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

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Advanced Queuing Theory Analysis Probability and Statistical Research Age of Information Optimization

Abstract

In a Jackson-type queuing network with feedback, the equilibrium state distribution of each queue is that of anM/M/ssystem. 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 equilibriumM/M/1queuing 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.

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1990An Anti-PASTA Result for Markovian Systems 1983Filtering of Markov renewal queues, I: Feedback queues 1981Stationary Distributions of Flows in Jackson Networks 1978Streams of a M/M/1 feedback queue in statistical equilibrium 1982On the equivalence of flows in networks of queues

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