Proof of a Conjecture on the Interarrival-Time Distribution in an M/M/1 Queue with Feedback

In a Jackson-type queuing network with feedback, the equilibrium state distribution of each queue is that of an
M/M/s
system. In support of a previous conjecture that nevertheless the input processes in such a network are not Poisson, the marginal interarrival-time distribution for an equilibrium
M/M/1
queuing system with feedback, counting both fed-back and exogenous customers as arrivals, is calculated. Since this distribution is a mixture of two exponentials, the total input to such a system is not Poisson.