Monotone Stochastic Recursions and their Duals

Asmussen, Søren; Sigman, Karl

doi:10.1017/s0269964800004137

Cited by 41 publications

(53 citation statements)

References 19 publications

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“…The main step in this approach is the computation of the success parameter of that distribution. This is again established by results in [5].…”

Section: {R(t)supporting

confidence: 72%

“…Equivalence is then shown using the machinery developed in [5]. The second proof, given in Subsection 2.2, establishes a link between the loss rate and the cycle maximum using an insightful regenerative argument.…”

Section: {R(t)mentioning

confidence: 94%

“…We will also give another proof based on a regenerative argument. Both proof techniques strongly rely on a powerful duality theory for stochastic recursions, which has been developed by Asmussen & Sigman [5], and dates back to Lindley [16], Loynes [17J, and Siegmund [20]. For a recent textbook treatment, see Asmussen [8].…”

Section: Introductionmentioning

confidence: 99%

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On an Equivalence Between Loss Rates and Cycle Maxima in Queues and Dams

Bekker

Zwart²

2005

Prob. Eng. Inf. Sci.

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We consider the loss probability of a customer in a single-server queue with finite buffer and partial rejection and show that it can be identified with the tail distribution of the cycle maximum of the associated infinite-buffer queue. This equivalence is shown to hold for the GI/G/1 queue and for dams with state-dependent release rates. To prove this equivalence, we use a duality for stochastically monotone recursions, developed by Asmussen and Sigman (1996). As an application, we obtain several exact and asymptotic results for the loss probability and extend Takács' formula for the cycle maximum in the M/G/1 queue to dams with variable release rate.

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“…The main step in this approach is the computation of the success parameter of that distribution. This is again established by results in [5].…”

Section: {R(t)supporting

confidence: 72%

Section: {R(t)mentioning

confidence: 94%

See 1 more Smart Citation

On an Equivalence Between Loss Rates and Cycle Maxima in Queues and Dams

Bekker

Zwart²

2005

Prob. Eng. Inf. Sci.

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“…Another transform via stochastic monotonicity was introduced by Asmussen and Sigman [4]. We call this the cone dual, since we will imbed this dual in a more general approach using cones with unique integral representations.…”

Section: The Cone Dualmentioning

confidence: 99%

“…Asmussen and Sigman introduced a dual GWP, (V n ) n∈N 0 , in [4] using the following formula for the transition probabilities:…”

Section: F C Klebaner Et Almentioning

confidence: 99%

Transformations of Galton-Watson processes and linear fractional reproduction

2007

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By establishing general relationships between branching transformations (HarrisSevastyanov, Lamperti-Ney, time reversals, and Asmussen-Sigman) and Markov chain transforms (Doob's h-transform, time reversal, and the cone dual), we discover a deeper connection between these transformations with harmonic functions and invariant measures for the process itself and its space-time process. We give a classification of the duals into Doob's h-transform, pathwise time reversal, and cone reversal. Explicit results are obtained for the linear fractional offspring distribution. Remarkably, for this case, all reversals turn out to be a Galton-Watson process with a dual reproduction law and eternal particle or some kind of immigration. In particular, we generalize a result of Klebaner and Sagitov (2002) in which only a geometric offspring distribution was considered. A new graphical representation in terms of an associated simple random walk on N 2 allows for illuminating picture proofs of our main results concerning transformations of the linear fractional Galton-Watson process.

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Queueing Theory

Sigman

2004

Encyclopedia of Actuarial Science

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The mathematical theory of congestion is associated with delays while waiting in a line or queue for service in a system . Examples of such systems include banks, post offices, and supermarkets as well as telecommunication systems involving telephones, computer networks, the Internet/the World Wide Web, inventory, and manufacturing processes. The early development of the field was motivated by telephone systems and the ways in which they could be made to operate more efficiently. Much of the theory relies heavily on the use of probability theory and stochastic processes , and as such is widely viewed as a subfield of stochastic modeling and applied probability. There is also significant interplay with other fields such as scheduling theory, inventory theory, and insurance risk theory; it is a central part of operations research .

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Monotone Stochastic Recursions and their Duals

Cited by 41 publications

References 19 publications

On an Equivalence Between Loss Rates and Cycle Maxima in Queues and Dams

On an Equivalence Between Loss Rates and Cycle Maxima in Queues and Dams

Transformations of Galton-Watson processes and linear fractional reproduction

Queueing Theory

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