Two M/M/1 Queues with Incongruent Arrivals and Services with Random Feedback

SYNOPTIC ABSTRACT

The current study examines a queuing system with two incongruent arrivals and two services. In this regard, two types of customers enter the system by a Poisson process and the service times are assumed to have exponential distributions. After the first service is completed, the system may provide feedback to repeat the first service, leave the system, or continue to give the second service. The same policy is utilized for the other kind of customers. The whole stochastic processes involved in the system are considered as independent random variables. A probability generating function is derived for each type and for the system that yield the performance measures. We examine the validity of the results through numerical approaches.

KEY WORDS AND PHRASES:

• $M/M/1$ Queue
• incongruent arrivals and services
• random feedback
• mean queue size