In this paper, the analysis of Semi-Markovian single server retrial queues by means of Markov Regenerative Stochastic Petri Nets (MRSPN) is considered. We propose MRSPN models for the two retrial queues M/G/1/N/N and M/G/1/N/N with orbital search. By inspecting the reduced reachability graph of both MRSPN models, the qualitative analysis is obtained. The quantitative analysis is carried out after constructing their one step transition probability matrix and computing the steady state probability distribution of each tangible marking. As an example, the queue M/H ypo 2 /1/2/2 is treated in order to illustrate the functionality of the MRSPN approach. The exact performance measures (mean number of customers in the system, mean response time, mean waiting time,…) are computed for different parameters of the two systems by an algorithm elaborated in Matlab environment.

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We study a finite source retrial queue with deterministic service times using an approach based on the theory of Markov Regenerative Process (MRP). A Deterministic Stochastic Petri Net (DSPN) model which copes with the complexity of this queue is given. For the steady state of this model, we construct the one step transition probability matrix of embedded Markov chain and the conversion matrix. As an example the retrial system M/D/1/2/2 is detailed. We establish an algorithm in Matlab environment based on the theoretic results obtained in order to compute efficiently various performance measures and to study the effect of system parameter's on the characteristics of the DSPN models the retial queue M/D/1/N/N.

In this article, a queueing inventory system with finite sources of demands, retrial demands, service time, lead time, ( s , S ) \left(s,S) replenishment policy, and demands search from the orbit was studied. When the lead time is exponentially distributed (resp. lead time is generally distributed), generalized stochastic Petri net (GSPN) (resp. Markov regenerative stochastic Petri net [MRSPN]) is proposed for this inventory system. The quantitative analysis of this stochastic Petri net model was obtained by continuous time Markov chain for the GSPN model (resp. the supplementary variable method for the MRSPN model). The probability distributions are obtained, witch allowed us to compute performance measures and the expected cost rate of the studied system.