An M/G/1 queue with multiple types of feedback, gated vacations and FCFS policy

We consider an M/G/1 queueing system with multiple types of feedback, gated vacations and FCFS policy where the first service of a new customer is either successful (and then the customer leaves the system) or unsuccessful (and then the customer joins to the end of the queue for another service as old customer with different Bernoulli feedback parameter and different service distribution), and customers are served in the order of joining the tail of the queue. By applying a new method developed by authors (Queueing system with fixed feedback policy, J. Aust. Math. Soc. B, to appear, Comput. Oper. Res. 27 (2000) 269) we obtain joint probability generating function of system sizes of new and old customers at steady state and Laplace Stieltjes transform of total response time. We also give algorithms for calculation of moments of system size and total response time. The polling system in computer network can be modeled as queue with gated vacation. In order to include transmission's error, the polling system is modeled as queue with gated vacation and feedback where old customers have different feedback parameter and different service time distribution compared to new customers. For M/G/1 queueing system with multiple types of feedback, analysis of FCFS policy is more difficult than that of priority policy. In fact, classical embedded Markov chain methods can be applied to priority queue with multiple types of feedback, but they cannot be applied to FCFS queue. Until now, system size distribution and total response time distribution are known for M/G/1 priority queues with multiple types of feedback (J. Appl. Prob. 34 (1997) 773; Queueing Systems 8 (1991) 183), but only mean system size and mean total response time are known for M/G/1 FCFS queues with multiple types of feedback (Oper. Res. 42 (1994) 380; Commun. Stat. Stochastic Model 5(1) (1989) 115; Oper. Res. Lett. 25 (1999) 137; Queueing Theory and its Applications, CWI Monographs 7, North-Holland, Amsterdam; J. ACM 31 (1984) 13). In order to analyze the queue with feedback and gated vacation completely, we need to find the distribution of the system size and total response time in addition to mean system size and mean total response time. The purpose of this paper is to obtain joint probability generating function of system sizes of new and old customers at steady state and Laplace Stieltjes transform of total response time of an M/G/1 queueing system with multiple types of feedback, gated vacations and FCFS policy.