Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

Hanschke, Thomas

doi:10.1017/s0021900200031120

Cited by 13 publications

(15 citation statements)

References 6 publications

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“…. , l(N)} whose infinitesimal generator is given by (9). Phung-Duc [19] presents an algorithm with the computational complexity of O(K) for solving this Markov chain.…”

Section: Traffic Intensity (ρ * )mentioning

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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“…. , l(N)} whose infinitesimal generator is given by (9). Phung-Duc [19] presents an algorithm with the computational complexity of O(K) for solving this Markov chain.…”

Section: Traffic Intensity (ρ * )mentioning

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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“…From 17, we can express ν 1 π 1 (z) in terms of π 0 (z). Substituting this expression into (16) with K = 1 yields a differential equation for π 0 (z) as follows.…”

Section: Analytical Solution For Single Server Casementioning

confidence: 99%

“…An explicit solution for the joint stationary distribution of the numbers of busy servers and customers in the orbit is obtained only for the M/M/1/1 retrial queue [14] without nonpersistent customer. The joint stationary distribution for M/M/1/1 retrial queue with nonpersistent customer and M/M/2/2 retrial queue without nonpersistent customers are expressed in terms of confluent hypergeometric functions and hypergeometric functions, respectively [14,16,24].…”

Section: Introductionmentioning

confidence: 99%

Multiserver Retrial Queues With Two Types of Nonpersistent Customers

Phung-Duc

2014

Asia Pac. J. Oper. Res.

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We consider M/M/c/K(K ≥ c ≥ 1) retrial queues with two types of nonpersistent customers, which are motivated from modeling of service systems such as call centers. Arriving customers that see the system fully occupied either join the orbit or abandon receiving service forever. After an exponentially distributed time in the orbit, each customer either abandons the system forever or retries to occupy a server again. For the case of K = c = 1, we present an analytical solution for the generating functions in terms of confluent hypegeometric functions. In the general case, the number of customers in the system and that in the orbit form a level-dependent quasi-birth-and-death (QBD) process whose structure is sparse. Based on this sparse structure, we develop a numerically stable algorithm to compute the joint stationary distribution. We show that the computational complexity of the algorithm is linear to the capacity of the queue. Furthermore, we present a simple fixed point approximation model for the case where the algorithm is time consuming. Numerical results show various insights into the system behavior.

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“…. are given in the last section of Hanschke (1987). in which the singularities of the system are the eigenvalues 0, which is irregular, andρ, which is regular.…”

Section: Two Serversmentioning

confidence: 99%

On Multiserver Retrial Queues: History, Okubo-Type Hypergeometric Systems and Matrix Continued-Fractions

Avram

Matei

Zhao

2014

Asia Pac. J. Oper. Res.

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We study two families of QBD processes with linear rates: (A) the multiserver retrial queue and its easier relative; and (B) the multiserver M/M/∞ Markov modulated queue.The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a "minimal/non-dominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods.Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems.We provide a differential system for our generating function that unifies problems (A) and (B), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest.The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: 1) the irregular singularity at 0; 2) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and 3) the point 1, whose Taylor series coefficients are the factorial moments.

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Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts

Abstract: In this paper we study the M/M/2/2 queue with repeated attempts. It is shown that the part generating functions of the steady state probabilities can be expressed in of generalized hypergeometric unctions.

Cited by 13 publications

References 6 publications

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

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

Multiserver Retrial Queues With Two Types of Nonpersistent Customers

On Multiserver Retrial Queues: History, Okubo-Type Hypergeometric Systems and Matrix Continued-Fractions

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