Queues where customers of one queue act as servers of the other queue

We consider a system comprised of two connected M/M/?/? type queues, where customers of one queue act as servers for the other queue. One queue, Q 1, operates as a limited-buffer M/M/1/N?1 system. The other queue, Q 2, has an unlimited-buffer and receives service from the customers of Q 1. Such analytic models may represent applications like [email protected], where idle computers of users are used to process data collected by space radio telescopes. Let L 1 denote the number of customers in Q 1. Then, two models are studied, distinguished by their service discipline in Q 2: In Model 1, Q 2 operates as an unlimited-buffer, single-server M/M/1/? queue with Poisson arrival rate ? 2 and dynamically changing service rate μ 2 L 1. In Model 2, Q 2 operates as a multi-server M/M/L 1/? queue with varying number of servers, L 1, each serving at a Poisson rate of μ 2. We analyze both models and derive the Probability Generating Functions of the system's steady-state probabilities. We then calculate the mean total number of customers present in each queue. Extreme cases are indicated.