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

We consider a system comprised of two connected M/M/center dot/center dot 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 SETI@home, 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/infinity queue with Poisson arrival rate lambda (2) and dynamically changing service rate mu (2) L (1). In Model 2, Q (2) operates as a multi-server M/M/L-1/infinity queue with varying number of servers, L (1), each serving at a Poisson rate of mu(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.