Priority queue with batch arrival, balking, threshold recovery, unreliable server and optimal service

Jain, Madhu

RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 2, pp. 417-432.

The threshold policy for the restoration of an unreliable server in a service system with bulk input and balking is investigated. The arriving customers in the queueing system are classified into two categories i.e. priority and ordinary customers. The priority customers are assumed to join the system in groups according to Poisson process. The ordinary customers join the system singly and require the essential service as well as optional service on demand and only a limited number of customers can wait in the queue when the server is busy. The service times of both types of customers and life time as well as repair time of the server are governed by the exponential distribution. When the server fails during the service of the ordinary customer, the repair is done following a threshold recovery rule according to which the repair of the failed server is started only when at least q ordinary customers are accumulated in the system. In case of failure while rendering the service to the priority customers, the server is immediately sent for the repair. The matrix geometric method (MGM) has been used to establish the queue size distribution and other performance indices. To validate the suggested MGM approach, numerical simulation is carried out by taking an illustration.

B. Avi-Itzhak and P. Naor, On a problem of preemptive priority queueing. Oper. Res. 9 (1961) 664–672. | DOI | MR | Zbl

G. Bitran and R. Caldentey, Two class priority system with state dependent arrivals. Queueing Syst., Theory Application 40(4) (2000) 355–382. | DOI | MR | Zbl

W. Chang, Preemptive priority queues. Oper. Res. 13 (1965) 820–827. | DOI | MR | Zbl

M.L. Chaudhary and G.C. Templeton, A First Course in Bulk Queues. John Wiley and Sons, New York (1983). | MR | Zbl

A. Derbala,Priority queueing in an operating system. Comput. Oper. Res. 32 (2005) 229-238. | DOI | MR | Zbl

I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations. Appl. Math. Modell. 37 (2013) 1295–1309. | DOI | MR | Zbl

I. Dimitriou, A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers. TOP 21 (2013) 542–571. | DOI | MR | Zbl

S. Drekic and D.G. Woolford, A preemptive priority queue with balking. Eur. J. Oper. Res. 164 (2005) 387–401. | DOI | MR | Zbl

D.V. Efrosinin and O.V. Semenova, An M/M/1 system with an unreliable device and a threshold recovery policy. J. Commun. Technology Electron. 55 (2010) 1526. | DOI

D. Efrosinin and A. Winkler, Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery. Eur. J. Oper. Res. 210 (2011) 594–605. | DOI | MR | Zbl

R. Groenevelt, G. Koole and P. Nain, On the bias vector of a two class preemptive priority queue. Math. Methods Oper. Res. 55 (2002) 107–120. | DOI | MR | Zbl

S. Halfin and M. Segal, A priority queueing model for a mixture of two types of customers. SIAM J. Appl. Math. 23 (1972) 369–379. | DOI | MR | Zbl

A.G. Hawkes, Time dependent solution of a priority queue with bulk arrivals. Oper. Res. 13 (1965) 586–596. | DOI | MR | Zbl

M. Jain, Transient analysis of machining systems with service interruption, mixed standbys and priority. Int. J. Math. Oper. Res. 5 (2013) 604–625. | DOI | MR

M. Jain and P.K. Agrawal, Optimal policy for bulk queue with multiple types of server breakdown. Int. J. Oper. Res. 4 (2009) 35–54. | DOI | MR | Zbl

M. Jain and A. Bhagat, Finite population retrial queueing model with threshold recovery, geometric arrivals and impatient customers. J. Inf. Oper. Manage. 3 (2012) 162–165.

M. Jain and A. Bhagat, Transient analysis of retrial queues with double orbits and priority customers.Proc. 8th International Conference on Queuing Theory and Network Applications,Taichung, Taiwan (2013) 235–240.

