Tail asymptotics for M/M/c retrial queues with non-persistent customers

Liu, Bin; Wang, Xi; Zhao, Yiqiang Q.

doi:10.1007/s12351-011-0106-6

Cited by 23 publications

(30 citation statements)

References 15 publications

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“…Liu and Zhao [17] use this property to obtain upper and lower asymptotic bounds for the stationary distribution of the fundamental retrial model without guard channels. 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.…”

Section: Introductionmentioning

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: 99%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiserver Queue with Guard Channel for Priority and Retrial Customers

Kajiwara

Phung-Duc

2016

International Journal of Stochastic Analysis

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“…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. To this end, they derive a few essential expansion formulae (up to three terms) for some elements of the rate matrices, which are enough for their purpose.…”

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confidence: 99%

“…Liu and Zhao [11] remark that it seems that there is no unified pattern for the higher order expansions. In this paper, motivated by Liu et al [10,11], we present an exhaustive perturbation analysis for the M/M/c/K retrial queue with two types of nonpersistent customers, which is recently introduced and numerically analyzed in [19]. This model extends those with and without one type of nonpersistent customers [10,11].Using a unified simple approach, we are able to derive Taylor series expansion for all nonzero elements of the rate matrices.…”

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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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“…Kim, Kim and Kim extended the study on the M/G/1 retrial queue in [11] by Kim, Kim and Ko to a M AP/G/1 retrial queue, and obtained tail asymptotics for the queue size distribution in [12]. By adopting matrix-analytic theory and the censoring technique in [19], Liu, Wang and Zhao studied the M/M/c retrial queues with non-persistent customers and obtained tail asymptotics for the joint stationary distribution of the number of retrial customers in the orbit and the number of busy servers.Most of the studies on retrial queues assumed a single type of customers flowing into the system, and references on retrial systems with multi-class customers are quite limited. The model studied by Avrachenkov, Nain and Yechiali in [3] and again in this paper is such a system.…”

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confidence: 99%

Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits

2015

Self Cite

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We revisit a single-server retrial queue with two independent Poisson streams (corresponding to two types of customers) and two orbits. The size of each orbit is infinite. The exponential server (with a rate independent of the type of customers) can hold at most one customer at a time and there is no waiting room. Upon arrival, if a type i customer (i = 1, 2) finds a busy server, it will join the type i orbit. After an exponential time with a constant (retrial) rate µ i , an type i customer attempts to get service. This model has been recently studied by Avrachenkov, Nain and Yechiali [3] by solving a Riemann-Hilbert boundary value problem. One may notice that, this model is not a random walk in the quarter plane. Instead, it can be viewed as a random walk in the quarter plane modulated by a two-state Markov chain, or a two-dimensional quasi-birth-and-death (QBD) process. The special structure of this chain allows us to deal with the fundamental form corresponding to one state of the chain at a time, and therefore it can be studied through a boundary value problem. Inspired by this fact, in this paper, we focus on the tail asymptotic behaviour of the stationary joint probability distribution of the two orbits with either an idle or busy server by using the kernel method, a different one that does not require a full determination of the unknown generating function. To take advantage of existing literature results on the kernel method, we identify a censored random walk, which is an usual walk in the quarter plane. This technique can also be used for other random walks modulated by a finite-state Markov chain with a similar structure property.Keywords: Retrial queue · Random walks in the quarter plane · Random walks in the quarter plane modulated by a finite-state Markov chain · Censored Markov 1 exponential rate for the customers of type i. Such a queueing system could serve as a model for two competing job streams in a carrier sensing multiple access system, and it has an application in a local area computer network (LAN) as explained in [3].Retrial queueing systems have been attracting researchers' attention for many years (e.g., [1,2,5,25] and references therein). Much of the previous work lays the emphasis on performance measures, such as the mean size of the orbit, the average number of the customers in the system, the average waiting time among others. We also notice that stationary tail asymptotic analysis has recently become one of the central research topics for retrial queues due to not only its own importance, but also its applications in approximation and performance bounds. For example, in [24], Shang, Liu and Li proved that the stationary queue length of the M/G/1 retrial queue has a subexponential tail if the queue length of the corresponding M/G/1 queue has a tail of the same type. Kim, Kim and Kim extended the study on the M/G/1 retrial queue in [11] by Kim, Kim and Ko to a M AP/G/1 retrial queue, and obtained tail asymptotics for the queue size distribution in [12]. By adopting matrix-analytic th...

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Tail asymptotics for M/M/c retrial queues with non-persistent customers

Cited by 23 publications

References 15 publications

Multiserver Queue with Guard Channel for Priority and Retrial Customers

Multiserver Queue with Guard Channel for Priority and Retrial Customers

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

Exact tail asymptotics: revisit of a retrial queue with two input streams and two orbits

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