Networks with Customers, Signals, and Product Form Solution

Chao, Xiuli

doi:10.1007/978-1-4419-6472-4_5

Cited by 12 publications

(21 citation statements)

References 35 publications

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“…In Appendix A, we show that ( 10) satisfies (8). It is furthermore straightforward to verify that (10) satisfies (9). This guarantees that (10) represents the unique stationary distribution.…”

Section: Derivation Of the Stationary Distributionmentioning

confidence: 80%

“…These networks demonstrate that models with multiple types of customers and general service time distributions could also have a product-form distribution. Since then, further studies have shown that networks with negative arrivals, instantaneous signals and blocking might have a product-form distribution, see [9] for an overview. Figure c) then sketches what happens when the first customer would leave: token t 1 scans the queue for the next compatible customer, and is then claimed by the oldest compatible customer, which is of type c 1 .…”

Section: Introductionmentioning

confidence: 99%

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A Token-Based Central Queue with Order-Independent Service Rates

Ayesta

Bodas

Dorsman

et al. 2022

Operations Research

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Section: Derivation Of the Stationary Distributionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

A Token-Based Central Queue with Order-Independent Service Rates

Ayesta

Bodas

Dorsman

et al. 2022

Operations Research

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“…It was shown in Section 2 that if the condition pΛ 1 = λ 1 p is fulfilled, then, in the stationary mode, the MAP generated by the first node is Poisson with rate λ 1 (see also [9,23]). Because of the PASTA property (Poisson Arrivals See Time Averages) of the Poisson process, the stationary distribution of the second node is equal to the stationary distribution of the Markov chain embedded at times before the customer arrivals.…”

Section: Interacting Markovian Queueing Systemsmentioning

confidence: 99%

“…Suppose that the stationary distribution π(i, k) has the product form. From (23), it follows that if one of the vectors pΛ 1 − λ 1 p, qM 1 − µ 1 q, pP 1 − p, qQ 1 − q is a zero vector then there is another zero vector in this group such that one of the conditions (17)-(20) is fulfilled. It remains to consider the case when all these vectors are non-zero.…”

Section: Interacting Markovian Queueing Systemsmentioning

confidence: 99%

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On Two Interacting Markovian Queueing Systems

Naumov¹,

Gaidamaka

Samouylov

2019

Mathematics

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In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution to be of product form and consider some particular cases of the subsystem interaction for which these conditions can be easily verified.

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Pass-and-swap queues

Comte

Dorsman

2021

Queueing Syst

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Order-independent (OI) queues, introduced by Berezner et al. (Queueing Syst 19(4):345–359, 1995), expanded the family of multi-class queues that are known to have a product-form stationary distribution by allowing for intricate class-dependent service rates. This paper further broadens this family by introducing pass-and-swap (P&S) queues, an extension of OI queues where, upon a service completion, the customer that completes service is not necessarily the one that leaves the system. More precisely, we supplement the OI queue model with an undirected graph on the customer classes, which we call a swapping graph, such that there is an edge between two classes if customers of these classes can be swapped with one another. When a customer completes service, it passes over customers in the remainder of the queue until it finds a customer it can swap positions with, that is, a customer whose class is a neighbor in the graph. In its turn, the customer that is ejected from its position takes the position of the next customer it can be swapped with, and so on. This is repeated until a customer can no longer find another customer to be swapped with; this customer is the one that leaves the queue. After proving that P&S queues have a product-form stationary distribution, we derive a necessary and sufficient stability condition for (open networks of) P&S queues that also applies to OI queues. We then study irreducibility properties of closed networks of P&S queues and derive the corresponding product-form stationary distribution. Lastly, we demonstrate that closed networks of P&S queues can be applied to describe the dynamics of new and existing load-distribution and scheduling protocols in clusters of machines in which jobs have assignment constraints.

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Networks with Customers, Signals, and Product Form Solution

Cited by 12 publications

References 35 publications

A Token-Based Central Queue with Order-Independent Service Rates

A Token-Based Central Queue with Order-Independent Service Rates

On Two Interacting Markovian Queueing Systems

Pass-and-swap queues

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