##
**Blocking probability for M/G/1 vacation systems with occupancy level dependent schedules.**
*(English)*
Zbl 0666.60097

An M/G/1 queue with finite buffer capacity and server vacation schedules dependent on occupancy level is studied. The blocking probability is expressed simply in terms of the ergodic queue length probabilities for the infinite buffer case.

The steady-state distribution of the number of customers in a system is derived. This measure differs from one of a system with infinite waiting room, basically, by only a scale factor. The analysis is based on the Theorem 1, a renewal theoretic result. In particular, the existence of a multiplicative constant \(\theta_ L\) is established relating ergodic probabilities for a finite capacity system to the corresponding, more accessible, probabilities for the infinite capacity system.

Vacation may start at the completion of a service or vacation. All vacations are independent and identically distributed. The main assumption of the results are the following: after a service (vacation task) completion, with probability \(p_ n\) \((q_ n)\), a customer starts service, and with probability \(1-p_ n\) \((1-q_ n)\), a vacation task starts. \(p_ 0=q_ 0=0\). For \(L\leq n<\infty\), \(p_ n=1\), \(q_ n=q_ L\). Here, \(L<\infty\) is the given, finite, capacity of the system and n is the number of customers in the system after a service (vacation task) completion.

Such vacation schedules arise, for example, in telecommunication switches of LAN’s. An extensive literature on single server vacation systems is listed at the end of the article.

The steady-state distribution of the number of customers in a system is derived. This measure differs from one of a system with infinite waiting room, basically, by only a scale factor. The analysis is based on the Theorem 1, a renewal theoretic result. In particular, the existence of a multiplicative constant \(\theta_ L\) is established relating ergodic probabilities for a finite capacity system to the corresponding, more accessible, probabilities for the infinite capacity system.

Vacation may start at the completion of a service or vacation. All vacations are independent and identically distributed. The main assumption of the results are the following: after a service (vacation task) completion, with probability \(p_ n\) \((q_ n)\), a customer starts service, and with probability \(1-p_ n\) \((1-q_ n)\), a vacation task starts. \(p_ 0=q_ 0=0\). For \(L\leq n<\infty\), \(p_ n=1\), \(q_ n=q_ L\). Here, \(L<\infty\) is the given, finite, capacity of the system and n is the number of customers in the system after a service (vacation task) completion.

Such vacation schedules arise, for example, in telecommunication switches of LAN’s. An extensive literature on single server vacation systems is listed at the end of the article.

Reviewer: J.Tanko

### MSC:

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

68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |