Note on the output of a queuing system

Consider a queuing system that has c servers and d waiting positions. Assume that the input is Poisson with rate α and the service times are exponential with mean β –1. Further assume the following: (i) a customer arriving when all servers are busy and all waiting positions are occupied is “cleared” from the system; (ii) a customer arriving when all servers are busy and not all waiting positions are occupied waits with probability 1 – ζ and “balks” or “clears” with probability ζ; (iii) a customer arriving when not all servers are busy commences service immediately (never balks); and, (iv) a customer who is waiting for service may “defect”, the distribution of time until a waiting customer defects being given by an exponential distribution with mean γ–1. Also, the usual independence assumptions, which make the process that is described by the number in the system at time t Markov, are assumed. An “output” of this queuing system is defined to occur whenever a service completion occurs, or whenever an arrival “clears” or “balks”, or whenever a waiting customer “defects”. Thus the output is a pooling of service completion epochs, the epochs when arrivals are cleared, the epochs when arrivals balk, and the defection epochs.