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# REPEAT mode: condition (13) is violated, condition (12) holds, all orbits are stable It is worth mentioning that the results demonstrated by Fig. 3 and Fig. 1 seem to be unexpected, because in both cases the sufficient stability condition is violated, while all orbits remain stable. This result has been first detected in the work [16] and it supports our conjecture that condition (13) is indeed redundant and the necessary stability condition (12) is close

Source publication

In this work, we investigate the stability conditions of a multiclass retrial system with coupled orbit queues and service interruptions. We consider a single server system accepting N classes of customers according to independent Poisson inputs and with class-dependent, arbitrarily distributed service times. An arriving customer who finds the serv...

## Contexts in source publication

**Context 1**

... 1/¯ r i = α i the rate of the setup time when a class-i customer is being served. Fig. 3 shows dynamics of the orbits for the REPEAT interruptions model and the following input ...

**Context 2**

... a result, these parameters give ρ = 0.85, see (11). On the other hand, it is easy to calculate, that the r.h.s. of conditions (12) and (13), equals 0.9 and 0.8, respectively, so only condition (12) hold while condition (13) is violated. As we see, despite breaking the sufficient condition, all orbits are stable as shown at Fig. ...

**Context 3**

... this case we obtain ρ = 0.9, while the r.h.s of (12) equals 0.83. Thus, condition (13) is violated while condition (12) is satisfied, and Fig. 1 indicates that dynamics of the orbits is indeed similar to a stable dynamics observed on Fig. 3. (12) holds, all orbits are stable Finally, for the RESUME interruptions system with the ...

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## Citations

... Finally, we would like to mention a series of recent works devoted to regenerative stability analysis of the multiclass retrial systems with coupled orbits (or state-dependent retrial rates), being a far-reaching generalization of the constant retrial rate systems, in which the retrial rate of each orbit depends on the binary state (busy or idle) of all other orbits, see [24,25,29,26,27,28]. In particular, this analysis is based on PASTA and a coupling procedure connecting the real processes of the retrial with the independent Poisson processes corresponding to various 'configurations' of the (binary) states of the orbits. ...

We establish stability criterion for a two-class retrial system with Poisson inputs, general class-dependent service times and class-dependent constant retrial rates. We also characterise an interesting phenomenon of partial stability when one orbit is tight but the other orbit goes to infinity in probability. All theoretical results are illustrated by numerical experiments.

In this paper, we consider a retrial queuing system with unreliable servers and analyze the distribution of the stationary generalized service time which includes also the unavailable periods (setup times) occurring during service of the customer. We consider three service interruption disciplines: preemptive resume (PRS), preemptive repeat different (PRD), and preemptive repeat identical (PRI); and provide the stationary distribution of the generalized service time and the remaining generalized service time for these disciplines in Laplace transform (LT) domain.

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In this paper, we consider a multiserver retrial queuing system with unreliable servers class-dependent retrial rates and N classes of customers following Poisson input processes. We analyze the distribution of the stationary generalized remaining service time which includes all unavailable periods (setup times) occurring during service of the customer. During service of a class-i customer, the interruptions occur according to the i-dependent Poisson process and the following i-dependent random setup time of the server. We consider two following disciplines caused by the service interruptions: preemptive repeat different and preemptive resume. Using coupling method and regenerative approach, we derive the stationary distribution of the generalized remaining service time in an arbitrary server. For each class i, this distribution is expressed as a convolution of the corresponding original service times and setup times, and in general is available in the terms of the Laplace-Stieltjes transform allowing to calculate the moments of the target distribution. Some numerical examples are included as well.