The Steady-State Queueing Time Distribution for the M/G/1 Finite Capacity Queue

Stephen S. Lavenberg

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

Abstract

We derive an expression for the Laplace-Stieltjes transform of the steady-state distribution of the queueing time for the M/G/1 finite capacity queue. The derivation proceeds in terms of a related 2-stage closed cyclic queueing network. The resulting expression is a rational function of the steady-state probabilities of the imbedded Markov chain at departure epochs and of the Laplace-Stieltjes transform of the service time distribution. The expression can be differentiated readily in order to obtain moments of the steady-state queueing time and some numerical results for the mean and coefficient of variation are presented.

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References

1983The Single Server Queue.1966The Ergodic Queue Length Distribution for Queueing Systems with Finite Capacity

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1986The influence of service-time variability in a closed network of queues 1983The cyclic queue with one general and one exponential server 1989On the Cycle Time Distribution in a Two-Stage Cyclic Network with Blocking 1985Analysis of a finite-capacity M/G/1 queue with a resume level 1982On response time and cycle time distributions in a two-stage cyclic queue 1984Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity 1986A maximum entropy queue length distribution for the G/G/1 finite capacity queue 1993Transform-Free Analysis of M/G/1/K and Related Queues

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