两个平行生产线的超额积压概率
Excessive Backlog Probabilities of Two Parallel Queues
摘要：
令$X$是${\mathbb Z}_+^2$上的约束随机游动，增量为$(1,0)$、$(-1,0)$、$(0,1)$和$(0,-1)$ ; $X$表示在到达和服务完成时并行工作的两个队列的长度，其服务和到达间隔时间随到达率$λ_i$和服务率$μ_i$、$i=1,2$呈指数分布;我们假设$λ_i < μ_i$、$i=1,2$，即，假定$X$稳定。不失一般性，我们假设$ρ_1 =λ_1/μ_1 \ge ρ_2 = λ_2/μ_2$。令$τ_n$为$X$第一次命中行$\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$。令$Y$是与$X$相同的随机游走，但仅约束在$\{y \in {\mathbb Z}^2: y(2)=0\}$上，并且其第一个分量的跳跃概率颠倒。设$\partial B =\{y \in {\mathbb Z}^2: y(1) = y(2) \}$和$τ$为$Y$第一次命中$\partial B$。概率$p_n = P_x(τ_n < τ_0)$是由$X$（在系统的第一个忙周期期间共享缓冲器溢出的概率）表示的排队系统的关键性能度量。$X$的稳定性意味着$p_n$在$n$中以指数方式衰减，当过程从$\partial A_n.$开始时，我们表明，对于$x_n= \lfloor nx \rfloor$，$x \in {\mathbb R}_+^2$，$x(1)+x(2) \le 1$，$x(1) > 0$，$P_{(n-x_n(1),x_n(2))}( τ< \infty)$近似于$P_{x_n}(τ_n < τ_0)$，相对误差呈指数消失。设$r = (λ_1 + λ_2)/(μ_1 + μ_2)$ ;对于$r^2 < ρ_2$和$ρ_1 \neq ρ_2$，我们从$Y$的一个特征曲面上的单点和共轭点构造了一类调和函数，用它可以有界的相对误差逼近$P_y(τ< \infty)$。对于$r^2 = ρ_1 ρ_2$，我们得到$P_y(τ< \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-ρ_2}\left( ρ_1^{y(1)} - r^{y(1)-y(2)} ρ_1^{y(2)}\right).$
英文摘要：
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 working in parallel whose service and interarrival times are exponentially distributed with arrival rates $λ_i$ and service rates $μ_i$, $i=1,2$; we assume $λ_i < μ_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $ρ_1 =λ_1/μ_1 \ge ρ_2 = λ_2/μ_2$. Let $τ_n$ be the first time $X$ hits the line $\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$. 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 $τ$ be the first time $Y$ hits $\partial B$. The probability $p_n = P_x(τ_n < τ_0)$ is a key performance measure of the queueing system represented by $X$ (probability of overflow of a shared buffer during system's first busy cycle). Stability of $X$ implies $p_n$ decays exponentially in $n$ when the process starts off $\partial A_n.$ We show that, for $x_n= \lfloor nx \rfloor$, $x \in {\mathbb R}_+^2$, $x(1)+x(2) \le 1$, $x(1) > 0$, $P_{(n-x_n(1),x_n(2))}( τ< \infty)$ approximates $P_{x_n}(τ_n < τ_0)$ with exponentially vanishing relative error. Let $r = (λ_1 + λ_2)/(μ_1 + μ_2)$; for $r^2 < ρ_2$ and $ρ_1 \neq ρ_2$, we construct a class of harmonic functions from single and conjugate points on a characteristic surface of $Y$ with which $P_y(τ< \infty)$ can be approximated with bounded relative error. For $r^2 = ρ_1 ρ_2$, we obtain $P_y(τ< \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-ρ_2}\left( ρ_1^{y(1)} - r^{y(1)-y(2)} ρ_1^{y(2)}\right).$
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