Summary: Let \(X\) be the constrained random walk on \(\mathbb {Z}_+^2\) with increments \((1, 0)\), \((-1,0)\), \((0, 1)\) and \((0,-1)\); \(X\) represents, at arrivals and service completions, the lengths of two queues (or two stacks in computer science applications) working in parallel whose service and interarrival times are exponentially distributed with arrival rates \(\lambda_i\) and service rates \(\mu_i\), \(i=1,2\); we assume \(\lambda_i < \mu_i\), \(i=1,2\), i.e., \(X\) is assumed stable. Without loss of generality we assume \(\rho_1 =\lambda_1/\mu_1 \geqslant \rho_2 = \lambda_2/\mu_2\). Let \(\tau_n\) be the first time \(X\) hits the line \(\partial A_n = \{x \in \mathbb {Z}^2:x(1)+x(2) = n \} \), i.e., when the sum of the components of \(X\) equals \(n\) for the first time. Let \(Y\) be the same random walk as \(X\) but only constrained on \(\{y \in \mathbb{Z}^2: y(2)=0\}\) and its jump probabilities for the first component reversed. Let \(\partial B =\{y \in \mathbb{Z}^2: y(1) = y(2) \}\) and let \(\tau\) be the first time \(Y\) hits \(\partial B\). The probability \(p_n = P_x(\tau_n < \tau_0)\) is a key performance measure of the queueing system (or the two stacks) represented by \(X\) (if the queues/stacks share a common buffer, then \(p_n\) is the probability that this buffer overflows during the system’s first busy cycle). Stability of the process implies that \(p_n\) decays exponentially in \(n\) when the process starts off the exit boundary \(\partial A_n.\) We show that, for \(x_n= \lfloor nx \rfloor\), \(x \in \mathbb{R}_+^2\), \(x(1)+x(2) \leqslant 1\), \(x(1) > 0\), \(P_{(n-x_n(1),x_n(2))}( \tau < \infty )\) approximates \(P_{x_n}(\tau_n < \tau_0)\) with exponentially vanishing relative error. Let \(r = (\lambda_1 + \lambda_2)/(\mu_1 + \mu_2)\); for \(r^2 < \rho_2\) and \(\rho_1 \ne \rho_2\), we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for \(Y\) with which the probability \(P_y(\tau < \infty )\) can be approximated with bounded relative error. For \(r^2 = \rho_1 \rho_2\), we obtain the exact formula \(P_y(\tau < \infty ) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho_2}\left( \rho_1^{y(1)} - r^{y(1)-y(2)} \rho_1^{y(2)}\right) .\)