On queueing systems with renewal departure processes

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classical GI/G/s queueing systems. In particular, alternative proofs of known characterizations of the M/G/1 and GI/M/1 systems are given, as well as a characterization of the GI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both the GI/G/1 and GI/G/∞ systems are derived.

[1]  W. Burnside.  Theory of Functions , 1899, Nature.

[2]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[3]  D. V. Lindley,et al.  The theory of queues with a single server , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[4]  J. Doob.  Stochastic processes , 1953 .

[5]  Walter L. Smith,et al.  Regenerative stochastic processes , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[6]  J. Kiefer,et al.  On the theory of queues with many servers , 1955 .

[7]  P. Burke.  The Output of a Queuing System , 1956 .

[8]  E. Reich.  Waiting Times When Queues are in Tandem , 1957 .

[9]  Walter L. Smith.  Renewal Theory and its Ramifications , 1958 .

[10]  O. L. Smith,et al.  ON THE CUMULANTS OF RENEWAL PROCESSES , 1959 .

[11]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[12]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[13]  J. Kingman.  A FIRST COURSE IN STOCHASTIC PROCESSES , 1967 .

[14]  David Vere-Jones,et al.  Some Applications of Probability Generating Functionals to the Study of Input-Output Streams , 1968 .

[15]  Charles J. Stone,et al.  On a Theorem by Dobrushin , 1968 .

[16]  D. Daley.  The Correlation Structure of the Output Process of Some Single Server Queueing Systems , 1968 .

[17]  Daryl J. Daley,et al.  Weakly Stationary Point Processes and Random Measures , 1971 .

[18]  W. Whitt.  Embedded renewal processes in the GI/G/s queue , 1972, Journal of Applied Probability.

[19]  S. Stidham.  Regenerative processes in the theory of queues, with applications to the alternating-priority queue , 1972, Advances in Applied Probability.

[20]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[21]  Ralph L. Disney,et al.  A Characterization of M/G/1 Queues with Renewal Departure Processes , 1973 .

[22]  D. Daley,et al.  Independent Inter‐Departure Times in M/G/1/N Queues , 1975 .

[23]  P. S. Puri.  A LIMIT THEOREM FOR POINT PROCESSES WITH APPLICATIONS , 1978 .

[24]  K. Meyer.  The Output of a Queueing System , 1981 .