English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A common assumption when modeling queuing systems is that arrivals behavelike a Poisson process with constant parameter. In practice, however, callarrivals are often observed to be significantly overdispersed. This motivatesthat in this paper we consider a mixed Poisson arrival process with arrivalrates that are resampled every $N^{a}$ time units, where $a> 0$ and $N$ ascaling parameter. In the first part of the paper we analyse the asymptotictail distribution of this doubly stochastic arrival process. That is, for large$N$ and i.i.d. arrival rates $X_1, \dots, X_N$, we focus on the evaluation of$P_N(A)$, the probability that the scaled number of arrivals exceeds $NA$.Relying on elementary techniques, we derive the exact asymptotics of $P_N(A)$:For $a< \frac{1}{3}$ and $a > 3$ we identify (in closed-form) a function$\tilde{P}_N(A)$ such that $P_N(A) / P_N(A)$ tends to $1$ as $N \to \infty$.For $a \in [\frac{1}{3},\frac{1}{2})$ and $a\in [2, 3)$ we find a partialsolution in terms of an asymptotic lower bound. For the special case that the$X_i$s are gamma distributed, we establish the exact asymptotics across all $a>0$. In addition, we set up an asymptotically efficient importance samplingprocedure that produces reliable estimates at low computational cost. Thesecond part of the paper considers an infinite-server queue assumed to be fedby such a mixed Poisson arrival process. Applying a scaling similar to the onein the definition of $P_N(A)$, we focus on the asymptotics of the probabilitythat the number of clients in the system exceeds $NA$. The resultingapproximations can be useful in the context of staffing. Our numericalexperiments show that, astoundingly, the required staffing level can actuallydecrease when service times are more variable.

Rare-event analysis of mixed Poisson random variables, and applications in staffing