minimizing loss probability in queuing systems with heterogeneous servers

the probability of losing a customer in m/g/n/0 and gi/m/n/0 loss queuing systems withheterogeneous servers is minimized. the first system uses a queue discipline in which a customer who arriveswhen there are free servers chooses any one of them with equal probability, but is lost otherwise. providedthat the sum of the servers rates are fixed, loss probability in this system attains minimum value when all theservice rates are equal. the second system uses queue discipline, in which a customer who enters into thesystem is assigned to the server with the lowest number. loss probability in this system takes the minimumvalue in the case when the fastest server rule is used in which an incoming customer is served by the freeserver with the shortest mean service time. if the mean of the arrival distribution is fixed, then loss probabilityis minimized by deterministic arrival distribution