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Journal ArticleDOI

Performance of large-scale polling systems with branching-type and limited service

01 Sep 2019-Performance Evaluation (Elsevier)-Vol. 133, pp 1-24
TL;DR: It is found that the behavior of individual queues simplifies to that of a discrete-time bulk service queue in the limit, so that the marginal queue length and waiting-time distributions become considerably easier to analyze.
About: This article is published in Performance Evaluation.The article was published on 2019-09-01 and is currently open access. It has received 8 citations till now. The article focuses on the topics: Queue & Polling.

Summary (5 min read)

1. Introduction

  • In the present paper the authors investigate the performance of large-scale symmetric polling systems.
  • The particular application that motivated the present study is the so-called BACnet (Building Automation and Control networks) protocol, which is specifically designed to meet the communication needs of the latter building automation and control systems [4].
  • Another well-known communication protocol that uses token-passing for its medium access control is the Token Ring mechanism [5].
  • Moreover, the authors assume that the server visits the queues in cyclic order, requiring some switch-over time to move from one queue to the next.

2. Model description and preliminaries

  • The server visits the queues in a cyclic non-idling manner.
  • The switch-over times of the server for moving between Qi and the next queue are i.i.d. random variables with first moment si, second moment s (2) i and LST Si(·).
  • All interarrival, service and switch-over times are assumed to be independent.
  • Additionally, it allows that multiple customers simultaneously join the system, after a customer has received service.
  • Again, one can think of communication networks where some packets may require a response from several nodes.

3. Branching-type service disciplines

  • In this section the authors will consider polling systems that employ a branching-type service discipline at all queues.
  • Subsequently, the authors use that expression to study the large-n asymptotics of the cycle time variance (cf. Proposition 3).
  • It does permit that a customer immediately returns to the queue where it has just received service.
  • Performing this procedure n times and deconditioning gives the desired result.

3.1. Queue length covariance

  • For the most common service disciplines satisfying Properties 1 and 2 including (binomial-) gated and (binomial-) exhaustive, the authors have φ = y+ψ andψ = µΛ/n for y := E[M1,1] (which is zero for standard gated and exhaustive service) and µ := − dduθ (u)|u=0.
  • Writing li,i = l1,1 + b(i − 1) + c(i − 1)2 and substituting then allows us to solve the resulting equations and determine the remaining unknowns, confirming the linear and quadratic relations alluded to above.

3.2. Station-time covariance

  • This will then allow us to give an explicit expression for Var[C] (cf. Proposition 2).
  • As the authors will see in the next section however, r1,i will be positive for traditional branching-type service policies, such as the binomial-gated and binomial-exhaustive service disciplines, as n grows large.
  • Finally, the authors find an explicit formula for the variance of the cycle time.
  • For some service disciplines, for example exhaustive service (cf. [13,14]), it is more natural to consider C∗, the time between successive visit completions at Q1.
  • Obviously it holds that E[C] = E[C∗], but higher moments generally differ.

3.3. Many-queue asymptotics

  • Wewill now apply the results of the previous subsections to analyze the limiting behavior of symmetric polling systems satisfying Properties 1 and 2 as the number of queues grows large.
  • The main result of this subsection is an explicit expression for the limit of the scaled cycle time variance nVar[C/n] (cf. Proposition 3).
  • Consider a system in which the queues get served according to the usual gated service discipline, but whenever a customer completes its service, Y new customers join the same queue at which the customer received its service, where Y is a non-negative integer-valued random variable with PGF GY (z).
  • Finally, the authors obtain the following result for the scaled variance of the cycle time.
  • It is noteworthy that for some service disciplines, the mean waiting time can be expressed in terms of the mean residual cycle time.

3.4. Binomial service

  • As an example, the authors will examine the binomial-gated and binomial-exhaustive service disciplines, which are, respectively, generalizations of the gated and exhaustive disciplines.
  • Besides giving an explicit expression for the limiting value of the scaled cycle time variance, Eq. (31) has another interesting consequence.
  • Ai, (34) where the two terms on the right-hand side are independent, the (possibly random) function f (·) is determined by the actual service discipline, and Ai is a Poisson distributed random variable with parameter Λs1−ρ .
  • Often, such a model is much easier to analyze than the pre-limit polling system.
  • Note that the same limit holds for binomial-exhaustive service, since the probability that a customer joins a queue while the server is serving that queue, is negligible for large n. Examining Eqs. (31) and (35), the authors see that when choosing p, there is actually a trade-off between a small variance of the cycle time and short mean waiting times.

