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, Q1, operates as a limited-buffer M/M/1/N − 1 system. The other queue, Q2, has an unlimited-buffer and receives service from the customers of Q1. 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 L1 denote the number of customers in Q1. Then, two models are studied, distinguished by their service discipline in Q2: In Model 1, Q2 operates as an unlimited-buffer, single-server M/M/1/∞ queue with Poisson arrival rate λ2 and dynamically changing service rate μ2L1. In Model 2, Q2 operates as a multi-server M/M/L1/∞ queue with varying number of servers, L1, each serving at a Poisson rate of μ2. 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.