Study of Preemptive Priority Single-server Queue with Peaked Arrival Flow

Mirtchev, S. T.

doi:10.1109/electronica.2019.8825636

Cited by 3 publications

(3 citation statements)

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“…there does not exist a random variable Q with such that as , where denotes convergence in distribution. That contradicts various conclusions about steady-state performance in [31, 32, 33]. We elaborate on stability and discuss ways to stabilize performance in queues with Pólya arrival processes in Section 6, but our main goal is to obtain a tractable approximation for transient performance in a class of models.…”

Section: Generalized Pólya Point Process With Stationary Increments: -Gppmentioning

confidence: 93%

“…(Index of dispersion) To better understand the impact of the variability as a function of time in an arrival process on the performance of a queueing model, we have shown in [17, 38, 39] that it is often helpful to look at the index of dispersion for counts, which for a -GPP is From this equation, we see that the variability increases without bound as t increases by this measure, consistent with Theorem 3. The index of dispersion for counts is also considered in [31, 32, 33] under the name ‘peakedness’, which is often used to describe traffic variability, but more commonly in a different way; see [28] and references therein.…”

Section: Generalized Pólya Point Process With Stationary Increments: -Gppmentioning

confidence: 99%

“…We conclude this introduction by discussing related work. First, we note that queues with Pólya arrival processes have been previously considered as a way to capture exceptional variability [31, 32, 33]; we discuss that earlier work in Remark 2. Second, the heavy-traffic limit of the queue can be regarded as a Gaussian queue with a net input process that is a -GMP with drift.…”

Section: Introductionmentioning

confidence: 99%

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Queues with path-dependent arrival processes

Fendick

Whitt

2021

J. Appl. Probab.

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We study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

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