On Single Server Queues with Batch Arrivals

Abstract

We consider a queuing system GIν|M|1|∞ with arrival of customers in batches, general renewal arrivals, exponential service times, single service channels and an infinite number of waiting positions, where customers are serviced in the order of their arrival. In the stationary case, new forms of the probability generating functions of the number of clients in the system are derived. These new forms are written in terms of the p.g.f. of the tail distribution function of the number of customers per group and of the p.g.f. of an embedded discrete time homogeneous Markov chain. In a queuing system with a batch Poisson arrival flow Mλν|Mμ|1|∞, the number of customers in the system can be obtained from the normalized tail distribution.

Keywords

• queueing system
• infinite capacity
• server
• batch arrivals
• renewal process
• probability generating functions