The shorter queue polling model

Adan, I.J.B.F.; Boxma, Onno; Kapodistria, Stella; Kulkarni, Vidyadhar G.

doi:10.1007/s10479-013-1495-0

Cited by 22 publications

(42 citation statements)

References 45 publications

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“…, which is both a pole and a branch point. The following theorem proves that there are three types of detailed asymptotic properties of π (1) Theorem 3: The behavior of π (1) 1 (x) at the dominant singularity is given as…”

Section: If μ 2 > λ 2 and Xmentioning

confidence: 97%

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A Two-Class Retrial System With Coupled Orbit Queues

Dimitriou

2017

Prob. Eng. Inf. Sci.

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We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, typeicustomer,i=1, 2, is routed to a separate typeiorbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queueitries to re-dispatch a blocked customer of typeito the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μito$\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

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Section: If μ 2 > λ 2 and Xmentioning

confidence: 97%

“…(y) + c(1) (x, y)π(1) (0, 0), (3.17) at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0269964816000528 Downloaded from https://www.cambridge.org/core.…”

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confidence: 99%

A Two-Class Retrial System With Coupled Orbit Queues

Dimitriou

2017

Prob. Eng. Inf. Sci.

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“…Remark 9.1 Next to queue-length-dependent server behavior, one could also allow queue-length-dependent customer behavior. In Adan et al (2016), a two-queue polling model is analyzed in which customers join the shortest queue. The joint queue-length distribution is determined both via the compensation approach and via reduction to a Riemann-Hilbert boundary value problem.…”

Section: Queue-length-dependent Server Behaviormentioning

confidence: 99%

Polling: past, present, and perspective

Borst

Boxma

2018

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This is a survey on polling systems, focussing on the basic single-server multi-queue polling system in which the server visits the queues in cyclic order. The main goals of the paper are: (i) to discuss a number of the key methodologies in analyzing polling models; (ii) to give an overview of recent polling developments; and (iii) to present a number of challenging open problems. Assumption 7The service order within each queue is First-Come First-Served (FCFS). This assumption was almost universally made in the polling literature until the work of Wierman et al. (2007). In Sect. 7, we will discuss non-FCFS service orders.Assumption 8 The times to switch from Q i , i = 1, 2, . . . , n, to the next queue are assumed to be i.i.d. random variables, generically denoted by S i , with distribution S i (·) and LST σ i (·). All switchover times are assumed to be independent of each other and of the interarrival and service times. When the switchover times between successive queues are all zero, a special situation arises. If the system has become empty after a visit to, say, Q i in the case of zero switchover times, then the server is assumed to visit queues Q i+1 , . . . , Q n (which now takes zero time) and stay in front of Q 1 (see Sect. 4). In the case of non-zero switchover times, the server is assumed to keep switching in an empty system. Assumption 9As soon as a customer has been served, it leaves the system. At some places, we briefly mention the case of customer routing; a served customer might rejoin the same queue, or join another one.

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“…The compensation approach is developed by Adan et al in a series of papers [1,2,5] and aims at a direct solution for the sub-class of two-dimensional random walks on the lattice of the first quadrant that obey the conditions for meromorphicity. The compensation approach can also be effectively used in cases that the random walk at hand does not satisfy the aforementioned conditions, but the equilibrium distribution can still be written in the form of series of product-forms [3,4,34]. This is due to the fact that this approach exploits the structure of the equilibrium equations in the interior of the quarter plane by imposing that linear (finite or infinite) combinations of product-forms satisfy them.…”

Section: Compensation Approachmentioning

confidence: 99%

Matrix geometric approach for random walks: Stability condition and equilibrium distribution

Kapodistria

Palmowski

2017

Stochastic Models

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In this paper, we analyse a sub-class of two-dimensional homogeneous nearest neighbour (simple) random walk restricted on the lattice using the matrix geometric approach. In particular, we first present an alternative approach for the calculation of the stability condition, extending the result of Neuts drift conditions [30] and connecting it with the result of Fayolle et al. which is based on Lyapunov functions [13]. Furthermore, we consider the sub-class of random walks with equilibrium distributions given as series of product-forms and, for this class of random walks, we calculate the eigenvalues and the corresponding eigenvectors of the infinite matrix R appearing in the matrix geometric approach. This result is obtained by connecting and extending three existing approaches available for such an analysis: the matrix geometric approach, the compensation approach and the boundary value problem method. In this paper, we also present the spectral properties of the infinite matrix R.

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The shorter queue polling model

Cited by 22 publications

References 45 publications

A Two-Class Retrial System With Coupled Orbit Queues

A Two-Class Retrial System With Coupled Orbit Queues

Polling: past, present, and perspective

Matrix geometric approach for random walks: Stability condition and equilibrium distribution

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