Gated polling models with customers in orbit. (English) Zbl 1042.60542
Summary: A gated polling model with n stations and switchover times is considered. The primary customers (those who are present at the polling instant) are served in the usual way, while the secondary customers (those who arrive in the meantime) do not wait in a queue, but they depart and start to make retrials until they succeed to find a position for service. The customers are of n different types and arrive to the system according to the Poisson distribution, in batches of random size. Each batch may contain customers of different types, while the numbers of customers belonging to each type in a batch are distributed according to a multivariate general distribution. The server, upon finishing the service of all primary customers in a station, stays there for an exponential period of time and if a customer asks for service before this time expires, the customer is served and a new stay period begins. Finally, the service times and the switchover times are both arbitrarily distributed with different distributions for the different stations.
For such a model we obtain formulae for the expected number of customers in each station in a steady state. Our formulae hold also for zero switchover periods and can easily be adapted to hold for the ordinary gated polling model with/without switchover times and correlated batch arrivals. In all cases, the results are obtained by solving a final set of only n linear equations. Numerical calculations are also used to observe systems performance.
For such a model we obtain formulae for the expected number of customers in each station in a steady state. Our formulae hold also for zero switchover periods and can easily be adapted to hold for the ordinary gated polling model with/without switchover times and correlated batch arrivals. In all cases, the results are obtained by solving a final set of only n linear equations. Numerical calculations are also used to observe systems performance.
MSC:
| 60K25 | Queueing theory (aspects of probability theory) |
| 90B22 | Queues and service in operations research |
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