucherie RJ, Boxma OJ (1996) The workload in the \(M/G/1\) queue with work removal. Probab Eng Inform Sci 10:261–277
Chen A, Renshaw E (1997) The \(M/M/1\) queue with mass exodus and mass arrivals when empty. J Appl Probab 34(1):192–207
Cohen JW (1982) The single server queue. North-Holland, Amsterdam
Gelenbe E (1991) Product-form queueing networks with negative and positive customer. J Appl Probab 28(3):656–663
Heyman DP (1968) Optimal operating policies for \(M/G/1\) queueing system. Oper Res 16:362–382
Kella ONS, Chaudhary ML (1984) The threshold policy in the \(M/G/1\) queue with server vacations. Nav Res Log 36:111–123
Lee DH, Yang WS (2013) The N-policy of a discrete time Geo/G/1 queue with disasters and its application to wireless sensor networks. Appl Math Model 37(23):9722–9731
Lim DE, Lee DH, Yang WS, Chae KC (2013) Analysis of the GI/Geo/1 queue with N-policy. Appl Math Model 37(7):4643–4652
Nelson R (1995) Probability, stochastic processes and queueing theory. The mathematics of computer performance modeling, 3rd edn. Springer, New York
Parthasarathy PR, Sudhesh R (2008) Transient solution of a multiserver Poisson queue with N-policy. Comput Math Appl 55(3):550–562
Singh CJ, Jain M, Kumar B (2014) Analysis of \(M^x/G/1\) queueing model with balking and vacation. IJOR 19(2):154–173
Sudhesh R, Vijayashree KV (2013) Stationary and transient analysis of \(M/M/1\) G-queues. Int J Math Oper Res 5(2):282–299
Suhasini AVS, Srinivasa Rao K, Rajasekhara Reddy P (2014) Queueing model with non-homogeneous bulk arrivals having state-dependent service rates. IJOR 21(1):84–99
Vasiliadis G (2014) Transient analysis of the \(M/M/k/N/N\) queue using a continuous time homogeneous Markov system with finite state size capacity. Commun Stat Theory Methods 43(7):1548–1562
Yadin M, Naror P (1963) Queueing systems with a removable service station. Oper Res Q 14:393–405
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sudhesh, R., Sebasthi Priya, R. & Lenin, R.B. Analysis of N-policy queues with disastrous breakdown. TOP 24, 612–634 (2016). https://doi.org/10.1007/s11750-016-0411-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-016-0411-6
Keywords
- System size probabilities
- Transient and steady state
- Generating functions
- Modified Bessel functions
- Chebyshev polynomials
- Disaster
- Mean and variance
- Busy period