Conference Agenda

In this paper, we investigate M/G/1 systems with server vacations using the well-established supplementary variable method, combined with purely probabilistic reasoning.

The novel probabilistic approach simultaneously considers the system at two distinct time points: one at the moment of observation during service, and the other at the start of that service. This dual-time analysis provides new insights into the system's behavior, yielding solutions to the problem.

This innovative approach bypasses the need to solve partial differential equations (Kolmogorov forward equations) for the non-classical boundary value problem with non-local boundary conditions. Instead, it directly derives the system's solution using operational calculus.

Acknowledgements The designated project has been fulfilled by financial support of the Shota Rustaveli National Science Foundation of Georgia (Grant: STEM-22-340)

Motivated by the surveillance of utility meters in Germany, we assume that every few years, a population of devices must be replaced if reliability for the next few years cannot be demonstrated in a test. To demonstrate reliability, the producer (or the operator) of the devices shall initiate acceptance sampling by attributes at every testing time. Sampling can be sequential. The resulting stopping problem for the producer may depend on previous results. If the current sample does not demonstrate reliability, the producer may or may not continue sampling. If the producer stops sampling, the population must be replaced. With the continuation, the producer hopes to demonstrate reliability with a larger sample. If the current sample demonstrates reliability, the population continues to operate until the next testing time, when a further test may be conducted.

To find an optimal strategy one needs to know the consequences of the decisions. We apply Bayesian analysis to predict the next sample. The analysis utilizes a lifetime model for the devices of the population in order to take samples from all previous testing times into account. Having the probability for the next sample we apply Bellman’s principle to calculate a cost minimizing strategy. We consider simple and easily adaptable cost functions. Considering the surveillance of water meters, it is reasonable to assume a small number of possible test results and a small maximal number of testing times. This enables the calculation of cost minimizing sampling plans using the Bellman principle. Compared to single sampling, the resulting optimal strategy can decrease the costs for water meter surveillance by almost 30%.