Analysis of $D$-policy discrete-time Geo/G/1 queue with second J-optional service and unreliable server

Lan, Shaojun; Tang, Yinghui

RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 1, pp. 101-122.

This paper is concerned with a discrete-time $Geo/G/1$ queueing system with $D$-policy and $J$-optional services in which the service station may be subject to failures at random during serving the customers. All the arriving customers require the first essential service, whereas some of them may opt for a second service from the $J$ additional services with some probability. As soon as the system becomes empty, the server will not restart the service until the sum of the service times of the waiting customers in the system reaches or exceeds some given positive integer $D$. Applying the total probability decomposition law, renewal theory, and probability generating function technique, the queueing indices and reliability measures are investigated simultaneously in our work. Both the probability generating function of the transient queue length distribution and the explicit formulas of the steady-state queue length distribution at time epoch $n^{+}$ are derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and steady-state unavailability, the expected number of breakdowns during $(0^{+},n^{+}]$, and the equilibrium failure frequency, are discussed. Finally, the optimum value of $D$ for minimizing the system cost is numerically discussed under a given cost structure.

Classification : 60K25, 68M20, 90B22
Mots clés : Discrete-time queue, D-policy, unreliable server, secondJ-optional service, cost optimization

I. Atencia and P. Moreno, $Geo/G/1$ retrial queue with 2nd optional service. Int. J. Oper. Res. 1 (4) (2006) 340–362. | DOI | MR | Zbl

I. Atencia and P. Moreno, A discrete-time $Geo/G/1$ retrial queue with server breakdowns. Asia-Pac. J. Oper. Res. 23 (2006) 247–271. | DOI | MR | Zbl

H. Bruneel and B.G. Kim, Discrete-Time Models for Communication Systems including ATM. Kluwer Academic Publishers, Boston (1993).

J.H. Cao and K. Cheng, Introduction to Reliability Mathematics. Higher Education Press, Beijing (2006).

G. Choudhury, An $M/G/1$ queueing system with two phase service under $D$-Policy. Int. J. Inf. Manag. Sci. 16 (2005) 1–17. | MR | Zbl

S. Gao and Z. Liu, A repairable $Ge{o}^X/G/1$ retrial queue with Bernoulli feedback and impatient customers. Acta Math. Appl. Sin., English Series 30 (2014) 205–222. | DOI | MR | Zbl

S. Gao, Z. Liu and H. Dong, A repairable discrete-time retrial queue with recurrent customers, Bernoulli feedback and general retrial times. Oper. Res. 12 (2012) 367–383. | Zbl

J.J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2: Discrete Time Models: Techniques and Applications. Academic Press, New York (1983). | MR

M. Jain and S. Upadhyaya, Optimal repairable $M^x/G/1$ queue with multi-optional services and Bernoulli vacation. Int. J. Oper. Res. 7 (2010) 109–132. | DOI | MR | Zbl

M. Jain, G.C. Sharma and R. Sharma, Unreliable server $M/G/1$ queue with multi-optional services and multi-optional vacations. Int. J. Math. Oper. Res. 5 (2013) 145–169. | DOI | MR | Zbl

E.I. Jury, Theory and Application of the $Z$-Transform Method. John Wiley & Sons, New York (1964).

J.C. Ke, An $M^{\left[x\right]}/G/1$ system with startup server and $J$ additional options for service. Appl. Math. Model. 32 (2008) 443–458. | DOI | MR | Zbl

M.S. Kumar, A discrete-time $Ge{o}^{\left[X\right]}/G/1$ retrial queue with general retrial time and $M$-additional options for service. RAIRO: OR 45 (2011) 131–152. | DOI | Numdam | MR | Zbl

J. Li and J. Wang, An $M/G/1$ retrial queue with second multi-optional service, feedback and unreliable server. Appl. Math.-A J. Chin. Univ. 21 (2006) 252–262. | DOI | MR | Zbl

C.H. Lin and J.C. Ke, On the discrete-time system with server breakdowns: Computational algorithm and optimization algorithm. Appl. Math. Comput. 218 (2011) 3624–3634. | MR | Zbl

K.C. Madan, An $M/G/1$ queue with second optional service. Queueing Syst. 34 (2000) 37–46. | DOI | MR | Zbl

Y. Tang, X. Yun and S. Huang, Discrete-time $Ge{o}^X/G/1$ queue with unreliable server and multiple adaptive delayed vacations. J. Comput. Appl. Math. 220 (2008) 439–455. | DOI | MR | Zbl

J. Wang and P. Zhang, A discrete-time retrial queue with negative customers and unreliable server. Comput. Ind. Eng. 56 (2009) 1216–1222. | DOI

J. Wang and Q. Zhao, A discrete-time $Geo/G/1$ retrial queue with starting failures and second optional service. Comput. Math. Appl. 53 (2007) 115-127. | DOI | MR | Zbl

K.H. Wang, C.C. Kuo and J.C. Ke, Optimal control of the $D$-policy $M/G/1$ queueing system with server breakdowns. Amer. J. Appl. Sci. 5 (2008) 565–573. | DOI

T.Y. Wang, An unreliable $Geo/G/1$ queue with startup and closedown times under randomized finite vacations. Appl. Math. Model. 39 (2015) 1383–1399. | DOI | MR | Zbl

T.Y. Wang, J.C. Ke and F.M. Chang, Analysis of a discrete-time queue with server subject to vacations and breakdowns. J. Ind. Prod. Eng. 30 (2013) 54–66.

Y. Wei, M. Yu, Y. Tang and J. Gu, Queue size distribution and capacity optimum design for $N$-policy $Ge{o}^{\lambda {}_1,\lambda {}_2,\lambda {}_3}/G/1$ queue with setup time and variable input rate. Math. Comput. Model. 57 (2013) 1559–1571. | DOI | MR

R.W. Wolff, Stochastic Modeling and the Theory of Queues. Prentice Hall. Inc., New Jersey (1989). | MR | Zbl

M.E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues. IEEE Computer Society Press, Los Alamitos (1994). | Zbl

F. Zhang and Z. Zhu, A discrete-time unreliable $Geo/G/1$ retrial queue with balking customers, second optional service, and general retrial times. Math. Probl. Eng. 2013 (2013) 1–12. | DOI | MR | Zbl