Subexponential asymptotics of a Markov-modulated random walk with queueing applications
Subexponential asymptotics of a Markov-modulated random walk with queueing applications
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DOI：
10.1239/jap/1032192851
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发表时间：
1998-06
影响因子：
1
通讯作者：
P. Jelenkovic;A. Lazar
中科院分区：
文献类型：
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作者：
P. Jelenkovic;A. Lazar
设{(Xn，Jn)}是ℝxE(E是有限的)上的平稳马尔可夫调制随机游动，由其概率转移矩阵测度F={Fij}，Fij(B)=ℙ[X1∈B，J1=j|J0=i]，B∈B(ℝ)，I，j∈E定义，如果Fij([x，∞))/(1-H(X))→Wij∈[0，∞)，As x→∞，对于某一长尾分布函数H，则上升阶梯高度矩阵分布G+(X)(右Wiener-Hopf因子)具有长尾渐近性。如果𝔼X n 0，且H(X)是次指数分布函数，则该随机游动上确界的渐近行为与I.I.D.ℙ[sup n≥0 S n>；x]→(−𝔼X n)−1∫x∞ℙ[X n>；u]du as x→∞，其中S n=∑1 n X k，S 0=0.给出了这一结果的两个一般排队应用。首先，如果对马尔可夫调制的G/G/1排队施加相同的渐近条件，则等待时间分布与GI/GI/1排队的等待时间分布具有相同的渐近分布，即由服务时间分布函数的积分尾除以队列增量过程的负漂移给出。其次，通过在次指数更新过程中嵌入马尔可夫链构造的一类过程的自相关函数具有次指数尾部。当流体流动队列被这些过程馈送时，队列长度分布与其自相关函数渐近成正比。
Let {(X n ,J n )} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F ij }, F ij (B) = ℙ[X 1 ∈ B, J 1 = j | J 0 = i], B ∈ B (ℝ), i, j ∈ E. If F ij ([x,∞))/(1-H(x)) → W ij ∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G +(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X n 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup n≥0 S n > x] → (−𝔼X n )−1 ∫ x ∞ ℙ[X n > u]du as x → ∞, where S n = ∑1 n X k , S 0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.