4. General non-idling service disciplines

  • The authors will now briefly consider general non-idling service disciplines, which are not necessarily of the branching-type.
  • The authors present two main results in this section.
  • Consider, for example, a service discipline where no customers join the queue being served, except when zero customers have arrived in all the other queues.
  • For k-limited service, Cov[Le, I] ≥ 0 because Le > 0 automatically implies that the maximum number of customers has been served in the preceding visit period, which will immediately result in a longer intervisit period due to the large number of arrivals in the other queues.

4.1. Limited service

  • Consider now the well-known k-limited service discipline.
  • A major benefit of the k-limited service discipline is that it in a way bounds the cycle time, which can be of vital importance for deadline-critical applications.
  • In Section 6.3 the authors conduct numerical experiments that confirm this relation in the limit.
  • Furthermore, it is easily seen that during a cycle at most (n+ ℓ)k customers will be served: the server then serves (ℓ+ 1)k customers at some queue and k customers at the following n − 1 queues.
  • In the remainder of this section the authors will present an approximate analysis of the queue length distribution under the flexible k-limited service policy, focusing their attention on large systems with many queues.

5.1. Performance for large systems

  • Therefore, the authors can argue that, under the flexible k-limited service discipline, each individual queue asymptotically behaves as an M/D/1 queue with bulk service and varying capacity.
  • In fact, the numerical examples in Section 6 seem to confirm that, in the limit, the steady-state queue-length distribution of the actual polling model converges to the steady-state distribution of this Markov chain.
  • Here the probabilities pi determining the function φk(ℓ+1)(z) and the zeros zi remain to be found.

5.1.1. The case ℓ = 1

  • Supported by their analysis in Section 3, the authors assume asymptotic independence between the queue lengths of two neighboring queues.
  • It is possible to determine them using the following iterative approach.
  • Using these estimates, one can determine the roots zi of the denominator of (45), giving a new estimate for the generating function Π (z) and the probabilities πi.
  • Implementing this approach gives good and fast results for reasonable values of k (k < 20) and ν not too close to k, converging after only a few iterations.
  • In the next section the authors will investigate how well the actual queue length distribution is approximated using this approach.

5.1.2. The case ℓ > 1

  • Let X−3, X−2 and X−1 denote the number of customers the server found at the last three queues that it visited and let K−3, K−2 and K−1 denote the K values at those queues during those visits.
  • Note that in order to avoid dependencies between queue lengths and K ’s of neighboring queues and to simplify the analysis, one could also consider a randomized flexible k-limited service policy.
  • By randomized the authors mean that instead of looking at how many customers were served during the last ℓ visits, the service limit K (i, j) is determined by how many customers were served at ℓ random visits during the last cycle.
  • Such a policy admits an easier analysis, since the probabilities pi can be expressed more easily in terms of the πj.

6. Numerical results

  • This section contains the results of extensive numerical experiments.
  • In Section 6.1 the authors evaluate how well the limiting many-queue results of the previous sections for the marginal queue length distribution can serve as approximations for polling systems of finite size.
  • The authors focus on the binomial-gated, k-limited and flexible k-limited service disciplines.
  • In Section 6.2 the authors compare these disciplines with respect to the mean queue length and the cycle time variance.
  • Finally, in Section 6.3, the authors numerically study the limiting behavior of Cov[Le, I] for these disciplines.

6.1. Marginal queue length distribution convergence

  • The authors first consider the binomial-gated service discipline by investigating how well Eq. (33) approximates the steadystate queue length at the start of a server visit for finite n.
  • The authors assume that service and switch-over times are exponentially distributed with mean 2/3 and 1, respectively.
  • The authors find that even for small n the limiting distribution given by Eq. (33) already approximates the steady-state queue length distribution well.
  • Additionally, the authors see that as p becomes smaller, the approximation tends to become more accurate.
  • For n = 50 the differences have almost disappeared and the limiting probabilities would be a very good approximation for the true distribution.

