Steady-state analysis of the multi-server retrial queueing system with heterogeneous servers and phase type distribution of service times
Abstract
A multi-server retrial queueing system with heterogeneous servers is analysed. The service times have a phase-type distribution with different irreducible representations. Customer arrival to the system is defined by a Markovian arrival process. When all servers are busy at an arrival moment, the customer moves to the virtual place called orbit to retry to reach the servers in exponentially distributed periods of time. The total retrial rate from the orbit infinitely increases when the number of customers residing in orbit grows. Upon arrival or retrial from the orbit, a customer occupies the server having the minimal number among all idle servers, if any. The dynamics of the system states is described by a multidimensional Markov chain having the special block structure of the infinitesimal generator. The explicit expression for this is presented. Ergodicity condition is derived. The expressions for computation of the key performance characteristics of the system are given. Numerical results, which highlight dependencies of these measures on the mean arrival rate for the system and its particular cases, when the arrivals are described by the stationary Poisson process or (and) service times follow the exponential distribution, are presented.
Keywords
Markovian arrival process,
retrials,
heterogeneous servers,
phase-type distribution,
asymptotically quasi-Toeplitz Markov chains
Supporting agencies: This research has received support by the Belarusian Republican Foundation for Fundamental Research (grant F25UZB-016) and the Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan (grant FL-8824063218).
About the Authors
M. Liu
Belarusian State University
Belarus
Belarus
Minsk
A. N. Dudin
Belarusian State University
Belarus
Belarus
Minsk
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Review
For citations:
Liu M., Dudin A.N. Steady-state analysis of the multi-server retrial queueing system with heterogeneous servers and phase type distribution of service times. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(1):111-120.
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