TY - JOUR
T1 - A multiplicative version of the Lindley recursion
AU - Boxma, Onno
AU - Löpker, Andreas
AU - Mandjes, Michel
AU - Palmowski, Zbigniew
PY - 2021/8
Y1 - 2021/8
N2 - This paper presents an analysis of the stochastic recursion Wi+1=[ViWi+Yi]+ 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 Yi= Bi- Ai, for independent sequences of nonnegative i.i.d. random variables {Ai}i∈N0 and {Bi}i∈N0, and assuming {Vi}i∈N0 is an i.i.d. sequence as well (independent of {Ai}i∈N0 and {Bi}i∈N0), we then consider three special cases (i) Vi equals a positive value a with certain probability p∈ (0 , 1) and is negative otherwise, and both Ai and Bi have a rational LST, (ii) Vi attains negative values only and Bi has a rational LST, (iii) Vi is uniformly distributed on [0, 1], and Ai 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.
AB - This paper presents an analysis of the stochastic recursion Wi+1=[ViWi+Yi]+ 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 Yi= Bi- Ai, for independent sequences of nonnegative i.i.d. random variables {Ai}i∈N0 and {Bi}i∈N0, and assuming {Vi}i∈N0 is an i.i.d. sequence as well (independent of {Ai}i∈N0 and {Bi}i∈N0), we then consider three special cases (i) Vi equals a positive value a with certain probability p∈ (0 , 1) and is negative otherwise, and both Ai and Bi have a rational LST, (ii) Vi attains negative values only and Bi has a rational LST, (iii) Vi is uniformly distributed on [0, 1], and Ai 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.
KW - Autoregressive models
KW - Laplace transform
KW - Lindley recursion
KW - Wiener–Hopf boundary value problem
UR - http://www.scopus.com/inward/record.url?scp=85102682220&partnerID=8YFLogxK
U2 - 10.1007/s11134-021-09698-8
DO - 10.1007/s11134-021-09698-8
M3 - Article
AN - SCOPUS:85102682220
VL - 98
SP - 225
EP - 245
JO - Queueing Systems: Theory and Applications
JF - Queueing Systems: Theory and Applications
SN - 0257-0130
IS - 3-4
ER -