Time-dependent queue length distribution in queues fed by $K$ customers in a finite interval
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910734894497792 |
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| author | Hayashi, Kaito Inoue, Yoshiaki Takine, Tetsuya |
| author_facet | Hayashi, Kaito Inoue, Yoshiaki Takine, Tetsuya |
| contents | We consider queueing models, where customers arrive according to a continuous-time binomial process on a finite interval. In this arrival process, a total of $K$ customers arrive in the finite time interval $[0,T]$, where arrival times of those $K$ customers are independent and identically distributed according to an absolutely continuous distribution defined by its probability density function $f(t)$ on $(0,T]$. To analyze the time-dependent queue length distribution of this model, we introduce the auxiliary model with non-homogeneous Poisson arrivals and show that the time-dependent queue length distribution in the original model is given in terms of the time-dependent joint distribution of the numbers of arrivals and departures in the auxiliary model. Next, we consider a numerical procedure for computing the time-dependent queue length distribution in Markovian models with piecewise constant $f(t)$. A particular feature of our computational procedure is that the truncation error bound can be set as the input. Some numerical examples are also provided. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06452 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Time-dependent queue length distribution in queues fed by $K$ customers in a finite interval Hayashi, Kaito Inoue, Yoshiaki Takine, Tetsuya Probability 60K25, 60J22, 60J27 We consider queueing models, where customers arrive according to a continuous-time binomial process on a finite interval. In this arrival process, a total of $K$ customers arrive in the finite time interval $[0,T]$, where arrival times of those $K$ customers are independent and identically distributed according to an absolutely continuous distribution defined by its probability density function $f(t)$ on $(0,T]$. To analyze the time-dependent queue length distribution of this model, we introduce the auxiliary model with non-homogeneous Poisson arrivals and show that the time-dependent queue length distribution in the original model is given in terms of the time-dependent joint distribution of the numbers of arrivals and departures in the auxiliary model. Next, we consider a numerical procedure for computing the time-dependent queue length distribution in Markovian models with piecewise constant $f(t)$. A particular feature of our computational procedure is that the truncation error bound can be set as the input. Some numerical examples are also provided. |
| title | Time-dependent queue length distribution in queues fed by $K$ customers in a finite interval |
| topic | Probability 60K25, 60J22, 60J27 |
| url | https://arxiv.org/abs/2412.06452 |