The \(M/G/1\) queue with quasi-restricted accessibility. (English) Zbl 1178.60062
The quasi-restricted accessibility (QRA) is a new variation of queueing models with workload dependent admission policies. Under QRA the M/G/1 queue system model was considered. In this model, the service time \(S_{n}\) assigned to the \(n\)-th customer depends on his service requirement \(X_{n}\) and his waiting time \(W_{n}\). If \(W_{n} + X_{n}\) does not exceed a certain threshold \(b\), then the customer receives service time \(b - W_{n} + (W_{n} + X_{n} - b)F_{n}\) where \(F_{n}\) is a fraction of the service requirement beyond \(b\).
The authors of this paper have determined the stead-state distribution of the waiting time in the QRA M/G/1 queue. The case of exponential barriers \(b\) instead of one constant threshold was studied. In this case, barrier \(B_{n}\) for the \(n\)-th customer is exponentially distributed with a mean, say \(1/\xi\). Thus, the stead-state waiting time (and workload) in the model with the use the closed expression for the LST formula was obtained. For the case of an QRA M/M/1 and M/G/1 queues the LST formula of the busy period in terms of transformations for these phases was derived. Very important are the derivations of the stopping time for both of these queues. As the example of this system the non-Markovian Erlang \((2, \mu)/M/1\)-type QRA system with service requirement \(X_{n}\exp(\lambda)\), under QRA was considered. For this system the LST formula of the length of the busy period was obtained. Other expressions, for instance the stopping time and the LST of the residual time until the time in a given functional is given, were also computed.
In summation, the following results for QRA M/G/1 system were obtained: the steady-state distribution of the waiting time and also the steady-state distribution of the workload process, the explicit formula for the LST of the steady-state waiting time, the distribution of the length of a busy period in a closed form, the cycle maximum distribution under QRA, the distribution function of the length of a busy period for the non-Markovian case of the phase-type arrivals.
The authors of this paper have determined the stead-state distribution of the waiting time in the QRA M/G/1 queue. The case of exponential barriers \(b\) instead of one constant threshold was studied. In this case, barrier \(B_{n}\) for the \(n\)-th customer is exponentially distributed with a mean, say \(1/\xi\). Thus, the stead-state waiting time (and workload) in the model with the use the closed expression for the LST formula was obtained. For the case of an QRA M/M/1 and M/G/1 queues the LST formula of the busy period in terms of transformations for these phases was derived. Very important are the derivations of the stopping time for both of these queues. As the example of this system the non-Markovian Erlang \((2, \mu)/M/1\)-type QRA system with service requirement \(X_{n}\exp(\lambda)\), under QRA was considered. For this system the LST formula of the length of the busy period was obtained. Other expressions, for instance the stopping time and the LST of the residual time until the time in a given functional is given, were also computed.
In summation, the following results for QRA M/G/1 system were obtained: the steady-state distribution of the waiting time and also the steady-state distribution of the workload process, the explicit formula for the LST of the steady-state waiting time, the distribution of the length of a busy period in a closed form, the cycle maximum distribution under QRA, the distribution function of the length of a busy period for the non-Markovian case of the phase-type arrivals.
Reviewer: Jerzy Martyna (Kraków)
MSC:
| 60K25 | Queueing theory (aspects of probability theory) |
| 68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |
| 90B22 | Queues and service in operations research |
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