6.2. Comparison of service disciplines

  • In the previous subsection, the authors used simulations to examine how well asymptotic results can be used to approximate the performance of polling systems of finite size.
  • This trade-off is particularly important for deadline-critical applications, where upon returning to a queue, the server might find a high-priority customer that needs to be served immediately.
  • The last row of both tables corresponds to the exact asymptotic results from Section 3.4.
  • Comparing with the results in Tables 4 and 5, the authors find that the k-limited service discipline is better at reducing the cycle time variance than the binomial-gated service discipline, without increasing the mean queue length too much when compared with the gated service discipline, i.e. p = 1.
  • If the authors now compare the results for the flexible k-limited service discipline with k = 4 with the results for the k-limited service discipline without flexibility and the gated service discipline, they find that the flexible k-limited service discipline is able to achieve the best of both worlds.

7. Conclusion

  • In the present paper the authors have provided a detailed analysis of the variance of the cycle time, the covariance of visit times and the covariance of queue lengths in symmetric polling systems where the total number of queues grows large.
  • While these results are of theoretical merit in their own right, they also yield significant novel insights from a practical perspective that have important engineering implications.
  • Wireless communication bandwidth is also shared by increasingly large numbers of users and devices with quite demanding latency and reliability requirements, especially in Internet-of-Things applications and smart control environments.
  • As these two examples illustrate, in many of these scenarios the number of contending users is large, with each individual user only requiring a relatively small portion of the overall capacity, but possibly involving quite stringent delay requirements.
  • Numerical experiments indicate that the discrete-time bulk service queue in fact provides a surprisingly accurate approximation even for a fairly moderate number of queues.

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Citations
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Journal ArticleDOI
01 Oct 2018-Top
TL;DR: 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.
Abstract: 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

34 citations


Cites background from "Performance of large-scale polling ..."

  • ...In Meyfroyt et al. (2018), another type of scaling with a large number of queues is studied....

    [...]

  • ...Motivated by token passing algorithms for communication channels with medium access control and a large number of nodes, Meyfroyt et al. (2018) consider the following scenario: the number of queues grows large, while the total arrival rate is kept fixed and the individual switchover time and…...

    [...]

01 Jan 1991
TL;DR: If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration.
Abstract: The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.

17 citations

Journal ArticleDOI
07 Jan 2021
TL;DR: A review of papers on stochastic polling systems published in 2007–2020 and an investigation of how the customer service order within a queue affects the performance characteristics are presented.
Abstract: The paper presents a review of papers on stochastic polling systems published in 2007–2020. Due to the applicability of stochastic polling models, the researchers face new and more complicated polling models. Stochastic polling models are effectively used for performance evaluation, design and optimization of telecommunication systems and networks, transport systems and road management systems, traffic, production systems and inventory management systems. In the review, we separately discuss the results for two-queue systems as a special case of polling systems. Then we discuss new and already known methods for polling system analysis including the mean value analysis and its application to systems with heavy load to approximate the performance characteristics. We also present the results concerning the specifics in polling models: a polling order, service disciplines, methods to queue or to group arriving customers, and a feedback in polling systems. The new direction in the polling system models is an investigation of how the customer service order within a queue affects the performance characteristics. The results on polling systems with correlated arrivals (MAP, BMAP, and the group Poisson arrivals simultaneously to all queues) are also considered. We briefly discuss the results on multi-server, non-discrete polling systems and application of polling models in various fields.

15 citations

Book ChapterDOI
23 Sep 2019
TL;DR: Using the probability generating function method, the system of the linear algebraic equations for the first and second order moments of the number of packets at an abonent station is obtained which allows calculating the mean sojourn time and other performance characteristics.
Abstract: In the paper, we consider a stochastic polling system with a cyclic adaptive polling order adequately modelling the behavior of the broadband wireless networks with centralized control. Using the probability generating function method, we obtain the system of the linear algebraic equations for the first and second order moments of the number of packets at an abonent station which allows calculating the mean sojourn time and other performance characteristics. The obtained results are illustrated by numerical estimation of the performance of the IEEE802.11 broadband wireless networks with centralized control.

