On mean-field (GI/GI/1) queueing model: existence and uniqueness

Abstract

A mean-field extension of the queueing system (GI/GI/1) is considered. The process is constructed as a Markov solution of a martingale problem. Uniqueness in distribution is also established under a slightly different set of assumptions on intensities in comparison with those required for existence.

Appendix

The following celebrated Lemma is stated for the convenience of the reader [2, Theorem 25.6].

Lemma 1

(Skorokhod [16, Ch.1, §6]) Let $\xi ^n_t$ ($t\ge 0$, $n=0,1,\ldots$) be some d-dimensional stochastic processes defined on some probability space, and let the following hold true for any $T>0$, $\epsilon > 0$:

$$\begin{aligned} \lim _{c\rightarrow \infty } \sup _n \sup _{t\le T}{\mathbb {P}}(|\xi ^n_t|>c)= & {} 0, \\ \lim _{h\downarrow 0} \sup _n \sup _{t,s\le T; \, |t-s|\le h}{\mathbb {P}}(|\xi ^n_t - \xi ^n_s|>\epsilon )= & {} 0. \end{aligned}$$

Then, there exists a subsequence $n'\rightarrow \infty$ and a new probability space can be constructed with processes ${\tilde{\xi }}^{n'}_t, \, t\ge 0$, and ${\tilde{\xi }}^{}_t, \, t\ge 0$, such that all finite-dimensional distributions of ${\tilde{\xi }}^{n'}_{\cdot }$ coincide with those of $\xi ^{n'}_{\cdot }$ and such that, for any $\epsilon >0$ and all $t\ge 0$,

$$\begin{aligned} {\mathbb {P}} (|{\tilde{\xi }}^{n'}_t - {\tilde{\xi }}^{}_t|> \epsilon ) \rightarrow 0, \quad n'\rightarrow \infty . \end{aligned}$$