A Single Server Queue with Two Types of Customers and a Gate Mechanism

SYNOPTIC ABSTRACT

We consider a single server queue with two types(I and II) of customers in which arrivals occur according to a Poisson process and service times of customers are independent and exponentially distributed random variables. The server works according to a gated discipline. Upon arrival customers join a queue in front of a gate. Whenever all customers present at the service area have received service, the gate opens. The customers waiting for service enter the service area and the gate closes. During the service time of a type I customer, only customers of type I are admitted into the system and type II are lost to the system. When a type II customer is served, customers of both type I and type II are permitted in the system.

Exact expressions for the performance characteristics, such as the total number of customers served and proportion of each type in the n-th batch are obtained. The connections between queues, branching processes and continued fractions are exploited to obtain these results. Numerical illustrations are presented.

Key Words and Phrases:

• continued fractions
• branching processes
• busy period
• gated service