First exit time for a discrete-time parallel queue

Excerpt

We consider a discrete-time parallel queue, which is a two-queue network, with batch arrivals and services. At the beginning of the nth time slot, \(A_n\) customers arrive to both queues. Then after this, \(S_n^i\in {\mathbb {N}}\cup \{0\}\) customers can be served in slot n by server i for \(i=1,2\). We assume that \(\{A_n\}_{\{n\in {\mathbb {N}}\cup \{0\}\}}\), \(\{S_n^i\}_{\{n\in {\mathbb {N}}\cup \{0\}\}}\) are independent sequences of i.i.d. random variables with support in \({\mathbb {N}}\cup \{0\}\) and \({\mathbb {E}}A_n<{\mathbb {E}}S_n^i\) for \(i=1,2\). Let \(Q_n^i\) with \(i=1,2\) be the queue length after the service \(S_{n-1}^i\) and before the arrival \(A_n\). Then, it satisfies the Lindley recursion \(Q_{n+1}^i=(Q_n^i+A_n-S_n^i)_+\) for \(i=1,2\) and its stationary law \((Q_\infty ^1, Q_\infty ^2)\) is given by \((\max _{n\in {\mathbb {N}}\cup \{0\}}T_n^1, \max _{n\in {\mathbb {N}}\cup \{0\}} T_n^2)\) where \(T_n^i=\sum _{k=1}^{n} (A_k-S_k^i),\quad T_0^i=0, \quad i=1,2.\) For and \(x,y\in {\mathbb {N}}\cup \{0\}\), we are interested in Note that H(x, y) equals the probability that the first entrance time of the two-dimensional random walk \(\{({\mathscr {A}}_n-{\mathscr {S}}_n^1, {\mathscr {A}}_n-{\mathscr {S}}_n^2), n\in {\mathbb {N}}\cup \{0\}\}\) to the set \([x, \infty ]\times [y, \infty ]\) is finite. In the context of actuarial science, H(x, y) corresponds to ruin probability for a discrete-time two-dimensional insurance risk model where each business line faces common simultaneous losses \(A_k\) and \(S_k^i\), for \(i=1,2\), is the premium derived within the kth period of times (see [1] for similar considerations). …