2 citations

Proceedings ArticleDOI
13 May 2022
TL;DR: This article combines the idea of parallel optimization with the polling system, changing the traditional pipeline scheduling method, and optimizing the transfer process and service process in the voting system in parallel, which improves the efficiency of polling.
Abstract: Aiming at the problems of long operation cycle and low system operation efficiency in the two-level priority exhaustive polling service system, this paper proposed a two-level priority polling system based on parallel exhaustive services mode. First of all, this article combines the idea of parallel optimization with the polling system, changing the traditional pipeline scheduling method, and optimizing the transfer process and service process in the polling system in parallel, which improves the efficiency of polling. Then, through the mathematical method of probability generating function and Markov chain, the two-level exhaustive polling service system is analyzed mathematically, and the first-order partial derivative of the probability generating function of the system state variables is obtained to obtain the equations of its characteristic parameters, which are solved simultaneously get the complete mathematical expression of the average queue length. Finally, the computer simulation experiment analysis of the system is carried out, and the experimental results are consistent with the theory, which proves the reliability of the theoretical analysis.
References
More filters
Journal ArticleDOI
TL;DR: In this article, the authors considered a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time and showed that the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables.
Abstract: This paper considers a class of M/G/1 queueing models with a server who is unavailable for occasional intervals of time. As has been noted by other researchers, for several specific models of this type, the stationary number of customers present in the system at a random point in time is distributed as the sum of two or more independent random variables, one of which is the stationary number of customers present in the standard M/G/1 queue i.e., the server is always available at a random point in time. In this paper we demonstrate that this type of decomposition holds, in fact, for a very general class of M/G/1 queueing models. The arguments employed are both direct and intuitive. In the course of this work, moreover, we obtain two new results that can lead to remarkable simplifications when solving complex M/G/1 queueing models.

664 citations

Journal ArticleDOI
31 May 2005
TL;DR: The task of building automation and the systems and communications infrastructure necessary to address it is introduced and an overview of relevant standards is given, including BACnet, LonWorks and EIB/KNX as open systems of key significance in the building automation domain.
Abstract: Building automation systems (BAS) provide automatic control of the conditions of indoor environments. The historical root and still core domain of BAS is the automation of heating, ventilation and air-conditioning systems in large functional buildings. Their primary goal is to realize significant savings in energy and reduce cost. Yet the reach of BAS has extended to include information from all kinds of building systems, working toward the goal of "intelligent buildings". Since these systems are diverse by tradition, integration issues are of particular importance. When compared with the field of industrial automation, building automation exhibits specific, differing characteristics. The present paper introduces the task of building automation and the systems and communications infrastructure necessary to address it. Basic requirements are covered as well as standard application models and typical services. An overview of relevant standards is given, including BACnet, LonWorks and EIB/KNX as open systems of key significance in the building automation domain.

542 citations

Journal ArticleDOI
TL;DR: In this paper, the joint queue length process at polling instants of a fixed queue is shown to be a multitype branching process (MTBP) with immigration, and sufficient conditions for ergodicity and moment calculations are given.
Abstract: The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.

226 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered single-server, multi-queue systems with cyclic service and derived a pseudo-conservation law for a weighted sum of the mean waiting times at the various queues.
Abstract: This paper considers single-server, multi-queue systems with cyclic service. Non-zero switch-over times of the server between consecutive queues are assumed. A stochastic decomposition for the amount of work in such systems is obtained. This decomposition allows a short derivation of a ‘pseudo-conservation law' for a weighted sum of the mean waiting times at the various queues. Thus several recently proved conservation laws are generalised and explained.

225 citations

Frequently Asked Questions (2)
Q1. What have the authors contributed in "Performance of large-scale polling systems with branching- type and limited service" ?

Motivated by emerging Internet-of-Things ( IoT ) applications and smart building environments, the authors analyze the performance of large-scale symmetric polling systems where the number of queues grows large. The authors consider a scenario in which the total arrival rate is kept fixed and the individual switch-over time and service time distributions remain the same. The authors show that for most traditional service policies the scaled cycle times converge to a deterministic value in the limit, which in turn implies that the queue lengths at the various nodes become asymptotically independent. Using these insights, the authors find that the behavior of individual queues simplifies to that of a discretetime bulk service queue in the limit, so that the marginal queue length and waiting-time distributions become considerably easier to analyze. Additionally, the authors propose a new flexible k-limited service discipline aimed at striking a good balance between short mean queue lengths and predictable cycle times for deadline-critical applications. 

Proving a rigorous result along these lines remains as an interesting topic for further research. These novel findings indicate that a complicated high-dimensional optimization problem, such as the selection of service limits, can be decoupled into a collection of one-dimensional optimization problems.