Useful stochastic bounds in time-varying queues with service and patience times having general joint distribution
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Bodas, Shreehari Anand Jacobovic, Royi |
| author_facet | Bodas, Shreehari Anand Jacobovic, Royi |
| contents | Consider a first-come, first-served single server queue with an initial workload $x>0$ and customers who arrive according to an inhomogeneous Poisson process with rate function $λ:[0,\infty)\rightarrow[0,λ_h ]$ for some $λ_h>0$. For each $i\in\mathbb{N}$, let $S_i$ (resp., $Y_i$) be the service (resp., patience) time of the $i$'th customer and assume that $(S_1,Y_1),(S_2,Y_2),\ldots$ is an iid sequence of bivariate random vectors with non-negative coordinates. A customer joins if and only if his patience time is not less than his prospective waiting time (i.e., the left-limit of the workload process at his arrival epoch). Let $τ(x)$ be the first time when the system becomes empty and let $N^*_λ(\cdot)$ be the arrival process of those who join the queue. In the present work we suggest a novel coupling technique which is applied to derive stochastic upper bounds for the functionals: \begin{equation*}
\int_0^{τ(x)}g\circ W_x(t){\rm d}t\ \ \text{and}\ \ \int_0^{τ(x)}g\circ W_x(t){\rm d}N^*_λ(t)\,, \end{equation*} where $W_x(\cdot)$ is the workload process in the queue and $g(\cdot)$ is any lower semi-continuous function. We also demonstrate how to utilise these bounds via some examples under the additional assumption that $λ(\cdot)$ is periodic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12745 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Useful stochastic bounds in time-varying queues with service and patience times having general joint distribution Bodas, Shreehari Anand Jacobovic, Royi Probability 90B22, 60K30, 60K25 Consider a first-come, first-served single server queue with an initial workload $x>0$ and customers who arrive according to an inhomogeneous Poisson process with rate function $λ:[0,\infty)\rightarrow[0,λ_h ]$ for some $λ_h>0$. For each $i\in\mathbb{N}$, let $S_i$ (resp., $Y_i$) be the service (resp., patience) time of the $i$'th customer and assume that $(S_1,Y_1),(S_2,Y_2),\ldots$ is an iid sequence of bivariate random vectors with non-negative coordinates. A customer joins if and only if his patience time is not less than his prospective waiting time (i.e., the left-limit of the workload process at his arrival epoch). Let $τ(x)$ be the first time when the system becomes empty and let $N^*_λ(\cdot)$ be the arrival process of those who join the queue. In the present work we suggest a novel coupling technique which is applied to derive stochastic upper bounds for the functionals: \begin{equation*} \int_0^{τ(x)}g\circ W_x(t){\rm d}t\ \ \text{and}\ \ \int_0^{τ(x)}g\circ W_x(t){\rm d}N^*_λ(t)\,, \end{equation*} where $W_x(\cdot)$ is the workload process in the queue and $g(\cdot)$ is any lower semi-continuous function. We also demonstrate how to utilise these bounds via some examples under the additional assumption that $λ(\cdot)$ is periodic. |
| title | Useful stochastic bounds in time-varying queues with service and patience times having general joint distribution |
| topic | Probability 90B22, 60K30, 60K25 |
| url | https://arxiv.org/abs/2406.12745 |