Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue

Kim, Jerim; Kim, Bara; Ko, Sung-Seok

doi:10.1017/s0021900200003788

Cited by 12 publications

(12 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof is similar to that of Lemma 1 of [8]. The function f (z) = N (λ − λz), |z| < 1 + γ/λ has a simple zero at z = 1 since f (1) < 0.…”

Section: Light Tailed Approximationmentioning

confidence: 73%

Light tailed asymptotics in an unreliable $M/G/1$ retrial queue

Aïssani

2013

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

Abstract. We consider the standard unreliable M/G/1 retrial queuing system with active and passive breakdowns. The explicit expressions of the probability generating functions of distribution of server state and orbit size are well known from early works. However, some problems particularly related to Cybernetic and Artificial Intellect need to save computational effort. So, we give here another look to solve this problem more simply, but under some light tailed assumptions. IntroductionQueueing Systems with breakdowns have been the subject of many investigations since they are interesting modeling tools in many modern systems: production, and telecommunication, computer science, finance and so on. Such studies are interesting since they create a gap between Queueing Theory and Reliability theory in the context of probabilistic framework, which is one of the logics for uncertainty problems [7], [6].In recent years, particular attention has been devoted to Retrial Queueing Systems for their interest in many modern systems (call centers, mobile systems, CSMA/CD protocol with star topology, random access protocols. . . ); see, for example, [11] and the references in the didactical book [9], or bibliographical surveys of [3,4].Early works were concerned with analytical or algorithmic resolution of the underlying problems; see, for example, [6,5]. Asymptotic methods are interesting in the sense that we can save computational effort for the resolution of the problem under study; see [2]. We refer also to some alternative approaches to study more qualitative properties such as stability (see [10]) or monotonicity and comparability methods; see [12]. The last paper gives also a short survey about recent advances in Unreliable Retrial Queues.In this paper, we consider the approximation of the queue size of the unreliable M/G/1 retrial queue under light tailed assumptions. In the next section we describe the mathematical model and in section 3 we provide some preliminary results. The light tailed approximation is given in section 4.2010 Mathematics Subject Classification. Primary 60K25; Secondary 68M20, 90B22.

show abstract

“…The proof is similar to that of Lemma 1 of [8]. The function f (z) = N (λ − λz), |z| < 1 + γ/λ has a simple zero at z = 1 since f (1) < 0.…”

Section: Light Tailed Approximationmentioning

confidence: 73%

Light tailed asymptotics in an unreliable $M/G/1$ retrial queue

Aïssani

2013

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

show abstract

“…More precisely, the analytic function N (λ − λz) − z, |z| ≤ 1 + γ/λ has simple zero at 1 and σ. Furthermore, it has no other zeros on {z ∈ C : |z| ≤ σ}, whereσ = σ(λ, μ, c, F (•), H(•))is the unique root of the equation(8)N(λ − λz) − z = σ, 1 < σ < 1 + γ λ .The proof is similar to that of Lemma 1 of[8].The function f (z) = N (λ − λz), |z| < 1 + γ/λ has a simple zero at z = 1 since f (1) < 0. Since f (z) is strictly convex in z ∈ (0, 1 + γ/λ) and f (1) = f (σ) = 0, then f (z) < 0 for 1 < z < σ.…”

mentioning

confidence: 73%

Untitled

2012

Theor. Probability and Math. Statist.

View full text Add to dashboard Cite

We consider the standard unreliable M/G/1 retrial queuing system with active and passive breakdowns. The explicit expressions of the probability generating functions of distribution of server state and orbit size are well known from early works. However, some problems particularly related to Cybernetic and Artificial Intellect need to save computational effort. So, we give here another look to solve this problem more simply, but under some light tailed assumptions.

show abstract

“…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 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.…”

Section: Introductionmentioning

confidence: 99%

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

2015

View full text Add to dashboard Cite

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...

show abstract

Tail Asymptotics for the Queue Size Distribution in an M/G/1 Retrial Queue

Cited by 12 publications

References 14 publications

Light tailed asymptotics in an unreliable $M/G/1$ retrial queue

Light tailed asymptotics in an unreliable $M/G/1$ retrial queue

Untitled

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

Contact Info

Product

Resources

About