The M/G/∞ Queue in Random Environment

Infinite-server queues are often used to approximate the behavior of systems with sufficiently large number of servers such as banks, call-centers, supermarkets or digital distribution platforms. Such systems often become subjects to external influences which affect their performance, particularly their arrival rate and service-time distribution. In this article we consider a mathematical model of such a situation as an M/G/8 queue operating in a "random environment", for which the underlying process is a continuous-time Markov chain with finite number of states. The arrival rate and service-time distribution change randomly according to the environment state. Note that distribution of service-time of customers, which are currently being served, does not change until the service-time is finished. The approximate probability distribution of the number of customers under certain conditions is obtained.