Retrial queuing system MMPP|GI|1 researching by means of the second-order asymptotic analysis method under a heavy load condition

Abstract

Retrial queuing systems which are mathematical models of real processes in telecommunication systems are a new developing direction of the queuing theory. However the analytical formulas are obtained only for systems with Poisson arrival process. Most of the foreign scientists on queuing theory use different numerical methods for researching retrial queueing systems with not Poisson arrival process (e. g. ММРР, МАР, BMAP). Such methods have a natural limit of applicability related to solving the equations systems of large dimension (from 1000 to 500thousand states). Thus, the urgency of the reserach is caused by the need to develop analytical methods for studying RQ-systems with arrival MMP-process. The main aim of the study is to find the asymptotic probability distribution of the number of calls in the orbit in the retrial queueing system MMPP|GI|1 for a sufficiently large number of states of the system. The methods used in the study: second-order asymptotic analysis method under heavy load condition. The results: The authors have obtained the asymptotic (second-order) characteristic function of the probability distribution of the number of calls in the orbit in the retrial queueing system MMPP|GI|1. The paper introduces the formula for asymptotic distribution construction. The numerical analysis of the results showed that the proposed method can be used to load values ρ>0,8, whereas the first-order asymptotic analysis method is applied when load values ρ>0,95. When using the obtained asymptotic probability distribution the most important characteristics of the system can be calculated (e. g. the average number of calls in the orbit). They can be used in modeling or optimization of real economic and technical systems operation.

Keywords

• retrial queueing system
• orbit
• MMP-process
• heavy load
• second-order asymptotic analysis method