On the performance of state-dependent single server queues.

*(English)*Zbl 0607.60087This is an interesting paper on applied probability in which the authors deploy powerful analytical weapons from the armoury of singular perturbation techniques. The analysis concerns the M/G/1 queueing process in which the appropriately defined mean arrival and service rates depend on the momentary system state.

Approximations are calculated to the stationary system state statistics and to statistics of the busy period. The analysis departs from a representation of state probabilities using supplementary variable technique, and subsequently uses the Markov chain embedded at end-of- service epochs.

The paper is heavily technique-oriented and does not seriously concern itself with operational performance, though a slight attempt is made to demonstrate the efficacy of the approximations in a practical case. Justice is, however, hardly done to the quite considerable body of work that has been done during the past 25 years on state dependent queues and similar models embodying the concept of harmonisation of demand and supply. This paper should not have been published without a proper review of earlier work, a lead on which could have easily been obtained by recourse to one of the many data bases now available for interrogation.

Approximations are calculated to the stationary system state statistics and to statistics of the busy period. The analysis departs from a representation of state probabilities using supplementary variable technique, and subsequently uses the Markov chain embedded at end-of- service epochs.

The paper is heavily technique-oriented and does not seriously concern itself with operational performance, though a slight attempt is made to demonstrate the efficacy of the approximations in a practical case. Justice is, however, hardly done to the quite considerable body of work that has been done during the past 25 years on state dependent queues and similar models embodying the concept of harmonisation of demand and supply. This paper should not have been published without a proper review of earlier work, a lead on which could have easily been obtained by recourse to one of the many data bases now available for interrogation.

Reviewer: B.W.Conolly

##### MSC:

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |