Mass serving theory application to the analysis of maintenance system functioning

Abstract

This paper describes models and conditions for the application of the Mass Serving Theory in order to analyze relations between clients demanding the service and channels which provide the service as well as to design technological elements in the optimal regime for the given maintenance system. Based on the actual data collected and the statistical analysis of the expected intensity of combat vehicle arrivals and queuing at service for tehnical maintenance, the mathematical modeling of a real process of queuing was carried out and certain parameters quantified, in terms of determining the weaknesses of the existing models and the corrective actions needed.

Introduction

While solving many practical problems within the process of maintenance, the technological demands (TD) for maintenance appear with the characteristics of stochasticity and stationarity. These properties provide the ability of the Mass Serving Theory (MST) to be used, under certain conditions, for the dimensioning of technological elements (TE) in the reporting maintenance system.The analysis of the mass serving system (MSS) means the analysis of the input stream of clients, time and number of customers in a queue, time of serving and the output stream of clients as well.

Mahtemathical models of the mass serving system applicable to maintenance processes

There are many mathematical models developed in the MST to analyze the relationship between clients demanding the serving and channels that serve them. In the mathematical models of mass serving, the following parameters are commonly used as inputs: Input stream intensity,Serving intensity of the TE, Number of channels, i.e. TE; as outputs: Serving probability of TD,The average number of TD in a serving queue, and The average time of stay in the TD queue.

In practice, during the system sizing, the number of channels is usually required, i.e. TE (n) necessary to serve the TD, and in certain situations Input stream intensity and Serving intensity of the TE.

Applied model of the mass serving system for the description of service for tehnical maintenance functioning

The functioning of the technical workshop is analyzed in terms of serving combat vehicles (c/v) which , in most cases, come at random periods of time. The time period of three years in the selected unit has been observed.

The service for tehnical maintenance is considered as an MSS with queuing in which the client (TD)  does not get cancellation when all channels are occupied, but it gets into a queue and waits for the channel release, i.e.the required queuing time is greater than the period of patience at the beginning of serving.

The analysis of input streams, for tanks and transporters separately, should give the answer to the question whether the application of the MST is possible, i.e. whether the conditions of stochasticity and stationarity are primarily satisfied.

The time interval of one day is considered as relevant because it is assumed that the largest number of TD are executed on the day of occurrence, following daily based planning activities.

The obtained results show that, by applying the hypothesis of matching the empirical and theoretical distribution, the random variable Xi – the number of vehicles per day is subjected to Poisson's distribution in all cases.

It was shown that, by applying the hypothesis about matching the theoretical and empirical distributions for different cases of c/v serving by electro-mechanics for c/v, the empirical distributions match with some of the given theoretical distributions. On the other hand, when the c/v were serviced by the mechanics for c/v, there is no matching between the empirical and the theoretical distribution for all considered cases, which leads to the conclusion that the application of this model is limited to certain cases.

The optimization in the mass serving system

The MSS optimization could be presented by the dependency of the system functioning performance and the parameters that describe the process of serving within the given limited values of the parameters. The key indicators of effectiveness are: the number of serving channels; the average serving time and the average number of clients accessing the system per time unit.

The best way of analysing the functioning of MSS in the optimum regime is a comparison of the two opposing criteria of effectiveness: the probability of serving and the ratio of the serving channel occupancy, assuming that the initial parameters - the average time of serving and the intensity of the input stream of clients - are constant. Under these conditions, the optimal number of serving channels should be determined.

As a criterion of optimality, we accept the following fact: the MSS operates in the optimum regime if the probability of serving and the ratio of the serving channel occupancy are large enough, values so that the TE value is minimal.

It can be noticed that the values of both probabilities are similar when the number of serving channels is 2. In other cases, these deviations are larger. The improvement of one criterion causes the deterioration of the other one and vice versa. It can be concluded that the optimal number of serving channels is when Ps and Pzk have equal values, and the values of probability exceed 51%.

The analysis of the shown case of the optimization in the MSS indicates that the number of TE is oversized i.e. the largest number of technological requirements could be fulfilled with 2 TE (instead of 4 as in a real system).

Retention time in the mass serving system

When analyzing the retention time of technical means in the MSS, we can notice that, by increasing the number of serving channels, all values of time converge to a single value.

Using the previous results which show that the number of 2 serving channels is optimal, we have analysed the range of values of retention time in the MSS for different values of malfunction intensity.

The analyzed data show clearly how the retention time for technical means in the maintenance system changes  when the number of serving channels varies from one to ten. At the same time, we can see the change of the value of the probability of serving and the probability of channel occupancy.

Conclusion

This paper presents an attempt to analyze a maintenance unit as an MSS using mathematical models developed in the MST,i.e. to review the relationships between clients who demand serving and channels that serve them as well as functioning in the optimum regime. With respect to many assumptions, the application of the MST for such systems was found to be limited, while the optimization process indicates the oversizing of the real system.