Descendant set: an efficient approach for the analysis of polling systems

Konheim, Alan G.; Levy, Hanoch; Srinivasan, Mandyam M.

doi:10.1109/tcomm.1994.580233

Cited by 78 publications

(74 citation statements)

References 23 publications

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“…The term E[X k (X k 0 1)] can be calculated using the descendant-sets method discussed in Konheim et al (1994). The idea behind this method is to decompose the variable X k at a polling epoch at queue k as an infinite sum of independent populations of ''descendants'' generated by ''originator customers'' who arrived during all of the past setup times.…”

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confidence: 99%

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When Does Forced Idle Time Improve Performance in Polling Models?

Cooper¹,

Niu²,

Srinivasan³

1998

Management Science

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S arkar and Zangwill (1991) showed by numerical examples that reduction in setup times can, surprisingly, actually increase work in process in some cyclic production systems (that is, reduction in switchover times can increase waiting times in some polling models). We present, for polling models with exhaustive and gated service disciplines, some explicit formulas that provide additional insight and characterization of this anomaly. More specifically, we show that, for both of these models, there exist simple formulas that define for each queue a critical value z* of the mean total setup time z per cycle such that, if z õ z*, then the expected waiting time at that queue will be minimized if the server is forced to idle for a constant length of time z* 0 z every cycle; also, for the symmetric polling model, we give a simple explicit formula for the expected waiting time and the critical value z* that minimizes it.

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confidence: 99%

“…Using the descendant-sets method, it can be shown (Konheim et al 1994;Equation (3.13), p. 1249) that…”

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confidence: 99%

When Does Forced Idle Time Improve Performance in Polling Models?

Cooper¹,

Niu²,

Srinivasan³

1998

Management Science

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“…The expression in Theorem 1 can be explained along the lines of the Descendant Set Approach (DSA) [6]. The DSA considers original customers (originators) and non-original customers, where an original customer arrives during a switch-over period, and a non-original customer arrives during the service of another customer; The children of a customer T arrive during the service of T .…”

Section: Remark 1 (Link With the Descendant Set Approach)mentioning

confidence: 99%

Transient analysis of cycle lengths in cyclic polling systems

Vis¹,

Bekker

Mei

2015

Performance Evaluation

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a b s t r a c tWe consider cyclic polling models with gated or globally gated service, and study the transient behavior of all cycle lengths. Our aim is to analyze the dependency structure between the different cycles, as this is an intrinsic property making polling models challenging to analyze. Moreover, the cycle structure is related to the output of a polling model and the current analysis may be useful to study networks of polling models. In addition, transient performance is of great interest in systems where disruptions or breakdowns may occur, leading to excessive cycle lengths. The time to recover from such events is a primary performance measure. For the analysis we assume that the distribution of the first cycle (globally gated) or N residence times (gated), where N is the number of queues, is known and that the arrivals are Poisson. The joint Laplace-Stieltjes transform (LST) of all x subsequent cycles (globally gated) or all x > N subsequent residence times (gated) is expressed in terms of the LST of the first cycle. From this joint LST, we derive first and second moments and correlation coefficients between different cycles. Finally, a heavy-tailed first cycle length or the heavy-traffic regime provides additional insights into the time-dependent behavior.

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“…This key observation (the starting point of the descendant-set approach in [12]) suggests that we can evaluate G 1 (z 1 , 1, · · · , 1) by considering, recursively, contributions to Y 1 from waiting customers at all queues at each of the past polling epochs, working backward from the reference point. Look backward in time and consider the cth cycle prior to the reference point, where c = 0, 1, · · ·.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…1125 and 1126) to cover disciplines other than exhaustive service. Our analysis in this paper will also be based on these decompositions: We will study the durations of vacations using the concept of "descendant set," as explicated recently by Konheim, Levy, and Srinivasan [12] for the analysis of the nonzero-switchover-times model. (See also Fuhrmann [7].)…”

Section: Introductionmentioning

confidence: 99%

Relating polling models with zero and nonzero switchover times

1995

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We consider a system of N queues served by a single server in cyclic order. Each queue has its own distinct Poisson arrival stream and its own distinct general service-time distribution (asymmetric queues); and each queue has its own distinct distribution of switchover time (the time required for the server to travel from that queue to the next). We consider two versions of this classical polling model: In the first, which we refer to as the zeroswitchover-times model, it is assumed that all switchover times are zero and the server stops traveling whenever the system becomes empty. In the second, which we refer to as the nonzero-switchover-times model, it is assumed that the sum of all switchover times in a cycle is nonzero and the server does not stop traveling when the system is empty. After providing a new analysis for the zero-switchover-times model, we obtain, for a host of service disciplines, transform results that completely characterize the relationship between the waiting times in these two, operationally-different, polling models. These results can be used to derive simple relations that express (all) waiting-time moments in the nonzeroswitchover-times model in terms of those in the zero-switchover-times model. Our results, therefore, generalize corresponding results for the expected waiting times obtained recently by Fuhrmann [8] and Cooper, Niu, and Srinivasan [4].

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Descendant set: an efficient approach for the analysis of polling systems

Cited by 78 publications

References 23 publications

When Does Forced Idle Time Improve Performance in Polling Models?

When Does Forced Idle Time Improve Performance in Polling Models?

Transient analysis of cycle lengths in cyclic polling systems

Relating polling models with zero and nonzero switchover times

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