Tail asymptotics of the queue size distribution in the M/M/m retrial queue

Kim, Jerim; Kim, Jeongsim; Kim, Bara

doi:10.1016/j.cam.2012.03.027

Cited by 22 publications

(17 citation statements)

References 18 publications

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“…Furthermore, substituting these explicit expressions into (20) and (23) and arranging the results, we obtain…”

Section: Corollarymentioning

confidence: 99%

“…Liu et al [18] further extend their analysis to the model with nonpersistent customers. B. Kim and J. Kim [19] and Kim et al [20] refine the tail asymptotic results in Liu and Zhao [17] and Liu et al [18], respectively. Phung-Duc [21] presents a perturbation analysis for a multiserver retrial queue with two types of nonpersistent customers.…”

Section: Introductionmentioning

confidence: 98%

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Multiserver Queue with Guard Channel for Priority and Retrial Customers

Kajiwara

Phung-Duc

2016

International Journal of Stochastic Analysis

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This paper considers a retrial queueing model where a group of guard channels is reserved for priority and retrial customers. Priority and normal customers arrive at the system according to two distinct Poisson processes. Priority customers are accepted if there is an idle channel upon arrival while normal customers are accepted if and only if the number of idle channels is larger than the number of guard channels. Blocked customers (priority or normal) join a virtual orbit and repeat their attempts in a later time. Customers from the orbit (retrial customers) are accepted if there is an idle channel available upon arrival. We formulate the queueing system using a level dependent quasi-birth-and-death (QBD) process. We obtain a Taylor series expansion for the nonzero elements of the rate matrices of the level dependent QBD process. Using the expansion results, we obtain an asymptotic upper bound for the joint stationary distribution of the number of busy channels and that of customers in the orbit. Furthermore, we develop an efficient numerical algorithm to calculate the joint stationary distribution.

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“…Furthermore, substituting these explicit expressions into (20) and (23) and arranging the results, we obtain…”

Section: Corollarymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Multiserver Queue with Guard Channel for Priority and Retrial Customers

Kajiwara

Phung-Duc

2016

International Journal of Stochastic Analysis

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“…Nevertheless, multi-server retrial queues are more flexible and applicable in practice than single server retrial queues. For the multi-server retrial queues, the readers can refer to Neuts and Rao [30], Choi et al [5], Artalejo and Pozo [2], Krishna Kumar and Raja [20], Lin and Ke [27], Kim et al [19], and others. Efrosinin and Sztrik [13] modelled a two-server heterogeneous retrial queue with threshold policy as a QBD process.…”

Section: Introductionmentioning

confidence: 99%

The multi-server retrial system with Bernoulli feedback and starting failures

Yang

2014

International Journal of Computer Mathematics

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In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probability p or to join the retrial orbit again for another service with complementary probability 1 − p. We analyse this queueing system as a quasi-birth-death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model.

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“…The methodology of [13,14] is based on an investigation of the analyticity of generating functions. However, the asymptotic formulae presented in [13,14] still contain some unknown coefficients.We recall that the number of customers in the system and that in the orbit form a leveldependent QBD process whose stationary distribution can be expressed in terms of a sequence of rate matrices [20]. Liu et al [10,11] focus on the asymptotic behavior of the joint stationary distribution.…”

mentioning

confidence: 99%

“…. The methodology of [13,14] is based on an investigation of the analyticity of generating functions. However, the asymptotic formulae presented in [13,14] still contain some unknown coefficients.…”

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confidence: 99%

Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers

Phung-Duc

2015

Applied Mathematics and Computation

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We consider Markovian multiserver retrial queues where a blocked customer has two opportunities for abandonment: at the moment of blocking or at the departure epoch from the orbit. In this queueing system, the number of customers in the system (servers and buffer) and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose stationary distribution is expressed in terms of a sequence of rate matrices. Using a simple perturbation technique and a matrix analytic method, we derive Taylor series expansion for nonzero elements of the rate matrices with respect to the number of customers in the orbit. We also obtain explicit expressions for all the coefficients of the expansion. Furthermore, we derive tail asymptotic formulae for the joint stationary distribution of the number of customers in the system and that in the orbit. Numerical examples reveal that the tail probability of the model with two types of nonpersistent customers is greater than that of the corresponding model with one type of nonpersistent customers.We refer to [12,4,8,15,16,17] for effort to find analytical solutions for M/M/c/c retrial queues with more than two servers by the generating function approach. Kim [12] and deal with the case of three servers while Choi and Kim [4] derive analytical solution for a model with feedback. It should be noted that some technical assumptions are imposed in these papers. Using an alternative approach, Phung-Duc et al. [16] show that the joint stationary distribution is expressed in terms of continued fractions for the cases of c = 3 and 4, without any technical assumption as presented in literature. The same authors in [17] further derive analytical solutions for the joint stationary distribution of state-dependent M/M/c/c + r retrial queues with Bernoulli abandonment, where c + r ≤ 4. Pearce [15] presents an expression for the joint stationary distribution in terms of generalized continued fractions for the M/M/c/c retrial queue with any c. Although, the formulae in [15] do not directly yield a numerical algorithm, this is one of the seminal papers providing the most general analytical results for the model. Recently, asymptotic analysis for multiserver retrial queues has been receiving considerable attention. Liu and Zhao [11] use a censoring technique and a level-dependent QBD approach to derive analytical solutions for the M/M/c/c retrial queues for the cases of c = 1 and 2 and investigate the asymptotic behavior for the stationary distribution of the general case with any c. Using the same approach, Liu et al. [10] extend their study to M/M/c/c retrial queues with one type of nonpersistent customers. Kim et al. [13] derive more detailed asymptotic formulae for the joint stationary distribution of M/M/c/c retrial queues in comparison with those obtained by Liu and Zhao [11]. Furthermore, Kim and Kim [14] refine the asymptotic result obtained by Liu et al. [10]. The methodology of [13,14] is based on an investigation of the analyticity of generating functions. However, the asymptoti...

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Tail asymptotics of the queue size distribution in the M/M/m retrial queue

Cited by 22 publications

References 18 publications

Multiserver Queue with Guard Channel for Priority and Retrial Customers

Multiserver Queue with Guard Channel for Priority and Retrial Customers

The multi-server retrial system with Bernoulli feedback and starting failures

Asymptotic analysis for Markovian queues with two types of nonpersistent retrial customers

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