Two approximations for the steady-state probabilities and the sojourn-time distribution of the M/D/c queue with state-dependent feedback

In the M/D/c queue with state-dependent feedback, a customer is only allowed to depart from the system if his service has been successful. Otherwise, the customer must be re-serviced immediately. The probability that a customer's service is successful depends on the number of customers in service at the moment the service is finished. The application behind this type of feedback queue is a real-time database where transactions must be rerun if their data was changed by other transactions during the execution. In this paper, two different approximations for the steady-state probabilities and the sojourn-time distribution of the M/D/c queue with state-dependent feedback are studied. The first approximation is based on an embedded Markov chain and uses the well-known residual-life approximation for the remaining service times of the customers in service. The second approximation is similar to the exact analysis of the ordinary M/D/c queue. Comparison with simulation shows, that both approximations are very accurate for a wide range of system parameters, even for heavily loaded systems.