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A multiplicative version of the Lindley recursion
Queueing Systems ( IF 1.2 ) Pub Date : 2021-03-13 , DOI: 10.1007/s11134-021-09698-8
Onno Boxma , Andreas Löpker , Michel Mandjes , Zbigniew Palmowski

This paper presents an analysis of the stochastic recursion \(W_{i+1} = [V_iW_i+Y_i]^+\) that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing \(Y_i=B_i-A_i\), for independent sequences of nonnegative i.i.d. random variables \(\{A_i\}_{i\in {\mathbb N}_0}\) and \(\{B_i\}_{i\in {\mathbb N}_0}\), and assuming \(\{V_i\}_{i\in {\mathbb N}_0}\) is an i.i.d. sequence as well (independent of \(\{A_i\}_{i\in {\mathbb N}_0}\) and \(\{B_i\}_{i\in {\mathbb N}_0}\)), we then consider three special cases (i) \(V_i\) equals a positive value a with certain probability \(p\in (0,1)\) and is negative otherwise, and both \(A_i\) and \(B_i\) have a rational LST, (ii) \(V_i\) attains negative values only and \(B_i\) has a rational LST, (iii) \(V_i\) is uniformly distributed on [0, 1], and \(A_i\) is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.



中文翻译:

Lindley递归的乘法版本

本文提出了对随机递归\(W_ {i + 1} = [V_iW_i + Y_i] ^ + \)的分析,该解释可以解释为1阶自回归过程,反映为0。模型的稳定性条件 写入\(Y_I = B_i-A_I \),对于非负IID随机变量独立序列\(\ {A_I \} _ {I \在{\ mathbb N} _0} \)\(\ {B_i \} _ { i \ in {\ mathbb N} _0} \)并假设\(\ {V_i \} _ {i \ in {\ mathbb N} _0} \)也是一个iid序列(独立于\(\ {A_i \} _ {i \ in {\ mathbb N} _0} \)\(\ {B_i \} _ {i \ in {\ mathbb N} _0} \))中,我们考虑了三种特殊情况(i)\ (V_i \)等于具有一定概率\(p \ in(0,1)\)的正值a,否则为负值,并且\(A_i \)\(B_i \)都具有有理LST,(ii)\(V_i \ )仅获得负值,并且\(B_i \)具有合理的LST,(iii)\(V_i \)均匀地分布在[0,1]上,并且((A_i \))呈指数分布。在所有这三种情况下,我们都获得了瞬态和平稳结果,其中,瞬态结果是根据几何分布的纪元进行变换的。

更新日期:2021-03-15
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