Analysis of a retrial queue with multiple vacations and state dependent arrivals

Jailaxmi, V.; Arumuganathan, R.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 49 (2015) no. 3, pp. 619-634.

This paper examines an M/G/1 retrial queueing system with multiple vacations and different arrival rates. Whenever the system is empty, the server immediately takes a vacation. At a vacation completion epoch, if the number of customers in the orbit is at least one the server remains in the system to activate service, otherwise the server avails multiple vacations until at least one customer is recorded in the orbit. The primary arrival rate is $\lambda {}_1$ when the server in idle and the primary arrival rate is $\lambda {}_2$ when the server is busy or on vacation $(\lambda {}_1>\lambda {}_2)$. The steady state queue size distribution of number of customers in the retrial group, expected number of customers in the retrial group and expected number of customers in the system are obtained. Some special cases are also discussed. Numerical illustrations are also provided.

A. Aissani and J.R. Artalejo, On the single server retrial queue subject to breakdowns. Queueing Syst. 30 (1998) 309–321. | DOI | MR | Zbl

J.R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations. Comput. Oper. Res. 24 (1997) 493–504. | DOI | MR | Zbl

J.R. Artalejo, Accessible bibliography on retrial queues. Math. Comput. Model. 30 (1999) 1–6. | DOI

J.R. Artelejo, A classified bibliography of research on retrial queues: progress in 1990–1999”. Top 7 (1999) 187–211. | DOI | MR | Zbl

J.R. Artalejo, Steady state analysis of an M/G/1 queue with repeated attempts and two phase service. Qual. Technol. Quant. Manage. 1 (2014) 189–199. | DOI | MR

J.R. Artalejo, A. Gomez-Corral and M.F. Neuts, Analysis of multiserver queues with constant retrial rate. Eur. J. Oper. Res. 135 (2001) 569–581. | DOI | MR | Zbl

J.R. Artalejo and Gomez-Corral, Retrial queueing systems, a computational apporach. Springer-Verlang, Berlin (2008). | MR | Zbl

R. Arumuganathan and S. Jeyakumar, A non-Markovian bulk queue with multiple vacations and control policy on request for re-service. Qual. Technol. Quant. Manage. 8 (2011) 253–269. | DOI

I. Atentia, I. Fortes, P. Moreno and S. Sanchez, An M/G/1 retrial queue with active breakdowns and Bernoulli schedule in the server. Inform. Manage. Serv. 17 (2006) 1–17. | MR | Zbl

I. Atentia, G. Bouza and P. Moreno, An M${}^X$/G/1 retrial queue with server breakdowns and constant rate of repeated attempts. Ann. Oper. Res. 157 (2008) 225–243. | DOI | MR | Zbl

Y. Baba, On the M${}^x$/G/1 queue with vacation time. Oper. Res. Lett. 5 (1986) 93–98. | DOI | MR | Zbl

B.D. Choi and K.K. Park, The M/G/1 retrial queue with Bernoulli schedule. Queueing Syst. 7 (1990) 219–227. | DOI | MR | Zbl

B.D. Choi and Y. Chang, Single server retrial queues with priority calls. Math. Comput. Model. 30 (1999) 7–32. | DOI | MR | Zbl

B.D. Choi, K.B. Choi and Y.W. Lee, M/G/1 retrial queueing system with two types of calls and finite capacity. Queueing Syst. 19 (1995) 215–229. | DOI | MR | Zbl

D. Efrosinin and J. Sztrik, Performance analysis of a two server heterogeneous retrial queue with threshold policy. Qual. Technol. Quant. Manage. 8 (2011) 211–236. | DOI

G.I. Falin and J.G.C. Templeton, Retrial queues. Chapman and Hall, London (1997). | Zbl

A. Gomez-Correl, Stochastic analysis of a single server retrial queue with general retrial times. Naval Res. Logist. 46 (1999) 561–581. | DOI | MR | Zbl

G.V. Krishna Reddy, R. Nadarajan and R. Arumuganathan, Analysis of a bulk queue with N-policy multiple vacations and setup times. Comput. Oper. Res. 25 (1998) 957–967. | DOI | MR | Zbl

H.S. Lee, Steady state probabilities for the server vacation model with group arrivals under control operation policy. J. Korean OR/MS Soc. 16 (1991) 36–48.

H.S. Lee and M.M. Srinivasan, Control policies for the M/G/1 queueing system. Manage. Sci. 35 (1989) 708–721. | DOI | MR | Zbl

S.S. Lee, H.W. Lee and K.C. Chae, Batch arrival queue with N-policy and single vacation. Comput. Oper. Res. 22 (1995) 173–189. | DOI | Zbl

H.W. Lee, S.S. Lee, J.O. Park and K.C. Chae, Analysis of M${}^x$/G/1 queue with N-policy and multiple vacations. J. Appl. Probab. 31 (1994) 467–496. | MR | Zbl

H. Li, and T. Yang, A single server retrial queue with server vacation and a finite number of input sources. Eur. J. Oper. Res. 85 (1995) 149–160. | DOI | Zbl

P. Moreno, An M/G/1 Retrial queue with recurrent customers and general retrial times. Appl. Math. Comput. 159 (2004) 651–666. | MR | Zbl

M. Senthil Kumar and R. Arumuganathan, On the single server batch arrival retrial queue with general vacation time under Bernoulli schedule and two phases of heterogeneous service. Qual. Technol. Quant. Manage. 5 (2008) 145–160. | DOI | MR

H. Takagi, Vacation and priority systems, Part I, Queueing analysis: A foundation of performance evaluation. North-Holland, Amsterdam (1991), Vol. I. | MR | Zbl