M. Jain and A. Bhagat, Double orbit finite retrial queues with priority customers and service interruption. Appl. Math. Comput. 253 (2015) 324–344. | MR | Zbl

M. Jain and C. Bhargava, Bulk arrival retrial queue with unreliable server and priority subscribers. Int. J. Oper. Res. 5 (2008) 242–259. | MR | Zbl

M. Jain and A. Jain, Working vacations queueing models with multiple types of server breakdowns. Appl. Math. Model. 34 (2010) 1–13. | DOI | MR | Zbl

F. Kamoun, Performance analysis of a non-preemptive priority queueing system subjected to a correlated Markovian interruption process. Comput. Opear. Res. 35 (2008) 3969–3988. | DOI | Zbl

G.V. Krishna Reddy, R. Nadarajan and P.R. Kandasamy, A non-preemptive priority mutiserver queueing system with general bulk service and heterogeneous arrivals. Comput. Oper. Res. 20 (1993) 447–453. | DOI | Zbl

Z. Liu, J. Wu and G. Yang, An M/G/1 retrial G-queue with preemptive resume and feedback under N-policy subject to the server breakdowns and repairs. Comput. Math. Appl. 58 (2009) 1792–1807. | DOI | MR | Zbl

S.A. Metwally and B.M. Zaki, Head-of-the-line priority discipline for the system with N-policy and single vacation. J. the Egyptian Math. Soci. 13 (2005) 159–172. | MR | Zbl

R.D. Miller, Computation of steady state probabilities for M/M/1 priority queues. Oper. Res. 29 (1981) 945–958. | DOI | MR | Zbl

G.S. Mokaddis, S.A. Metwally and B.M. Zaki, On a batch arrival queue with priority and single vacation. Int. J. Inform. Manage. Sci. 20 (2009) 519–534. | MR | Zbl

M.F. Neuts, Matrix Geometric Solutions in Stochastic Models-An Algorithmic Approach, Dover Publications, New York (1981). | MR | Zbl

Y. Padma, A.R. Reddy and L.V. Rao, Matrix geometric approach for M/M/C/N queue with two phase service. Int. J. Eng. Sci. Adv. Technol. 2 (2012) 166–175.

F. Papier and U.W. Thonemann, Capacity rationing in rental systems with two customer classes and batch arrivals. Omega 39(1) (2011) 73-85. | DOI

G. N.Purohit, M. Jain and S. Rani, M/M/1 retrial queue with constant retrial policy, unreliable server, threshold based recovery and state dependent arrival rates. Appl. Math. Sci. 6 (2012) 1837–1846. | MR | Zbl

N. Thillaigovindan and R. Kalyanaraman, A feedback retrial queueing system with two types of arrivals. Proc. 6th Int, Conf. Queueing Theory and Network Applications (2011) 177–181.

A.S. Vadivu, R. Vanayak, S. Dharmaraja and R. Arumuganathan, Performance analysis of voice over internet protocol via non-Markovian loss system with preemptive priority and server breakdown. OPSERACH 5 (2014) 50–75. | DOI | MR | Zbl

R. Vanayak, S. Dharmaraja and R. Arunachalam, On the study of simultaneous service by random number of servers with retrial and preemptive priority. Int. J. Oper. Res. 20 (2014) 68–90. | DOI | MR | Zbl

J. Walraevens, D. Fiems and H. Bruneel, Performance analysis of priority queueing systems in discrete time. Network Perform. Eng. 5233 (2011) 203–232. | DOI

H. White and L. Christie, Queueing with preemptive priorities or with breakdown. Oper. Res. 356 (1958) 79–96. | DOI | MR | Zbl

X. Xu, and L. Wang, Transient analysis of retrial queues with double orbits and priority customers. Proc. 8th International Conference on Queuing Theory and Network Applications, Taichung, Taiwan (2013) 311–315.

D.H. Yang, C.H. Yen and Y.C. Chiang, Numerical analysis for time-dependent machine repair model with threshold recovery policy and server vacations, Proceedings of the International MultiConference of Engineers and Computer Scientists, Hong Kong (2013).

J.A. Zhao, B. Li, X.R. Cao and I. Ahmad, A matrix analytic solution for the D-BMAP/PH/1 priority queue. Queueing Syst., Theory Application 53 (2006) 127–145. | DOI | MR | Zbl