Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

Nazarov, Anatoly; Phung-Duc, Tuan; Paul, Svetlana

doi:10.1007/s11518-018-5404-6

Cited by 8 publications

(6 citation statements)

References 13 publications

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“…A characteristic of two-way communication is the fact that unoccupied servers can perform outgoing calls to the source. In such systems, the server's utilization is always a critical issue; for an example, see [6][7][8][9]. Ayyappan and Gowthami [10] performed a stationary analysis of a feedback retrial queue with impatient customers, vacations, and two types of arrivals.…”

Section: Introductionmentioning

confidence: 99%

Multiserver Retrial Queue with Two-Way Communication and Synchronous Working Vacation

Liu,

Chiou,

Chen

et al. 2024

Mathematics

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This work investigates a two-way communication retrial queue with synchronous working vacation and a constant retrial policy. During the idle time, a server makes an outgoing call after a random length. The service time of the incoming call and outgoing call obeys exponential distribution with different rates. If the incoming call finds all servers to be unavailable, it may or may not enter orbit. All servers immediately go on vacation simultaneously as soon as they find an empty system after the service finishes. During vacation, the servers can provide a service to those incoming calls, but this is at a lower-speed rate. The stationary probability distribution and the ergodic condition are obtained utilizing the matrix geometric technique. Some system characteristics are developed. Using MATLAB software, the variation in average orbit length, idle ratio, and the average number of servers in different server states is plotted for different values of the incoming/outgoing call rate and retrial rate. We further propose a multi-objective optimization model from which the optimal rate of outgoing calls and optimal vacation rate are explicitly obtained.

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Section: Introductionmentioning

confidence: 99%

Multiserver Retrial Queue with Two-Way Communication and Synchronous Working Vacation

Liu,

Chiou,

Chen

et al. 2024

Mathematics

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“…Retrial queues with outgoing calls (also called two-way communication) have been extensively studied [7][8][9][10][11][12]. Generating functions for single-server retrial queues with outgoing calls were obtained in [7,8], while asymptotic results were investigated in [9,10,12]. However, in these models, the input is either Poisson or Markovian.…”

Section: Introductionmentioning

confidence: 99%

“…As for related asymptotic results, Sakurai and Phung-Duc [10] studied the asymptotic behavior of an M/G/1 retrial queue with outgoing calls under three regimes: (i) heavy incoming calls, (ii) heavy outgoing calls, and (iii) a low retrial rate. Nazarov et al [9,12] studied the low retrial and heavy outgoing call asymptotics for Markovian retrial queues under a random environment. However, in [9,10,12], only the stationary distribution was considered.…”

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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“…All the models in the papers mentioned above are assumed to have a Poisson arrival process, which allows us to obtain explicit expressions for the generating functions of the queue length distribution. Recently, Nazarov et al (2019) considered an MMPP/M/1 retrial queue with two way communication, where ingoing calls arrive at the system according to a Markov modulated Poisson process and the service times of ingoing calls and outgoing calls have different exponential distributions. The authors derived the first order (law of large numbers) and the second order (central limit theorem) asymptotics for the queue length distribution under the condition that the retrial rate is extremely low.…”

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confidence: 99%

Analysis of the waiting time distribution in M/G/1 retrial queues with two way communication

Lee

Kim

2020

Ann Oper Res

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We consider an M/G/1 retrial queue in which there are two types of calls: ingoing calls made by regular customers and outgoing calls made by the server in idle time. The service times of ingoing calls and outgoing calls have different arbitrary distributions. In this paper, we are interested in the analysis of the waiting time distribution. We obtain an equation for the joint transform of the number of ingoing calls in the orbit and the waiting time of an arbitrary ingoing call. Using this result, we can obtain the moments of the waiting time distribution of an arbitrary ingoing call. Keywords Retrial queue • Two way communication • Joint transform 1 IntroductionRetrial queues are queuing systems in which arriving customers who find all servers occupied may retry for service again after a random amount of time. Retrial queues have been widely used to model many problems in telephone systems, call centers, telecommunication networks, computer networks and computer systems, as well as in daily life. For an overview regarding retrial queues, refer to the surveys (

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Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

Cited by 8 publications

References 13 publications

Multiserver Retrial Queue with Two-Way Communication and Synchronous Working Vacation

Multiserver Retrial Queue with Two-Way Communication and Synchronous Working Vacation

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

Analysis of the waiting time distribution in M/G/1 retrial queues with two way communication

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