Consider a queueing system that has two classes of customers, two clerks providing service, and no queue. Potential customers from each class arrive according to a Poisson process, with a mean arrival rate of 10 customers per hour for class 1 and 5 customers per hour for class 2, but these arrivals are lost to the system if they cannot immediately enter service.
Each customer of class 1 that enters the system will receive service from either one of the clerks that is free, where the service times have an exponential distribution with a mean of 5 minutes.
Each customer of class 2 that enters the system requires the simultaneous use of both clerks (the two clerks work together as a single server), where the service times have an exponential distribution with a mean of 5 minutes. Thus, an arriving customer of this kind would be lost to the system unless both clerks are free to begin service immediately.
(a) Formulate the queueing model in terms of transitions that only involve exponential distributions by defining the appropriate states and constructing the rate diagram.
(b) Now describe how the formulation in part (a) can be fitted into the format of the birth-and-death process.
(c) Use the results for the birth-and-death process to calculate the steady-state joint distribution of the number of customers of each class in the system.
(d) For each of the two classes of customers, what is the expected fraction of arrivals who are unable to enter the system?