A Retrial Queueing System with Renewal Input and Phase Type Service Time Distribution

Klimenok, Valentina

doi:10.1007/978-3-319-44615-8_12

Cited by 2 publications

(4 citation statements)

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“…We refer to [1,2] for major results up to the 1990s. For single-server models, most studies focus on models with Poisson or Markovian arrival processes [3][4][5]. To the best of our knowledge, only a few papers have dealt with models with renewal input.…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty is that the embedded Markov chain of the model has a complicated transition structure that requires the transient analysis of a two-dimensional Markov chain. Furthermore, in [3][4][5], the total retrial rate of all customers in orbit was assumed to be constant. In contrast to [3][4][5], in this paper, we considered a GI/M/1 model with outgoing calls and the classical retrial rate where the total retrial rate is proportional to the number of customers in orbit, i.e., each customer in orbit retries independently of other customers.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, in [3][4][5], the total retrial rate of all customers in orbit was assumed to be constant. In contrast to [3][4][5], in this paper, we considered a GI/M/1 model with outgoing calls and the classical retrial rate where the total retrial rate is proportional to the number of customers in orbit, i.e., each customer in orbit retries independently of other customers. The methodology in [3][4][5][6] cannot be applied to GI/M/1 retrial queues with the classical retrial rate.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to [3][4][5], in this paper, we considered a GI/M/1 model with outgoing calls and the classical retrial rate where the total retrial rate is proportional to the number of customers in orbit, i.e., each customer in orbit retries independently of other customers. The methodology in [3][4][5][6] cannot be applied to GI/M/1 retrial queues with the classical retrial rate. More precisely, if we consider the embedded Markov chain at the pre-arrival epochs, we need the transient solution of a two-dimensional Markov chain representing the dynamics of the number of customers in orbit and the state of the servers.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion Limit for Single-Server Retrial Queues with Renewal Input and Outgoing Calls

et al. 2022

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This paper studies a single-server retrial queue with two types of calls (incoming and outgoing calls). Incoming calls arrive at the server according to a renewal process, and outgoing calls of N−1 (N≥2) categories occur according to N−1 independent Poisson processes. Upon arrival, if the server is occupied, an incoming call joins a virtual infinite queue called the orbit, and after an exponentially distributed time in orbit enters the server again, while outgoing calls are lost if the server is busy at the time of their arrivals. Although M/G/1 retrial queues and their variants are extensively studied in the literature, the GI/M/1 retrial queues are less studied due to their complexity. This paper aims to obtain a diffusion limit for the number of calls in orbit when the retrial rate is extremely low. Based on the diffusion limit, we built an approximation to the distribution of the number of calls in orbit.

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