Generalization of the M/G/1/∞ Queueing System Based on Markov Renewal Service Times

الملخص

The  queueing system is a fundamental model in queueing theory, used to analyze and evaluate the performance of systems that manage customer flows through a single server with service times following a general probabilistic distribution. However, there are scenarios requiring the handling of correlated random service times, as these systems provide various services and need different random service times. Thus, a new tool is needed to address such cases.

This research presents an integrated mathematical framework for the developed  model, generalizing the results through theoretical and applied analysis. It aims to generalize the theoretical foundations of the  model by representing service times in this model using two Markov renewal phases. This representation allows the derivation of general mathematical formulas for both the probability vector and the general probability vector based on stable vectors and generating functions . Mathematical relationships for the performance metrics of the proposed system are also derived.

Finally, the special case of service times with an exponential distribution for each customer type, with different arrival rates, is discussed. The results are illustrated through examples and applications, which also verify the suitability of the Poisson distribution for the arrival times for each customer type. It is concluded that the proposed  system is a generalization of the  service system, as it deals with linked service times and possesses its properties.