A multiplicative version of the Lindley recursion

Onno Boxma (), Andreas Löpker (), Michel Mandjes () and Zbigniew Palmowski ()
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Onno Boxma: Eindhoven University of Technology
Andreas Löpker: HTW Dresden, University of Applied Sciences
Michel Mandjes: University of Amsterdam
Zbigniew Palmowski: Wrocław University of Science and Technology

Queueing Systems: Theory and Applications, 2021, vol. 98, issue 3, No 3, 225-245

Abstract: Abstract This paper presents an analysis of the stochastic recursion $$W_{i+1} = [V_iW_i+Y_i]^+$$ W i + 1 = [ V i W 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 $$Y_i=B_i-A_i$$ Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 , and assuming $$\{V_i\}_{i\in {\mathbb N}_0}$$ { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 ), we then consider three special cases (i) $$V_i$$ V i equals a positive value a with certain probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) and is negative otherwise, and both $$A_i$$ A i and $$B_i$$ B i have a rational LST, (ii) $$V_i$$ V i attains negative values only and $$B_i$$ B i has a rational LST, (iii) $$V_i$$ V i is uniformly distributed on [0, 1], and $$A_i$$ 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.

Keywords: Lindley recursion; Autoregressive models; Wiener–Hopf boundary value problem; Laplace transform; Primary: 60K25; Secondary: 90B22 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s11134-021-09698-8

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