Abstract
The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated.
Highlights
Most of the papers on queuing theory have considered the systems without feedback, i.e. without re-service phenomena
The state of the system at an arbitrary moment of the time is defined by the three-dimensional vector (n, k, r), where n is the state of the Markov modulated Poisson process (MMPP)-flow, k is total number of busy channels and r is number of retrial calls in orbit
Exact values of the steady-state probabilities were calculated from the corresponding system of equilibrium equations (SEE) and the accuracy of their calculation might be estimated using various similarity norms
Summary
Most of the papers on queuing theory have considered the systems without feedback, i.e. without re-service phenomena. Queues with feedback occur in many practical situations, e.g., multiple access telecommunication systems, where data that is transmitted erroneously are sent again can be modeled as queue with feedback. We will distinguish between re-service of two types: instantaneous and delayed re-service. In the case of delayed re-service, after the customer is served completely, they will decide either to join the retrial group again for another service with some positive probability or to leave the system forever with complimentary probability. Such kinds of feedbacks are called Bernoulli feedback
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